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Calculated Shard: 152 (from laksa072)

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curl -X POST \
  'http://laksa152.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Mathematical_beauty\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Mathematical_beauty\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty","crawl_time":1773765144,"first_indexed_time":1375985647,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Mathematical_beauty s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Mathematical beauty - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"Mathematical beauty\nis a type of\naesthetic\nvalue that is experienced in doing or contemplating\nmathematics\n. The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such as\nG.H. Hardy\n, have characterized mathematics as an\nart\nform that seeks beauty.\nBeauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understanding\nbeauty\nin general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstract\nideas\nwhich can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than (for example) listening to music.\n[\n1\n]\nFurthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors identify mathematical elegance with mathematical beauty; others distinguish elegance as a separate aesthetic value, or as being, for instance, limited to the form mathematical exposition.\n[\n2\n]\n: 177–178 \nBeauty itself is often linked to, or thought to be dependent on, the abstractness, purity, simplicity, depth or order of mathematics.\nExamples of beautiful mathematics\n[\nedit\n]\nStarting at\ne\n0\n= 1, travelling at the velocity\ni\nrelative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an\nArgand diagram\n.)\nEuler's identity\nis often given as an example of a beautiful result:\n[\n3\n]\n: 1–3 \n[\n4\n]\n: 835–836 \n[\n5\n]\nThis expression ties together arguably the five most important\nmathematical constants\n(\ne\n,\ni\n, π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of\nEuler's formula\n, which the physicist\nRichard Feynman\ncalled \"our jewel\" and \"the most remarkable formula in mathematics\".\n[\n6\n]\nAnother example is\nFermat's theorem on sums of two squares\n, which says that any\nprime number\nsuch that\ncan be written as a sum of two square numbers (for example,\n,\n,\n), which both\nG.H. Hardy\n[\n7\n]\n: §12 \nand\nE.T. Bell\n[\n8\n]\n: ch.4 \nthought was a beautiful result.\nIn a survey in which mathematicians were asked to evaluate 24 theorems for their beauty, the top-rated three theorems were:\n[\n9\n]\nEuler's equation;\nEuler's polyhedron formula\n, which asserts that for a polyhedron with\nV\nvertices,\nE\nedges, and\nF\nfaces,\n; and\nEuclid's theorem\nthat there are infinitely many prime numbers, which was also given by\nHardy\nas an example of a beautiful theorem.\n[\n7\n]\n: §12 \nAn example of \"beauty in method\"—a simple and elegant visual descriptor of the\nPythagorean theorem\n.\nCantor's diagonal argument\n, which establishes that there are\ninfinite sets\nwhich cannot be put into\none-to-one correspondence\nwith the infinite set of\nnatural numbers\n, has been cited by both mathematicians\n[\n10\n]\nand philosophers\n[\n11\n]\nas an example of a beautiful proof.\nA proof without words for the sum of odd numbers theorem\nVisual proofs, such as the illustrated proof of the\nPythagorean theorem\n, and other\nproofs without words\ngenerally, such as the shown proof that the sum of all positive\nodd numbers\nup to 2\nn\n − 1 is a\nperfect square\n, have been thought beautiful.\n[\n12\n]\nThe mathematician\nPaul Erdős\nspoke of\nThe Book\n, an imaginary infinite book in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would proclaim it \"straight from The Book!\".\n[\n13\n]\n: 35 \nHis rhetorical device inspired the creation of\nProofs from THE BOOK\n, a collection of such proofs, including many suggested by Erdős himself.\n[\n14\n]\n: v \nIn\nPlato\n's\nTimaeus\n, the five\nregular convex polyhedra\n, called the\nPlatonic solids\nfor their role in this dialogue, are called the \"most beautiful\" (\"κάλλιστα\") bodies.\n[\n15\n]\n: 53e \nIn the\nTimaeus\n, they are described as having been used by the\ndemiurge\n, or creator-craftsman who builds the cosmos, for the four\nclassical elements\nplus the heavens, because of their beauty.\n[\n15\n]\n: 54e–55e \nKepler's Platonic solid model of the solar system\nIn his 1596 book\nMysterium Cosmographicum\n,\nJohannes Kepler\nargued that the orbits of the then-known planets in the\nSolar System\nhave been arranged by\nGod\nto correspond to a concentric arrangement of the five\nPlatonic solids\n, each orbit lying on the\ncircumsphere\nof one\npolyhedron\nand the\ninsphere\nof another. For Kepler, God had wanted to shape the universe according to the five regular solids because of their beauty, and this explained\nwhy\nthere were six planets (according to the knowledge of the time).\n[\n16\n]\n: ch.3 \n[\n4\n]\n: 280--285 \nPetrie projection of\nA more modern example is the exceptional\nsimple Lie group\n, which has been called \"perhaps the most beautiful structure in all of mathematics\".\n[\n17\n]\nScientific theories\n[\nedit\n]\nThe mathematical statements of scientific theories, especially in physics, are sometimes considered to be mathematically beautiful. For example,\nRoger Penrose\nthought there was a \"special beauty\" in\nMaxwell's equations\nof electromagnetism:\n[\n18\n]\n: 268 \nEinstein's theory of\ngeneral relativity\nhas been characterized as a work of art, and, among other aesthetic praise,\n[\n19\n]\n: 148 \nwas described by\nPaul Dirac\nas having \"great mathematical beauty\"\n[\n20\n]\n: 123 \nand by Penrose as having \"supreme mathematical beauty\".\n[\n21\n]\n: 1038 \n(There can be more to the beauty of a scientific theory than just its mathematical statement. For example, whether a theory is visualizable or deterministic might have an influence on whether it is seen as beautiful.\n[\n22\n]\n: 53 \n[\n4\n]\n: 873–877 \n)\nProperties of beautiful mathematics\n[\nedit\n]\nMany mathematicians and philosophers who have written about mathematical beauty have tried to identify properties or criteria that are conducive to the perception of beauty in a piece of mathematics. It is debated whether beauty can be clarified or explained by such properties:\nPaul Erdős\nthought that it was no more possible to convince someone of the beauty of a piece of mathematics than to convince them of the beauty of\nBeethoven's Ninth Symphony\n, if they couldn't see it for themselves.\n[\n23\n]\nIn his 1940 essay\nA Mathematician's Apology\n,\nG. H. Hardy\nsaid that a beautiful result, including its proof, possesses three \"purely aesthetic qualities\", namely \"inevitability\", \"unexpectedness\", and \"economy\". He particularly excluded enumeration of cases as \"one of the duller forms\nof mathematical argument\".\n[\n7\n]\n: §18 \nIn 1997,\nGian-Carlo Rota\ndisagreed with unexpectedness as a sufficient condition for beauty and proposed a counterexample:\nA great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago [from 1977] the proof of the existence of\nnon-equivalent differentiable structures\non spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or now.\n[\n2\n]\n: 172 \nIn contrast, Monastyrsky wrote in 2001:\nIt is very difficult to find an analogous invention in the past to\nMilnor\n's beautiful construction of the different differential structures on the seven-dimensional sphere... The original proof of Milnor was not very constructive, but later E. Briscorn showed that these differential structures can be described in an extremely explicit and beautiful form.\n[\n24\n]\n: 44 \nThis disagreement illustrates both the subjective nature of mathematical beauty,\nlike other forms of beauty in general\n, and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them.\nBesides Hardy's properties of \"unexpectedness\", \"inevitability\", \"economy\", which he applied to proofs as well as results, mathematicians have customarily thought beautiful proofs that are short and simple.\n[\n25\n]\n: 22 \nIn the search for an elegant proof, mathematicians often look for different independent ways to prove a result—as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly the\nPythagorean theorem\n, with hundreds of proofs having been published.\n[\n26\n]\nAnother theorem that has been proved in many different ways is the theorem of\nquadratic reciprocity\n. In fact,\nCarl Friedrich Gauss\nalone had eight different proofs of this theorem, six of which he published.\n[\n27\n]\nIn contrast, results that are logically correct but involve laborious calculations or consideration of many cases, are not usually considered beautiful, and may be even referred to as\nugly\nor\nclumsy\n. For example,\nKenneth Appel\nand\nWolfgang Haken\n's proof of the\nfour color theorem\nmade use of computer checking of over a thousand cases.\nPhilip J. Davis\nand\nReuben Hersh\nsaid that when they first heard that about the proof, they hoped it contained a new insight \"whose beauty would transform my day\", and were disheartened when informed the proof was by case enumeration and computer verification.\n[\n28\n]\n: 384 \nPaul Erdős\nsaid it was \"not beautiful\" because it gave no insight into why the theorem was true.\n[\n29\n]\n: 44 \nPhilosophical analysis\n[\nedit\n]\nAristotle\nthought that beauty was found especially in mathematics, writing in the\nMetaphysics\nthat\nthose who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a very great deal about them; for if they do not expressly mention them, but prove attributes which are their results or their formulae, it is not true to say that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.\n[\n30\n]\n: 1078a32–35 \nThe logician and philosopher\nBertrand Russell\nmade a now-famous statement characterizing mathematical beauty in terms of purity and austerity:\nMathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.\n[\n31\n]\nIn the twentieth century, some philosophers questioned whether there was genuinely beauty in mathematics. The philosopher of science\nRom Harré\nargued that there were no true aesthetic appraisals of mathematics, but only\nquasi-aesthetic\nappraisals. Any mathematical success described by an aesthetic term was a second-order success besides understanding and correctness. In contrast, aesthetic appraisal of a work of art was first-order. Harré considered this to be the difference between a quasi-aesthetic and a genuinely aesthetic appraisal.\n[\n32\n]\nNick Zangwill\nthought that there were no true aesthetic experiences of mathematics and that a proofs or theories could only be\nmetaphorically\nbeautiful. His argument had two bases. First, he thought that aesthetic properties depended on sensory properties, and so abstract entities could not have aesthetic properties. Second, he thought that proofs, theorems, theories, and so on had purposes such as demonstrating correctness or granting understanding, and that any praise of them reflected\nonly\nhow well they achieved their purpose.\n[\n33\n]\n: 140–142 \nScientific analysis\n[\nedit\n]\nInformation-theory model\n[\nedit\n]\nIn the 1970s,\nAbraham Moles\nand\nFrieder Nake\nanalyzed links between beauty,\ninformation processing\n, and\ninformation theory\n.\n[\n34\n]\n[\n35\n]\nIn the 1990s,\nJürgen Schmidhuber\nformulated a mathematical theory of observer-dependent subjective beauty based on\nalgorithmic information theory\n: the most beautiful objects among subjectively comparable objects have short\nalgorithmic\ndescriptions (i.e.,\nKolmogorov complexity\n) relative to what the observer already knows.\n[\n36\n]\n[\n37\n]\n[\n38\n]\nSchmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the\nfirst derivative\nof subjectively perceived beauty:\nthe observer continually tries to improve the\npredictability\nand\ncompressibility\nof the observations by discovering regularities such as repetitions and\nsymmetries\nand\nfractal\nself-similarity\n. Whenever the observer's learning process (possibly a predictive artificial\nneural network\n) leads to improved data compression such that the observation sequence can be described by fewer\nbits\nthan before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.\n[\n39\n]\n[\n40\n]\nBrain imaging experiments conducted by\nSemir Zeki\n,\nMichael Atiyah\nand collaborators show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial\norbito-frontal cortex\n(mOFC) of the brain and that this activity is parametrically related to the declared intensity of beauty. The location of the activity is similar to the location of the activity that correlates with the experience of beauty from other sources, such as visual art or music.\n[\n41\n]\nMoreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers.\n[\n42\n]\nMathematical beauty and the arts\n[\nedit\n]\nExamples of the use of mathematics in music include the\nstochastic music\nof\nIannis Xenakis\n,\n[\n43\n]\nthe\nFibonacci sequence\nin\nTool\n's\nLateralus\n,\n[\n44\n]\nand the\nMetric modulation\nof\nElliott Carter\n.\n[\n45\n]\nOther instances include the counterpoint of\nJohann Sebastian Bach\n,\npolyrhythmic\nstructures (as in\nIgor Stravinsky\n's\nThe Rite of Spring\n),\npermutation\ntheory in\nserialism\nbeginning with\nArnold Schoenberg\n, the application of Shepard tones in\nKarlheinz Stockhausen\n's\nHymnen\nand the application of\nGroup theory\nto transformations in music in the theoretical writings of\nDavid Lewin\n.\nDiagram from\nLeon Battista Alberti\n's 1435\nDella Pittura\n, with pillars in\nperspective\non a grid\nExamples of the use of mathematics in the visual arts include applications of\nchaos theory\nand\nfractal\ngeometry to\ncomputer-generated art\n, symmetry studies of\nLeonardo da Vinci\n,\nprojective geometries\nin development of the\nperspective\ntheory of\nRenaissance\nart,\ngrids\nin\nOp art\n, optical geometry in the\ncamera obscura\nof\nGiambattista della Porta\n, and multiple perspective in analytic\ncubism\nand\nfuturism\n.\nSacred geometry\nis a field of its own, giving rise to countless art forms including some of the best known mystic symbols and religious motifs, and has a particularly rich history in\nIslamic architecture\n.  It also provides a means of meditation and comtemplation, for example the study of the\nKaballah\nSefirot\n(Tree Of Life) and\nMetatron's Cube\n; and also the act of drawing itself.\nThe Dutch graphic designer\nM. C. Escher\ncreated mathematically inspired\nwoodcuts\n,\nlithographs\n, and\nmezzotints\n. These feature impossible constructions, explorations of\ninfinity\n, architecture, visual\nparadoxes\nand\ntessellations\n.\nSome painters and sculptors create work distorted with the mathematical principles of\nanamorphosis\n, including South African sculptor\nJonty Hurwitz\n.\nOrigami\n, the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the\nmathematics of paper folding\nby observing the\ncrease pattern\non unfolded origami pieces.\n[\n46\n]\nBritish constructionist artist\nJohn Ernest\ncreated reliefs and paintings inspired by group theory.\n[\n47\n]\nA number of other British artists of the constructionist and systems schools of thought also draw on mathematics models and structures as a source of inspiration, including\nAnthony Hill\nand\nPeter Lowe\n.\n[\n48\n]\nComputer-generated art is based on mathematical\nalgorithms\n.\nArgument from beauty\nFluency heuristic\nGolden ratio\nNeuroesthetics\nPhilosophy of mathematics\nProcessing fluency theory of aesthetic pleasure\nPythagoreanism\nSense of wonder\nTheory of everything\n^\nPhillips, George (2005).\n\"Preface\"\n.\nMathematics Is Not a Spectator Sport\n.\nSpringer Science+Business Media\n.\nISBN\n \n0-387-25528-1\n. Retrieved\n2008-08-22\n.\n\"...there is nothing in the world of mathematics that corresponds to an audience in a concert hall, where the passive listen to the active. Happily, mathematicians are all\ndoers\n, not spectators.\n^\na\nb\nRota, Gian-Carlo (May 1997). \"The Phenomenology of Mathematical Beauty\".\nSynthese\n.\n111\n(2):\n171–\n182.\ndoi\n:\n10.1023\/A:1004930722234\n.\n^\nWilson, Robin (2018).\nEuler's Pioneering Equation: The most beautiful theorem in mathematics\n. Oxford University Press.\nISBN\n \n978-0-19-879492-9\n.\n^\na\nb\nc\nCain, Alan J. (2024).\nForm & Number: A History of Mathematical Beauty\n. Lisbon: EBook.\n^\nGallagher, James (13 February 2014).\n\"Mathematics: Why the brain sees maths as beauty\"\n.\nBBC News online\n. Retrieved\n13 February\n2014\n.\n^\nFeynman, Richard P. (1977).\nThe Feynman Lectures on Physics\n. Vol. I. Addison-Wesley. p. 22-16.\nISBN\n \n0-201-02010-6\n.\n^\na\nb\nc\nHardy, G.H. (1967).\nA Mathematician's Apology\n. Cambridge University Press.\nISBN\n \n978-1-107-60463-6\n.\n^\nBell, E.T. (1937).\nMen of Mathematics\n. Simon & Schuster.\n^\nWells, David (June 1990). \"Are these the most beautiful?\".\nThe Mathematical Intelligencer\n.\n12\n(3):\n37–\n41.\ndoi\n:\n10.1007\/BF03024015\n.\n^\nPaulos, John Allen (1991).\nBeyond Numeracy\n. New York: Alfred A. Knopf. pp. \n125–\n127.\nISBN\n \n0-394-586409\n.\n^\nDutilh Novaes, Catarina (2019). \"The Beauty (?) of Mathematical Proofs\". In Aberdein, Andrew; Inglis, Matthew (eds.).\nAdvances in Experimental Philosophy of Logic and Mathematics\n. Bloomsbury Academic. pp. \n69–\n71.\nISBN\n \n978-1-350-03901-8\n.\n^\nPolster, Burkert (2004).\nQ.E.D.: Beauty in Mathematical Proof\n. Walker & Company. pp. \n32–\n33.\nISBN\n \n978-0-8027-1431-2\n.\n^\nSchechter, Bruce (2000).\nMy Brain is Open: The Mathematical Journeys of Paul Erdős\n. New York:\nSimon & Schuster\n.\nISBN\n \n0-684-85980-7\n.\n^\nAigner, Martin\n;\nZiegler, Günter M.\n(2018).\nProofs from THE BOOK\n(6th ed.). Springer.\nISBN\n \n978-3-662-57264-1\n.\n^\na\nb\nPlato (1929).\nTimaeus\n. Cambridge, MA: Harvard University Press.\nISBN\n \n978-0-674-99257-3\n.\n^\nField, J.V. (2013).\nKepler's Geometrical Cosmology\n. Bloomsbury.\nISBN\n \n9781472507037\n.\n^\nWhitfield, John.\n\"Journey to the 248th dimension\"\n.\nNature Online\n. Retrieved\n12 September\n2025\n.\n^\nPenrose, Roger (1974). \"The Rôle of Aesthetics in Pure and Applied Mathematical Research\".\nBulletin of the Institute of Mathematics and its Applications\n.\n10\n:\n226–\n271.\n^\nChandrasekhar, Subrahmanyan\n(1987).\nTruth and Beauty: Aesthetics and Motivation in Science\n. Chicago, London: University of Chicago Press.\nISBN\n \n978-0-226-10087-6\n.\n^\nDirac, P.A.M. (1940). \"The Relation between Mathematics and Physics\".\nProceedings of the Royal Society of Edinburgh\n.\n59\n:\n122–\n129.\ndoi\n:\n10.1017\/s0370164600012207\n.\n^\nPenrose, Roger (2004).\nThe Road to Reality\n. London: Jonathan Cape.\nISBN\n \n978-0-224-04447-9\n.\n^\nMcAllister, James W. (1996).\nBeauty & Revolution in Science\n. Ithaca: Cornell University Press.\nISBN\n \n978-0-8014-3240-8\n.\n^\nDevlin, Keith\n(2000). \"Do Mathematicians Have Different Brains?\".\nThe Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip\n.\nBasic Books\n. p. \n140\n.\nISBN\n \n978-0-465-01619-8\n. Retrieved\n2008-08-22\n.\n^\nMonastyrsky, Michael (2001).\n\"Some Trends in Modern Mathematics and the Fields Medal\"\n(PDF)\n.\nCan. Math. Soc. Notes\n.\n33\n(2 and 3).\n^\nMcAllister, James W. (2005). \"Mathematical Beauty and the Evolution of the Standards of Mathematical Proof\". In Emmer, Michele (ed.).\nThe Visual Mind II\n. MIT Press. pp. \n15–\n34.\nISBN\n \n978-0-262-05076-0\n.\n^\nLoomis, Elisha Scott\n(1968).\nThe Pythagorean Proposition\n. Washington, DC: National Council of Teachers of Mathematics.\nISBN\n \n978-0-873-53036-1\n.\n^\nWeisstein, Eric W.\n\"Quadratic Reciprocity Theorem\"\n.\nmathworld.wolfram.com\n. Retrieved\n2019-10-31\n.\n^\nDavis, Philip J.; Hersh, Reuben (1981).\nThe Mathematical Experience\n. Boston: Houghton Mifflin.\nISBN\n \n978-0-395-32131-7\n.\n^\nHoffman, Paul (1999).\nThe Man Who Loved Only Numbers\n. London: Fourth Estate.\nISBN\n \n978-1-85702-829-4\n.\n^\nAristotle\n(1995). \"Metaphysics\". In Barnes, Jonathan (ed.).\nThe Complete Works of Aristotle\n. Princeton University Press.\nISBN\n \n978-0-691-01650-4\n.\n^\nRussell, Bertrand\n(1919). \"The Study of Mathematics\".\nMysticism and Logic: And Other Essays\n.\nLongman\n. p. \n60\n. Retrieved\n2008-08-22\n.\nMathematics rightly viewed possesses not only truth but supreme beauty a beauty cold and austere like that of sculpture without appeal to any part of our weaker nature without the gorgeous trappings Russell.\n^\nHarré, Rom\n(1958).\n\"Quasi-Aesthetic Appraisals\"\n.\nPhilosophy\n.\n33\n:\n132–\n137.\n^\nZangwill, Nick\n(2001).\nThe Metaphysics of Beauty\n. Ithaca, London: Cornell University Press.\nISBN\n \n978-0-8014-3820-2\n.\n^\nA. Moles:\nThéorie de l'information et perception esthétique\n, Paris, Denoël, 1973 (\nInformation Theory\nand aesthetical perception)\n^\nF Nake (1974). Ästhetik als Informationsverarbeitung. (\nAesthetics\nas information processing). Grundlagen und Anwendungen der Informatik im Bereich ästhetischer Produktion und Kritik. Springer, 1974,\nISBN\n \n3-211-81216-4\n,\nISBN\n \n978-3-211-81216-7\n^\nJ. Schmidhuber.\nLow-complexity art\n.\nLeonardo\n, Journal of the International Society for the Arts, Sciences, and Technology (\nLeonardo\/ISAST\n), 30(2):97–103, 1997.\ndoi\n:\n10.2307\/1576418\n.\nJSTOR\n \n1576418\n.\n^\nJ. Schmidhuber. Papers on the theory of beauty and\nlow-complexity art\nsince 1994:\nhttp:\/\/www.idsia.ch\/~juergen\/beauty.html\n^\nJ. Schmidhuber. Simple Algorithmic Principles of Discovery, Subjective Beauty, Selective Attention, Curiosity & Creativity. Proc. 10th Intl. Conf. on Discovery Science (DS 2007) pp. 26–38, LNAI 4755, Springer, 2007. Also in Proc. 18th Intl. Conf. on Algorithmic Learning Theory (ALT 2007) p. 32, LNAI 4754, Springer, 2007. Joint invited lecture for DS 2007 and ALT 2007, Sendai, Japan, 2007.\narXiv\n:\n0709.0674\n.\n^\n.J. Schmidhuber. Curious model-building control systems. International Joint Conference on Neural Networks, Singapore, vol 2, 1458–1463. IEEE press, 1991\n^\nSchmidhuber's theory of beauty and curiosity in a German TV show:\nhttp:\/\/www.br-online.de\/bayerisches-fernsehen\/faszination-wissen\/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml\nArchived\nJune 3, 2008, at the\nWayback Machine\n^\nZeki, Semir; Romaya, John Paul; Benincasa, Dionigi M. T.; Atiyah, Michael F. (2014).\n\"The experience of mathematical beauty and its neural correlates\"\n.\nFrontiers in Human Neuroscience\n.\n8\n: 68.\ndoi\n:\n10.3389\/fnhum.2014.00068\n.\nISSN\n \n1662-5161\n.\nPMC\n \n3923150\n.\nPMID\n \n24592230\n.\n^\nZhang, Haoxuan; Zeki, Semir (May 2022).\n\"Judgments of mathematical beauty are resistant to revision through external opinion\"\n.\nPsyCh Journal\n.\n11\n(5):\n741–\n747.\ndoi\n:\n10.1002\/pchj.556\n.\nISSN\n \n2046-0252\n.\nPMC\n \n9790661\n.\nPMID\n \n35491015\n.\n^\nLuque, Sergio. 2009. \"The Stochastic Synthesis of Iannis Xenakis.\" Leonardo Music Journal (19): 77–84\n^\nNorris, Chris (2001).\n\"Hammer Of The Gods\"\n. Archived from\nthe original\non November 1, 2011\n. Retrieved\nApril 25,\n2007\n.\n^\nSchell, Michael (December 11, 2018).\n\"Elliott Carter (1908–2012): Legacy of a Centenarian\"\n.\nSecond Inversion\n.\nArchived\nfrom the original on December 24, 2018\n. Retrieved\nDecember 11,\n2018\n.\n^\nHull, Thomas. \"Project Origami: Activities for Exploring Mathematics\". Taylor & Francis, 2006.\n^\nJohn Ernest's use of mathematics and especially group theory in his art works is analysed in\nJohn Ernest, A Mathematical Artist\nby Paul Ernest in\nPhilosophy of Mathematics Education Journal\n, No. 24 Dec. 2009 (Special Issue on Mathematics and Art):\nhttp:\/\/people.exeter.ac.uk\/PErnest\/pome24\/index.htm\n^\nFranco, Francesca (2017-10-05).\n\"The Systems Group (Chapter 2)\"\n.\nGenerative Systems Art: The Work of Ernest Edmonds\n. Routledge.\nISBN\n \n9781317137436\n.\nCellucci, Carlo (2015), \"Mathematical beauty, understanding, and discovery\",\nFoundations of Science\n,\n20\n(4):\n339–\n355,\ndoi\n:\n10.1007\/s10699-014-9378-7\n,\nS2CID\n \n120068870\nHoffman, Paul\n(1992),\nThe Man Who Loved Only Numbers\n, Hyperion.\nLoomis, Elisha Scott\n(1968),\nThe Pythagorean Proposition\n, The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem.\nPandey, S.K.\n.\nThe Humming of Mathematics: Melody of Mathematics\n. Independently Published, 2019.\nISBN\n \n1710134437\n.\nPeitgen, H.-O., and Richter, P.H. (1986),\nThe Beauty of Fractals\n, Springer-Verlag.\nReber, R.\n; Brun, M.; Mitterndorfer, K. (2008). \"The use of heuristics in intuitive mathematical judgment\".\nPsychonomic Bulletin & Review\n.\n15\n(6):\n1174–\n1178.\ndoi\n:\n10.3758\/PBR.15.6.1174\n.\nhdl\n:\n1956\/2734\n.\nPMID\n \n19001586\n.\nS2CID\n \n5297500\n.\nStrohmeier, John, and Westbrook, Peter (1999),\nDivine Harmony, The Life and Teachings of Pythagoras\n, Berkeley Hills Books, Berkeley, CA.\nZeki, S.\n; Romaya, J. P.; Benincasa, D. M. T.;\nAtiyah, M. F.\n(2014), \"The experience of mathematical beauty and its neural correlates\",\nFrontiers in Human Neuroscience\n,\n8\n: 68,\ndoi\n:\n10.3389\/fnhum.2014.00068\n,\nPMC\n \n3923150\n,\nPMID\n \n24592230\nAigner, Martin\n;\nZiegler, Günter M.\n(2018).\nProofs from THE BOOK\n(6th ed.). Springer.\nISBN\n \n978-3-662-57264-1\n.\nCain, Alan J. (2024).\nForm & Number: A History of Mathematical Beauty\n. Lisbon: Ebook.\nHadamard, Jacques\n(1949).\nAn Essay on the Psychology of Invention in the Mathematical Field\n(2nd enlarged ed.). Princeton University Press.\nISBN\n \n0-486-20107-4\n.\nHardy, G.H.\n(1967) [1st published 1940].\nA Mathematician's Apology\n. Cambridge University Press.\nISBN\n \n978-1-107-60463-6\n.\nHuntley, H.E. (1970).\nThe Divine Proportion: A Study in Mathematical Beauty\n. New York: Dover Publications.\nISBN\n \n978-0-486-22254-7\n.\nStewart, Ian (2007).\nWhy beauty is truth : a history of symmetry\n. New York: Basic Books, a member of the Perseus Books Group.\nISBN\n \n978-0-465-08236-0\n.\nOCLC\n \n76481488\n.\nMathematics, Poetry and Beauty\nIs Mathematics Beautiful?\ncut-the-knot.org\nJustin Mullins.com\nEdna St. Vincent Millay (poet):\nEuclid alone has looked on beauty bare\nTerence Tao\n,\nWhat is good mathematics?\nMathbeauty Blog\nThe\nAesthetic Appeal\ncollection at the\nInternet Archive\nA Mathematical Romance\nJim Holt\nDecember 5, 2013 issue of\nThe New York Review of Books\nreview of\nLove and Math: The Heart of Hidden Reality\nby\nEdward Frenkel","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#bodyContent)\n\nMain menu\n\nMain menu\n\nmove to sidebar\n\nhide\n\nNavigation\n\n- [Main page](https:\/\/en.wikipedia.org\/wiki\/Main_Page \"Visit the main page [z]\")\n- [Contents](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contents \"Guides to browsing Wikipedia\")\n- [Current events](https:\/\/en.wikipedia.org\/wiki\/Portal:Current_events \"Articles related to current events\")\n- [Random article](https:\/\/en.wikipedia.org\/wiki\/Special:Random \"Visit a randomly selected article [x]\")\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About \"Learn about Wikipedia and how it works\")\n- [Contact us](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us \"How to contact Wikipedia\")\n\nContribute\n\n- [Help](https:\/\/en.wikipedia.org\/wiki\/Help:Contents \"Guidance on how to use and edit Wikipedia\")\n- [Learn to edit](https:\/\/en.wikipedia.org\/wiki\/Help:Introduction \"Learn how to edit Wikipedia\")\n- [Community portal](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Community_portal \"The hub for editors\")\n- [Recent changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChanges \"A list of recent changes to Wikipedia [r]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_upload_wizard \"Add images or other media for use on Wikipedia\")\n- [Special pages](https:\/\/en.wikipedia.org\/wiki\/Special:SpecialPages \"A list of all special pages [q]\")\n\n[![](https:\/\/en.wikipedia.org\/static\/images\/icons\/enwiki-25.svg) ![Wikipedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-tagline-en-25.svg)](https:\/\/en.wikipedia.org\/wiki\/Main_Page)\n\n[Search](https:\/\/en.wikipedia.org\/wiki\/Special:Search \"Search Wikipedia [f]\")\n\nAppearance\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Mathematical+beauty \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Mathematical+beauty \"You're encouraged to log in; however, it's not mandatory. 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[o]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty)\n- [1 Examples of beautiful mathematics](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Examples_of_beautiful_mathematics)\n  Toggle Examples of beautiful mathematics subsection\n  - [1\\.1 Results](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Results)\n  - [1\\.2 Proofs](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Proofs)\n  - [1\\.3 Objects](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Objects)\n  - [1\\.4 Scientific theories](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Scientific_theories)\n- [2 Properties of beautiful mathematics](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Properties_of_beautiful_mathematics)\n  Toggle Properties of beautiful mathematics subsection\n  - [2\\.1 Results](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Results_2)\n  - [2\\.2 Proofs](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Proofs_2)\n- [3 Philosophical analysis](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Philosophical_analysis)\n- [4 Scientific analysis](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Scientific_analysis)\n  Toggle Scientific analysis subsection\n  - [4\\.1 Information-theory model](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Information-theory_model)\n  - [4\\.2 Neural correlates](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Neural_correlates)\n- [5 Mathematical beauty and the arts](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Mathematical_beauty_and_the_arts)\n  Toggle Mathematical beauty and the arts subsection\n  - [5\\.1 Music](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Music)\n  - [5\\.2 Visual arts](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Visual_arts)\n- [6 See also](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#See_also)\n- [7 Notes](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Notes)\n- [8 References](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#References)\n- [9 Further reading](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#Further_reading)\n- [10 External links](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#External_links)\n\nToggle the table of contents\n\n# Mathematical beauty\n24 languages\n\n- [العربية](https:\/\/ar.wikipedia.org\/wiki\/%D8%AC%D9%85%D8%A7%D9%84_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA%D9%8A \"جمال رياضياتي – Arabic\")\n- [বাংলা](https:\/\/bn.wikipedia.org\/wiki\/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%B8%E0%A7%8C%E0%A6%A8%E0%A7%8D%E0%A6%A6%E0%A6%B0%E0%A7%8D%E0%A6%AF \"গাণিতিক সৌন্দর্য – Bangla\")\n- [Català](https:\/\/ca.wikipedia.org\/wiki\/Bellesa_matem%C3%A0tica \"Bellesa matemàtica – Catalan\")\n- [Dansk](https:\/\/da.wikipedia.org\/wiki\/Matematisk_sk%C3%B8nhed \"Matematisk skønhed – Danish\")\n- [Español](https:\/\/es.wikipedia.org\/wiki\/Belleza_matem%C3%A1tica \"Belleza matemática – Spanish\")\n- [فارسی](https:\/\/fa.wikipedia.org\/wiki\/%D8%B2%DB%8C%D8%A8%D8%A7%DB%8C%DB%8C_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C \"زیبایی ریاضی – Persian\")\n- [Suomi](https:\/\/fi.wikipedia.org\/wiki\/Matematiikan_kauneus \"Matematiikan kauneus – Finnish\")\n- [Français](https:\/\/fr.wikipedia.org\/wiki\/Beaut%C3%A9_math%C3%A9matique \"Beauté mathématique – French\")\n- [עברית](https:\/\/he.wikipedia.org\/wiki\/%D7%99%D7%95%D7%A4%D7%99_%D7%9E%D7%AA%D7%9E%D7%98%D7%99 \"יופי מתמטי – Hebrew\")\n- [हिन्दी](https:\/\/hi.wikipedia.org\/wiki\/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A5%8C%E0%A4%A8%E0%A5%8D%E0%A4%A6%E0%A4%B0%E0%A5%8D%E0%A4%AF \"गणितीय सौन्दर्य – Hindi\")\n- [Bahasa Indonesia](https:\/\/id.wikipedia.org\/wiki\/Keindahan_matematis \"Keindahan matematis – Indonesian\")\n- [Italiano](https:\/\/it.wikipedia.org\/wiki\/Bellezza_matematica \"Bellezza matematica – Italian\")\n- [日本語](https:\/\/ja.wikipedia.org\/wiki\/%E6%95%B0%E5%AD%A6%E7%9A%84%E3%81%AA%E7%BE%8E \"数学的な美 – Japanese\")\n- [한국어](https:\/\/ko.wikipedia.org\/wiki\/%EC%88%98%ED%95%99%EC%A0%81_%EB%AF%B8 \"수학적 미 – Korean\")\n- [Lietuvių](https:\/\/lt.wikipedia.org\/wiki\/Matematinis_gro%C5%BEis \"Matematinis grožis – Lithuanian\")\n- [မြန်မာဘာသာ](https:\/\/my.wikipedia.org\/wiki\/%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC%E1%80%86%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%9B%E1%80%AC%E1%80%A1%E1%80%9C%E1%80%BE%E1%80%A1%E1%80%95 \"သင်္ချာဆိုင်ရာအလှအပ – Burmese\")\n- [Polski](https:\/\/pl.wikipedia.org\/wiki\/Matematyka_a_estetyka \"Matematyka a estetyka – Polish\")\n- [Português](https:\/\/pt.wikipedia.org\/wiki\/Beleza_da_matem%C3%A1tica \"Beleza da matemática – Portuguese\")\n- [Русский](https:\/\/ru.wikipedia.org\/wiki\/%D0%9A%D1%80%D0%B0%D1%81%D0%BE%D1%82%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B8 \"Красота математики – Russian\")\n- [Српски \/ srpski](https:\/\/sr.wikipedia.org\/wiki\/%D0%9B%D0%B5%D0%BF%D0%BE%D1%82%D0%B0_%D1%83_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%86%D0%B8 \"Лепота у математици – Serbian\")\n- [Українська](https:\/\/uk.wikipedia.org\/wiki\/%D0%9A%D1%80%D0%B0%D1%81%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B8 \"Краса математики – Ukrainian\")\n- [Tiếng Việt](https:\/\/vi.wikipedia.org\/wiki\/V%E1%BA%BB_%C4%91%E1%BA%B9p_c%E1%BB%A7a_to%C3%A1n_h%E1%BB%8Dc \"Vẻ đẹp của toán học – Vietnamese\")\n- [粵語](https:\/\/zh-yue.wikipedia.org\/wiki\/%E6%95%B8%E5%AD%B8%E4%B9%8B%E7%BE%8E \"數學之美 – Cantonese\")\n- [中文](https:\/\/zh.wikipedia.org\/wiki\/%E6%95%B8%E5%AD%B8%E4%B9%8B%E7%BE%8E \"數學之美 – Chinese\")\n\n[Edit links](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q2248521#sitelinks-wikipedia \"Edit interlanguage links\")\n\n- [Article](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty \"View the content page [c]\")\n- [Talk](https:\/\/en.wikipedia.org\/wiki\/Talk:Mathematical_beauty \"Discuss improvements to the content page [t]\")\n\nEnglish\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=history \"Past revisions of this page [h]\")\n\nTools\n\nTools\n\nmove to sidebar\n\nhide\n\nActions\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=history)\n\nGeneral\n\n- [What links here](https:\/\/en.wikipedia.org\/wiki\/Special:WhatLinksHere\/Mathematical_beauty \"List of all English Wikipedia pages containing links to this page [j]\")\n- [Related 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The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such as [G.H. Hardy](https:\/\/en.wikipedia.org\/wiki\/G.H._Hardy \"G.H. Hardy\"), have characterized mathematics as an [art](https:\/\/en.wikipedia.org\/wiki\/Art \"Art\") form that seeks beauty.\n\nBeauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understanding [beauty](https:\/\/en.wikipedia.org\/wiki\/Beauty \"Beauty\") in general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstract *ideas* which can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than (for example) listening to music.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-1) Furthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors identify mathematical elegance with mathematical beauty; others distinguish elegance as a separate aesthetic value, or as being, for instance, limited to the form mathematical exposition.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-rota-phenomenology-2): 177–178  Beauty itself is often linked to, or thought to be dependent on, the abstractness, purity, simplicity, depth or order of mathematics.\n\n## Examples of beautiful mathematics\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=1 \"Edit section: Examples of beautiful mathematics\")\\]\n\n### Results\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=2 \"Edit section: Results\")\\]\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/3b\/EulerIdentity2.svg\/250px-EulerIdentity2.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:EulerIdentity2.svg)\n\nStarting at *e*0 = 1, travelling at the velocity *i* relative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an [Argand diagram](https:\/\/en.wikipedia.org\/wiki\/Argand_diagram \"Argand diagram\").)\n\n[Euler's identity](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity \"Euler's identity\") is often given as an example of a beautiful result:[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-3): 1–3 [\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-cain-form-4): 835–836 [\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-5)\n\ne i π \\+ 1 \\= 0 . {\\\\displaystyle \\\\displaystyle \\\\mathrm {e} ^{\\\\mathrm {i} \\\\pi }+1=0\\\\,.} ![{\\\\displaystyle \\\\displaystyle \\\\mathrm {e} ^{\\\\mathrm {i} \\\\pi }+1=0\\\\,.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fd5213bb3f233806ff3212142214fee997434a15)\n\nThis expression ties together arguably the five most important [mathematical constants](https:\/\/en.wikipedia.org\/wiki\/Mathematical_constant \"Mathematical constant\") (*e*, *i*, π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\"), which the physicist [Richard Feynman](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\") called \"our jewel\" and \"the most remarkable formula in mathematics\".[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-6)\n\nAnother example is [Fermat's theorem on sums of two squares](https:\/\/en.wikipedia.org\/wiki\/Fermat%27s_theorem_on_sums_of_two_squares \"Fermat's theorem on sums of two squares\"), which says that any [prime number](https:\/\/en.wikipedia.org\/wiki\/Prime_number \"Prime number\") such that p ≡ 1 ( mod 4 ) {\\\\displaystyle p\\\\equiv 1{\\\\pmod {4}}} ![{\\\\displaystyle p\\\\equiv 1{\\\\pmod {4}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/125d70c2c8848e96954eddfc7f9283ec4f676ed7) can be written as a sum of two square numbers (for example, 5 \\= 1 2 \\+ 2 2 {\\\\displaystyle 5=1^{2}+2^{2}} ![{\\\\displaystyle 5=1^{2}+2^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c4b288581682d73e457bc6829e15be203280852a), 13 \\= 2 2 \\+ 3 2 {\\\\displaystyle 13=2^{2}+3^{2}} ![{\\\\displaystyle 13=2^{2}+3^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8be5ea38d26942831bf630c465218ef5e3189181), 37 \\= 1 2 \\+ 6 2 {\\\\displaystyle 37=1^{2}+6^{2}} ![{\\\\displaystyle 37=1^{2}+6^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0f81cd69710c7c05bdc903449dc7e834540c797a)), which both [G.H. Hardy](https:\/\/en.wikipedia.org\/wiki\/G.H._Hardy \"G.H. Hardy\")[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-hardy-apology-7): §12  and [E.T. Bell](https:\/\/en.wikipedia.org\/wiki\/E.T._Bell \"E.T. Bell\")[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-8): ch.4  thought was a beautiful result.\n\nIn a survey in which mathematicians were asked to evaluate 24 theorems for their beauty, the top-rated three theorems were:[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-9) Euler's equation; [Euler's polyhedron formula](https:\/\/en.wikipedia.org\/wiki\/Euler_characteristic \"Euler characteristic\"), which asserts that for a polyhedron with *V* vertices, *E* edges, and *F* faces, V − E \\+ F \\= 2 {\\\\displaystyle V-E+F=2} ![{\\\\displaystyle V-E+F=2}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/759601e482258ff7a359a7db381abf60372c5b06); and [Euclid's theorem](https:\/\/en.wikipedia.org\/wiki\/Euclid%27s_theorem \"Euclid's theorem\") that there are infinitely many prime numbers, which was also given by [Hardy](https:\/\/en.wikipedia.org\/wiki\/G.H._Hardy \"G.H. Hardy\") as an example of a beautiful theorem.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-hardy-apology-7): §12\n\n### Proofs\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=3 \"Edit section: Proofs\")\\]\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/9\/9a\/Pythagorean_proof_%281%29.svg\/250px-Pythagorean_proof_%281%29.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Pythagorean_proof_\\(1\\).svg)\n\nAn example of \"beauty in method\"—a simple and elegant visual descriptor of the [Pythagorean theorem](https:\/\/en.wikipedia.org\/wiki\/Pythagorean_theorem \"Pythagorean theorem\").\n\n[Cantor's diagonal argument](https:\/\/en.wikipedia.org\/wiki\/Cantor%27s_diagonal_argument \"Cantor's diagonal argument\"), which establishes that there are [infinite sets](https:\/\/en.wikipedia.org\/wiki\/Infinite_set \"Infinite set\") which cannot be put into [one-to-one correspondence](https:\/\/en.wikipedia.org\/wiki\/Bijection \"Bijection\") with the infinite set of [natural numbers](https:\/\/en.wikipedia.org\/wiki\/Natural_number \"Natural number\"), has been cited by both mathematicians[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-10) and philosophers[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-11) as an example of a beautiful proof.\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/e8\/Proofwithoutwords.svg\/250px-Proofwithoutwords.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Proofwithoutwords.svg)\n\nA proof without words for the sum of odd numbers theorem\n\nVisual proofs, such as the illustrated proof of the [Pythagorean theorem](https:\/\/en.wikipedia.org\/wiki\/Pythagorean_theorem \"Pythagorean theorem\"), and other [proofs without words](https:\/\/en.wikipedia.org\/wiki\/Proofs_without_words \"Proofs without words\") generally, such as the shown proof that the sum of all positive [odd numbers](https:\/\/en.wikipedia.org\/wiki\/Parity_\\(mathematics\\) \"Parity (mathematics)\") up to 2*n* − 1 is a [perfect square](https:\/\/en.wikipedia.org\/wiki\/Square_number \"Square number\"), have been thought beautiful.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-12)\n\nThe mathematician [Paul Erdős](https:\/\/en.wikipedia.org\/wiki\/Paul_Erd%C5%91s \"Paul Erdős\") spoke of *The Book*, an imaginary infinite book in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would proclaim it \"straight from The Book!\".[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-13): 35  His rhetorical device inspired the creation of [*Proofs from THE BOOK*](https:\/\/en.wikipedia.org\/wiki\/Proofs_from_THE_BOOK \"Proofs from THE BOOK\"), a collection of such proofs, including many suggested by Erdős himself.[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-14): v\n\n### Objects\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=4 \"Edit section: Objects\")\\]\n\nIn [Plato](https:\/\/en.wikipedia.org\/wiki\/Plato \"Plato\")'s [*Timaeus*](https:\/\/en.wikipedia.org\/wiki\/Timaeus_\\(dialogue\\) \"Timaeus (dialogue)\"), the five [regular convex polyhedra](https:\/\/en.wikipedia.org\/wiki\/Platonic_solids \"Platonic solids\"), called the *Platonic solids* for their role in this dialogue, are called the \"most beautiful\" (\"κάλλιστα\") bodies.[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-plato-timaeus-15): 53e  In the *Timaeus*, they are described as having been used by the [demiurge](https:\/\/en.wikipedia.org\/wiki\/Demiurge \"Demiurge\"), or creator-craftsman who builds the cosmos, for the four [classical elements](https:\/\/en.wikipedia.org\/wiki\/Classical_elements \"Classical elements\") plus the heavens, because of their beauty.[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-plato-timaeus-15): 54e–55e\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/19\/Kepler-solar-system-1.png\/250px-Kepler-solar-system-1.png)](https:\/\/en.wikipedia.org\/wiki\/File:Kepler-solar-system-1.png)\n\nKepler's Platonic solid model of the solar system\n\nIn his 1596 book [*Mysterium Cosmographicum*](https:\/\/en.wikipedia.org\/wiki\/Mysterium_Cosmographicum \"Mysterium Cosmographicum\"), [Johannes Kepler](https:\/\/en.wikipedia.org\/wiki\/Johannes_Kepler \"Johannes Kepler\") argued that the orbits of the then-known planets in the [Solar System](https:\/\/en.wikipedia.org\/wiki\/Solar_System \"Solar System\") have been arranged by [God](https:\/\/en.wikipedia.org\/wiki\/God \"God\") to correspond to a concentric arrangement of the five [Platonic solids](https:\/\/en.wikipedia.org\/wiki\/Platonic_solid \"Platonic solid\"), each orbit lying on the [circumsphere](https:\/\/en.wikipedia.org\/wiki\/Circumscribed_sphere \"Circumscribed sphere\") of one [polyhedron](https:\/\/en.wikipedia.org\/wiki\/Polyhedron \"Polyhedron\") and the [insphere](https:\/\/en.wikipedia.org\/wiki\/Inscribed_sphere \"Inscribed sphere\") of another. For Kepler, God had wanted to shape the universe according to the five regular solids because of their beauty, and this explained *why* there were six planets (according to the knowledge of the time).[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-16): ch.3 [\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-cain-form-4): 280--285\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/14\/E8Petrie.svg\/250px-E8Petrie.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:E8Petrie.svg)\n\nPetrie projection of\n\nE\n\n8\n\n{\\\\displaystyle E\\_{8}}\n\n![{\\\\displaystyle E\\_{8}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/48479e96d90b4cfabc7784106cc3cfff907dda34)\n\nA more modern example is the exceptional [simple Lie group](https:\/\/en.wikipedia.org\/wiki\/Simple_Lie_group \"Simple Lie group\") E 8 {\\\\displaystyle E\\_{8}} ![{\\\\displaystyle E\\_{8}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/48479e96d90b4cfabc7784106cc3cfff907dda34), which has been called \"perhaps the most beautiful structure in all of mathematics\".[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-17)\n\n### Scientific theories\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=5 \"Edit section: Scientific theories\")\\]\n\nThe mathematical statements of scientific theories, especially in physics, are sometimes considered to be mathematically beautiful. For example, [Roger Penrose](https:\/\/en.wikipedia.org\/wiki\/Roger_Penrose \"Roger Penrose\") thought there was a \"special beauty\" in [Maxwell's equations](https:\/\/en.wikipedia.org\/wiki\/Maxwell%27s_equations \"Maxwell's equations\") of electromagnetism:[\\[18\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-18): 268  ∇ ⋅ E \\= ρ ε 0 ∇ ⋅ B \\= 0 ∇ × E \\= − ∂ B ∂ t ∇ × B \\= μ 0 ( J \\+ ε 0 ∂ E ∂ t ) {\\\\displaystyle {\\\\begin{aligned}\\\\nabla \\\\cdot \\\\mathbf {E} \\\\,\\\\,\\\\,&={\\\\frac {\\\\rho }{\\\\varepsilon \\_{0}}}\\\\\\\\\\\\nabla \\\\cdot \\\\mathbf {B} \\\\,\\\\,\\\\,&=0\\\\\\\\\\\\nabla \\\\times \\\\mathbf {E} &=-{\\\\frac {\\\\partial \\\\mathbf {B} }{\\\\partial t}}\\\\\\\\\\\\nabla \\\\times \\\\mathbf {B} &=\\\\mu \\_{0}\\\\left(\\\\mathbf {J} +\\\\varepsilon \\_{0}{\\\\frac {\\\\partial \\\\mathbf {E} }{\\\\partial t}}\\\\right)\\\\end{aligned}}} ![{\\\\displaystyle {\\\\begin{aligned}\\\\nabla \\\\cdot \\\\mathbf {E} \\\\,\\\\,\\\\,&={\\\\frac {\\\\rho }{\\\\varepsilon \\_{0}}}\\\\\\\\\\\\nabla \\\\cdot \\\\mathbf {B} \\\\,\\\\,\\\\,&=0\\\\\\\\\\\\nabla \\\\times \\\\mathbf {E} &=-{\\\\frac {\\\\partial \\\\mathbf {B} }{\\\\partial t}}\\\\\\\\\\\\nabla \\\\times \\\\mathbf {B} &=\\\\mu \\_{0}\\\\left(\\\\mathbf {J} +\\\\varepsilon \\_{0}{\\\\frac {\\\\partial \\\\mathbf {E} }{\\\\partial t}}\\\\right)\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1740d383a275f64105f457e209ff5c66eeeb21)\n\nEinstein's theory of [general relativity](https:\/\/en.wikipedia.org\/wiki\/General_relativity \"General relativity\") has been characterized as a work of art, and, among other aesthetic praise,[\\[19\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-19): 148  was described by [Paul Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\") as having \"great mathematical beauty\"[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-20): 123  and by Penrose as having \"supreme mathematical beauty\".[\\[21\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-21): 1038\n\n(There can be more to the beauty of a scientific theory than just its mathematical statement. For example, whether a theory is visualizable or deterministic might have an influence on whether it is seen as beautiful.[\\[22\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-22): 53 [\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-cain-form-4): 873–877 )\n\n## Properties of beautiful mathematics\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=6 \"Edit section: Properties of beautiful mathematics\")\\]\n\nMany mathematicians and philosophers who have written about mathematical beauty have tried to identify properties or criteria that are conducive to the perception of beauty in a piece of mathematics. It is debated whether beauty can be clarified or explained by such properties: [Paul Erdős](https:\/\/en.wikipedia.org\/wiki\/Paul_Erd%C5%91s \"Paul Erdős\") thought that it was no more possible to convince someone of the beauty of a piece of mathematics than to convince them of the beauty of [Beethoven's Ninth Symphony](https:\/\/en.wikipedia.org\/wiki\/Symphony_No._9_\\(Beethoven\\) \"Symphony No. 9 (Beethoven)\"), if they couldn't see it for themselves.[\\[23\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-23)\n\n### Results\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=7 \"Edit section: Results\")\\]\n\nIn his 1940 essay *[A Mathematician's Apology](https:\/\/en.wikipedia.org\/wiki\/A_Mathematician%27s_Apology \"A Mathematician's Apology\")*, [G. H. Hardy](https:\/\/en.wikipedia.org\/wiki\/G._H._Hardy \"G. H. Hardy\") said that a beautiful result, including its proof, possesses three \"purely aesthetic qualities\", namely \"inevitability\", \"unexpectedness\", and \"economy\". He particularly excluded enumeration of cases as \"one of the duller forms of mathematical argument\".[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-hardy-apology-7): §18\n\nIn 1997, [Gian-Carlo Rota](https:\/\/en.wikipedia.org\/wiki\/Gian-Carlo_Rota \"Gian-Carlo Rota\") disagreed with unexpectedness as a sufficient condition for beauty and proposed a counterexample:\n\n> A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago \\[from 1977\\] the proof of the existence of [non-equivalent differentiable structures](https:\/\/en.wikipedia.org\/wiki\/Exotic_sphere \"Exotic sphere\") on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or now.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-rota-phenomenology-2): 172\n\nIn contrast, Monastyrsky wrote in 2001:\n\n> It is very difficult to find an analogous invention in the past to [Milnor](https:\/\/en.wikipedia.org\/wiki\/Milnor \"Milnor\")'s beautiful construction of the different differential structures on the seven-dimensional sphere... The original proof of Milnor was not very constructive, but later E. Briscorn showed that these differential structures can be described in an extremely explicit and beautiful form.[\\[24\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-24): 44\n\nThis disagreement illustrates both the subjective nature of mathematical beauty, [like other forms of beauty in general](https:\/\/en.wikipedia.org\/wiki\/Beauty \"Beauty\"), and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them.\n\n### Proofs\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=8 \"Edit section: Proofs\")\\]\n\nBesides Hardy's properties of \"unexpectedness\", \"inevitability\", \"economy\", which he applied to proofs as well as results, mathematicians have customarily thought beautiful proofs that are short and simple.[\\[25\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-25): 22\n\nIn the search for an elegant proof, mathematicians often look for different independent ways to prove a result—as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly the [Pythagorean theorem](https:\/\/en.wikipedia.org\/wiki\/Pythagorean_theorem \"Pythagorean theorem\"), with hundreds of proofs having been published.[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-26) Another theorem that has been proved in many different ways is the theorem of [quadratic reciprocity](https:\/\/en.wikipedia.org\/wiki\/Quadratic_reciprocity \"Quadratic reciprocity\"). In fact, [Carl Friedrich Gauss](https:\/\/en.wikipedia.org\/wiki\/Carl_Friedrich_Gauss \"Carl Friedrich Gauss\") alone had eight different proofs of this theorem, six of which he published.[\\[27\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-27)\n\nIn contrast, results that are logically correct but involve laborious calculations or consideration of many cases, are not usually considered beautiful, and may be even referred to as *ugly* or *clumsy*. For example, [Kenneth Appel](https:\/\/en.wikipedia.org\/wiki\/Kenneth_Appel \"Kenneth Appel\") and [Wolfgang Haken](https:\/\/en.wikipedia.org\/wiki\/Wolfgang_Haken \"Wolfgang Haken\")'s proof of the [four color theorem](https:\/\/en.wikipedia.org\/wiki\/Four_color_theorem \"Four color theorem\") made use of computer checking of over a thousand cases. [Philip J. Davis](https:\/\/en.wikipedia.org\/wiki\/Philip_J._Davis \"Philip J. Davis\") and [Reuben Hersh](https:\/\/en.wikipedia.org\/wiki\/Reuben_Hersh \"Reuben Hersh\") said that when they first heard that about the proof, they hoped it contained a new insight \"whose beauty would transform my day\", and were disheartened when informed the proof was by case enumeration and computer verification.[\\[28\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-davis-experience-28): 384  [Paul Erdős](https:\/\/en.wikipedia.org\/wiki\/Paul_Erd%C5%91s \"Paul Erdős\") said it was \"not beautiful\" because it gave no insight into why the theorem was true.[\\[29\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-29): 44\n\n## Philosophical analysis\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=9 \"Edit section: Philosophical analysis\")\\]\n\n[Aristotle](https:\/\/en.wikipedia.org\/wiki\/Aristotle \"Aristotle\") thought that beauty was found especially in mathematics, writing in the [*Metaphysics*](https:\/\/en.wikipedia.org\/wiki\/Metaphysics_\\(Aristotle\\) \"Metaphysics (Aristotle)\") that\n\n> those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a very great deal about them; for if they do not expressly mention them, but prove attributes which are their results or their formulae, it is not true to say that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.[\\[30\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-30): 1078a32–35\n\nThe logician and philosopher [Bertrand Russell](https:\/\/en.wikipedia.org\/wiki\/Bertrand_Russell \"Bertrand Russell\") made a now-famous statement characterizing mathematical beauty in terms of purity and austerity:\n\n> Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.[\\[31\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-31)\n\nIn the twentieth century, some philosophers questioned whether there was genuinely beauty in mathematics. The philosopher of science [Rom Harré](https:\/\/en.wikipedia.org\/wiki\/Rom_Harr%C3%A9 \"Rom Harré\") argued that there were no true aesthetic appraisals of mathematics, but only *quasi-aesthetic* appraisals. Any mathematical success described by an aesthetic term was a second-order success besides understanding and correctness. In contrast, aesthetic appraisal of a work of art was first-order. Harré considered this to be the difference between a quasi-aesthetic and a genuinely aesthetic appraisal.[\\[32\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-32)\n\n[Nick Zangwill](https:\/\/en.wikipedia.org\/wiki\/Nick_Zangwill \"Nick Zangwill\") thought that there were no true aesthetic experiences of mathematics and that a proofs or theories could only be *metaphorically* beautiful. His argument had two bases. First, he thought that aesthetic properties depended on sensory properties, and so abstract entities could not have aesthetic properties. Second, he thought that proofs, theorems, theories, and so on had purposes such as demonstrating correctness or granting understanding, and that any praise of them reflected *only* how well they achieved their purpose.[\\[33\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-33): 140–142\n\n## Scientific analysis\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=10 \"Edit section: Scientific analysis\")\\]\n\n### Information-theory model\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=11 \"Edit section: Information-theory model\")\\]\n\nIn the 1970s, [Abraham Moles](https:\/\/en.wikipedia.org\/wiki\/Abraham_Moles \"Abraham Moles\") and [Frieder Nake](https:\/\/en.wikipedia.org\/wiki\/Frieder_Nake \"Frieder Nake\") analyzed links between beauty, [information processing](https:\/\/en.wikipedia.org\/wiki\/Data_processing \"Data processing\"), and [information theory](https:\/\/en.wikipedia.org\/wiki\/Information_theory \"Information theory\").[\\[34\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-34)[\\[35\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-35) In the 1990s, [Jürgen Schmidhuber](https:\/\/en.wikipedia.org\/wiki\/J%C3%BCrgen_Schmidhuber \"Jürgen Schmidhuber\") formulated a mathematical theory of observer-dependent subjective beauty based on [algorithmic information theory](https:\/\/en.wikipedia.org\/wiki\/Algorithmic_information_theory \"Algorithmic information theory\"): the most beautiful objects among subjectively comparable objects have short [algorithmic](https:\/\/en.wikipedia.org\/wiki\/Algorithmic_information_theory \"Algorithmic information theory\") descriptions (i.e., [Kolmogorov complexity](https:\/\/en.wikipedia.org\/wiki\/Kolmogorov_complexity \"Kolmogorov complexity\")) relative to what the observer already knows.[\\[36\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-36)[\\[37\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-37)[\\[38\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-38) Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the [first derivative](https:\/\/en.wikipedia.org\/wiki\/First_derivative \"First derivative\") of subjectively perceived beauty: the observer continually tries to improve the [predictability](https:\/\/en.wikipedia.org\/wiki\/Predictability \"Predictability\") and [compressibility](https:\/\/en.wikipedia.org\/wiki\/Data_compression \"Data compression\") of the observations by discovering regularities such as repetitions and [symmetries](https:\/\/en.wikipedia.org\/wiki\/Symmetries \"Symmetries\") and [fractal](https:\/\/en.wikipedia.org\/wiki\/Fractal \"Fractal\") [self-similarity](https:\/\/en.wikipedia.org\/wiki\/Self-similarity \"Self-similarity\"). Whenever the observer's learning process (possibly a predictive artificial [neural network](https:\/\/en.wikipedia.org\/wiki\/Neural_network \"Neural network\")) leads to improved data compression such that the observation sequence can be described by fewer [bits](https:\/\/en.wikipedia.org\/wiki\/Bit \"Bit\") than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.[\\[39\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-39)[\\[40\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-40)\n\n### Neural correlates\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=12 \"Edit section: Neural correlates\")\\]\n\nBrain imaging experiments conducted by [Semir Zeki](https:\/\/en.wikipedia.org\/wiki\/Semir_Zeki \"Semir Zeki\"), [Michael Atiyah](https:\/\/en.wikipedia.org\/wiki\/Michael_Atiyah \"Michael Atiyah\") and collaborators show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial [orbito-frontal cortex](https:\/\/en.wikipedia.org\/wiki\/Orbitofrontal_cortex \"Orbitofrontal cortex\") (mOFC) of the brain and that this activity is parametrically related to the declared intensity of beauty. The location of the activity is similar to the location of the activity that correlates with the experience of beauty from other sources, such as visual art or music.[\\[41\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-41) Moreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers.[\\[42\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-42)\n\n## Mathematical beauty and the arts\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=13 \"Edit section: Mathematical beauty and the arts\")\\]\n\nMain articles: [Mathematics and art](https:\/\/en.wikipedia.org\/wiki\/Mathematics_and_art \"Mathematics and art\"), [Mathematics and music](https:\/\/en.wikipedia.org\/wiki\/Mathematics_and_music \"Mathematics and music\"), and [Mathematics and architecture](https:\/\/en.wikipedia.org\/wiki\/Mathematics_and_architecture \"Mathematics and architecture\")\n\n### Music\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=14 \"Edit section: Music\")\\]\n\nExamples of the use of mathematics in music include the [stochastic music](https:\/\/en.wikipedia.org\/wiki\/Stochastic_music \"Stochastic music\") of [Iannis Xenakis](https:\/\/en.wikipedia.org\/wiki\/Iannis_Xenakis \"Iannis Xenakis\"),[\\[43\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-43) the [Fibonacci sequence](https:\/\/en.wikipedia.org\/wiki\/Fibonacci_number \"Fibonacci number\") in [Tool](https:\/\/en.wikipedia.org\/wiki\/Tool_\\(band\\) \"Tool (band)\")'s [Lateralus](https:\/\/en.wikipedia.org\/wiki\/Lateralus_\\(song\\) \"Lateralus (song)\"),[\\[44\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-44) and the [Metric modulation](https:\/\/en.wikipedia.org\/wiki\/Metric_modulation \"Metric modulation\") of [Elliott Carter](https:\/\/en.wikipedia.org\/wiki\/Elliott_Carter \"Elliott Carter\").[\\[45\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-45)\n\nOther instances include the counterpoint of [Johann Sebastian Bach](https:\/\/en.wikipedia.org\/wiki\/Johann_Sebastian_Bach \"Johann Sebastian Bach\"), [polyrhythmic](https:\/\/en.wikipedia.org\/wiki\/Polyrhythm \"Polyrhythm\") structures (as in [Igor Stravinsky](https:\/\/en.wikipedia.org\/wiki\/Igor_Stravinsky \"Igor Stravinsky\")'s *[The Rite of Spring](https:\/\/en.wikipedia.org\/wiki\/The_Rite_of_Spring \"The Rite of Spring\")*), [permutation](https:\/\/en.wikipedia.org\/wiki\/Permutation \"Permutation\") theory in [serialism](https:\/\/en.wikipedia.org\/wiki\/Serialism \"Serialism\") beginning with [Arnold Schoenberg](https:\/\/en.wikipedia.org\/wiki\/Arnold_Schoenberg \"Arnold Schoenberg\"), the application of Shepard tones in [Karlheinz Stockhausen](https:\/\/en.wikipedia.org\/wiki\/Karlheinz_Stockhausen \"Karlheinz Stockhausen\")'s *[Hymnen](https:\/\/en.wikipedia.org\/wiki\/Hymnen \"Hymnen\")* and the application of [Group theory](https:\/\/en.wikipedia.org\/wiki\/Group_theory \"Group theory\") to transformations in music in the theoretical writings of [David Lewin](https:\/\/en.wikipedia.org\/wiki\/David_Lewin \"David Lewin\").\n\n### Visual arts\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=15 \"Edit section: Visual arts\")\\]\n\n|   |   |\n|---|---|\n| [![icon](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/9\/99\/Question_book-new.svg\/60px-Question_book-new.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Question_book-new.svg) | This section **needs additional citations for [verification](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Verifiability \"Wikipedia:Verifiability\")**. Please help [improve this article](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Mathematical_beauty \"Special:EditPage\/Mathematical beauty\") by [adding citations to reliable sources](https:\/\/en.wikipedia.org\/wiki\/Help:Referencing_for_beginners \"Help:Referencing for beginners\") in this section. Unsourced material may be challenged and removed. *(September 2025)* *([Learn how and when to remove this message](https:\/\/en.wikipedia.org\/wiki\/Help:Maintenance_template_removal \"Help:Maintenance template removal\"))* |\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/38\/Della_Pittura_Alberti_perspective_pillars_on_grid.jpg\/250px-Della_Pittura_Alberti_perspective_pillars_on_grid.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Della_Pittura_Alberti_perspective_pillars_on_grid.jpg)\n\nDiagram from [Leon Battista Alberti](https:\/\/en.wikipedia.org\/wiki\/Leon_Battista_Alberti \"Leon Battista Alberti\")'s 1435 *[Della Pittura](https:\/\/en.wikipedia.org\/wiki\/Della_Pittura \"Della Pittura\")*, with pillars in [perspective](https:\/\/en.wikipedia.org\/wiki\/Perspective_\\(graphical\\) \"Perspective (graphical)\") on a grid\n\nExamples of the use of mathematics in the visual arts include applications of [chaos theory](https:\/\/en.wikipedia.org\/wiki\/Chaos_theory \"Chaos theory\") and [fractal](https:\/\/en.wikipedia.org\/wiki\/Fractal \"Fractal\") geometry to [computer-generated art](https:\/\/en.wikipedia.org\/wiki\/Digital_art \"Digital art\"), symmetry studies of [Leonardo da Vinci](https:\/\/en.wikipedia.org\/wiki\/Leonardo_da_Vinci \"Leonardo da Vinci\"), [projective geometries](https:\/\/en.wikipedia.org\/wiki\/Projective_geometry \"Projective geometry\") in development of the [perspective](https:\/\/en.wikipedia.org\/wiki\/Perspective_\\(graphical\\) \"Perspective (graphical)\") theory of [Renaissance](https:\/\/en.wikipedia.org\/wiki\/Renaissance \"Renaissance\") art, [grids](https:\/\/en.wikipedia.org\/wiki\/Grid_\\(page_layout\\) \"Grid (page layout)\") in [Op art](https:\/\/en.wikipedia.org\/wiki\/Op_art \"Op art\"), optical geometry in the [camera obscura](https:\/\/en.wikipedia.org\/wiki\/Camera_obscura \"Camera obscura\") of [Giambattista della Porta](https:\/\/en.wikipedia.org\/wiki\/Giambattista_della_Porta \"Giambattista della Porta\"), and multiple perspective in analytic [cubism](https:\/\/en.wikipedia.org\/wiki\/Cubism \"Cubism\") and [futurism](https:\/\/en.wikipedia.org\/wiki\/Futurism \"Futurism\").\n\n[Sacred geometry](https:\/\/en.wikipedia.org\/wiki\/Sacred_geometry \"Sacred geometry\") is a field of its own, giving rise to countless art forms including some of the best known mystic symbols and religious motifs, and has a particularly rich history in [Islamic architecture](https:\/\/en.wikipedia.org\/wiki\/Islamic_architecture \"Islamic architecture\"). It also provides a means of meditation and comtemplation, for example the study of the [Kaballah](https:\/\/en.wikipedia.org\/wiki\/Kaballah \"Kaballah\") [Sefirot](https:\/\/en.wikipedia.org\/wiki\/Sefirot \"Sefirot\") (Tree Of Life) and [Metatron's Cube](https:\/\/en.wikipedia.org\/wiki\/Metatron%27s_Cube \"Metatron's Cube\"); and also the act of drawing itself.\n\nThe Dutch graphic designer [M. C. Escher](https:\/\/en.wikipedia.org\/wiki\/M._C._Escher \"M. C. Escher\") created mathematically inspired [woodcuts](https:\/\/en.wikipedia.org\/wiki\/Woodcut \"Woodcut\"), [lithographs](https:\/\/en.wikipedia.org\/wiki\/Lithograph \"Lithograph\"), and [mezzotints](https:\/\/en.wikipedia.org\/wiki\/Mezzotint \"Mezzotint\"). These feature impossible constructions, explorations of [infinity](https:\/\/en.wikipedia.org\/wiki\/Infinity \"Infinity\"), architecture, visual [paradoxes](https:\/\/en.wikipedia.org\/wiki\/Paradox \"Paradox\") and [tessellations](https:\/\/en.wikipedia.org\/wiki\/Tessellation \"Tessellation\").\n\nSome painters and sculptors create work distorted with the mathematical principles of [anamorphosis](https:\/\/en.wikipedia.org\/wiki\/Anamorphosis \"Anamorphosis\"), including South African sculptor [Jonty Hurwitz](https:\/\/en.wikipedia.org\/wiki\/Jonty_Hurwitz \"Jonty Hurwitz\").\n\n[Origami](https:\/\/en.wikipedia.org\/wiki\/Origami \"Origami\"), the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the [mathematics of paper folding](https:\/\/en.wikipedia.org\/wiki\/Mathematics_of_paper_folding \"Mathematics of paper folding\") by observing the [crease pattern](https:\/\/en.wikipedia.org\/wiki\/Crease_pattern \"Crease pattern\") on unfolded origami pieces.[\\[46\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-46)\n\nBritish constructionist artist [John Ernest](https:\/\/en.wikipedia.org\/wiki\/John_Ernest \"John Ernest\") created reliefs and paintings inspired by group theory.[\\[47\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-47) A number of other British artists of the constructionist and systems schools of thought also draw on mathematics models and structures as a source of inspiration, including [Anthony Hill](https:\/\/en.wikipedia.org\/wiki\/Anthony_Hill_\\(artist\\) \"Anthony Hill (artist)\") and [Peter Lowe](https:\/\/en.wikipedia.org\/wiki\/Peter_Lowe_\\(artist\\) \"Peter Lowe (artist)\").[\\[48\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-48) Computer-generated art is based on mathematical [algorithms](https:\/\/en.wikipedia.org\/wiki\/Algorithm \"Algorithm\").\n\n## See also\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=16 \"Edit section: See also\")\\]\n\n- [Argument from beauty](https:\/\/en.wikipedia.org\/wiki\/Argument_from_beauty \"Argument from beauty\")\n- [Fluency heuristic](https:\/\/en.wikipedia.org\/wiki\/Fluency_heuristic \"Fluency heuristic\")\n- [Golden ratio](https:\/\/en.wikipedia.org\/wiki\/Golden_ratio \"Golden ratio\")\n- [Neuroesthetics](https:\/\/en.wikipedia.org\/wiki\/Neuroesthetics \"Neuroesthetics\")\n- [Philosophy of mathematics](https:\/\/en.wikipedia.org\/wiki\/Philosophy_of_mathematics \"Philosophy of mathematics\")\n- [Processing fluency theory of aesthetic pleasure](https:\/\/en.wikipedia.org\/wiki\/Processing_fluency_theory_of_aesthetic_pleasure \"Processing fluency theory of aesthetic pleasure\")\n- [Pythagoreanism](https:\/\/en.wikipedia.org\/wiki\/Pythagoreanism \"Pythagoreanism\")\n- [Sense of wonder](https:\/\/en.wikipedia.org\/wiki\/Sense_of_wonder \"Sense of wonder\")\n- [Theory of everything](https:\/\/en.wikipedia.org\/wiki\/Theory_of_everything \"Theory of everything\")\n\n## Notes\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=17 \"Edit section: Notes\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-1)**\n   Phillips, George (2005). [\"Preface\"](https:\/\/books.google.com\/books?id=psFwdN6V6icC&q=there+is+nothing+in+the+world+of+mathematics+that+corresponds+to+an+audience+in+a+concert+hall,+where+the+passive+listen+to+the+active.+Happily,+mathematicians+are+all+doers,+not+spectators.&pg=PR7). *Mathematics Is Not a Spectator Sport*. [Springer Science+Business Media](https:\/\/en.wikipedia.org\/wiki\/Springer_Science%2BBusiness_Media \"Springer Science+Business Media\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0-387-25528-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-387-25528-1 \"Special:BookSources\/0-387-25528-1\")\n   \n   . Retrieved 2008-08-22. \"\"...there is nothing in the world of mathematics that corresponds to an audience in a concert hall, where the passive listen to the active. Happily, mathematicians are all *doers*, not spectators.\"\n2. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-rota-phenomenology_2-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-rota-phenomenology_2-1)\n   Rota, Gian-Carlo (May 1997). \"The Phenomenology of Mathematical Beauty\". *Synthese*. **111** (2): 171–182\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1023\/A:1004930722234](https:\/\/doi.org\/10.1023%2FA%3A1004930722234).\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-3)**\n   Wilson, Robin (2018). *Euler's Pioneering Equation: The most beautiful theorem in mathematics*. Oxford University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-19-879492-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-19-879492-9 \"Special:BookSources\/978-0-19-879492-9\")\n   \n   .\n4. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-cain-form_4-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-cain-form_4-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-cain-form_4-2)\n   Cain, Alan J. (2024). [*Form & Number: A History of Mathematical Beauty*](https:\/\/archive.org\/details\/cain_formandnumber_ebook_large). Lisbon: EBook.\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-5)**\n   Gallagher, James (13 February 2014). [\"Mathematics: Why the brain sees maths as beauty\"](https:\/\/www.bbc.co.uk\/news\/science-environment-26151062). *BBC News online*. Retrieved 13 February 2014.\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-6)**\n   Feynman, Richard P. (1977). [*The Feynman Lectures on Physics*](https:\/\/feynmanlectures.caltech.edu\/I_22.html#Ch22-S6-p4). Vol. I. Addison-Wesley. p. 22-16. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0-201-02010-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-201-02010-6 \"Special:BookSources\/0-201-02010-6\")\n   \n   .\n7. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-hardy-apology_7-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-hardy-apology_7-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-hardy-apology_7-2)\n   Hardy, G.H. (1967). [*A Mathematician's Apology*](https:\/\/archive.org\/details\/hardy_annotated). Cambridge University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-1-107-60463-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-107-60463-6 \"Special:BookSources\/978-1-107-60463-6\")\n   \n   .\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-8)**\n   Bell, E.T. (1937). *Men of Mathematics*. Simon & Schuster.\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-9)**\n   Wells, David (June 1990). \"Are these the most beautiful?\". *The Mathematical Intelligencer*. **12** (3): 37–41\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/BF03024015](https:\/\/doi.org\/10.1007%2FBF03024015).\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-10)**\n    Paulos, John Allen (1991). *Beyond Numeracy*. New York: Alfred A. Knopf. pp. 125–127\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-394-586409](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-394-586409 \"Special:BookSources\/0-394-586409\")\n    \n    .\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-11)**\n    Dutilh Novaes, Catarina (2019). \"The Beauty (?) of Mathematical Proofs\". In Aberdein, Andrew; Inglis, Matthew (eds.). *Advances in Experimental Philosophy of Logic and Mathematics*. Bloomsbury Academic. pp. 69–71\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-350-03901-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-350-03901-8 \"Special:BookSources\/978-1-350-03901-8\")\n    \n    .\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-12)**\n    Polster, Burkert (2004). *Q.E.D.: Beauty in Mathematical Proof*. Walker & Company. pp. 32–33\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-8027-1431-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-8027-1431-2 \"Special:BookSources\/978-0-8027-1431-2\")\n    \n    .\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-13)**\n    Schechter, Bruce (2000). *My Brain is Open: The Mathematical Journeys of Paul Erdős*. New York: [Simon & Schuster](https:\/\/en.wikipedia.org\/wiki\/Simon_%26_Schuster \"Simon & Schuster\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-684-85980-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-684-85980-7 \"Special:BookSources\/0-684-85980-7\")\n    \n    .\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-14)**\n    [Aigner, Martin](https:\/\/en.wikipedia.org\/wiki\/Martin_Aigner \"Martin Aigner\"); [Ziegler, Günter M.](https:\/\/en.wikipedia.org\/wiki\/G%C3%BCnter_M._Ziegler \"Günter M. Ziegler\") (2018). [*Proofs from THE BOOK*](https:\/\/en.wikipedia.org\/wiki\/Proofs_from_THE_BOOK \"Proofs from THE BOOK\") (6th ed.). Springer. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-3-662-57264-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-662-57264-1 \"Special:BookSources\/978-3-662-57264-1\")\n    \n    .\n15. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-plato-timaeus_15-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-plato-timaeus_15-1)\n    Plato (1929). [*Timaeus*](https:\/\/www.loebclassics.com\/view\/LCL234\/1929\/volume.xml). Cambridge, MA: Harvard University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-674-99257-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-674-99257-3 \"Special:BookSources\/978-0-674-99257-3\")\n    \n    .\n    \n    `{{cite book}}`: ISBN \/ Date incompatibility ([help](https:\/\/en.wikipedia.org\/wiki\/Help:CS1_errors#invalid_isbn_date \"Help:CS1 errors\"))\n16. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-16)**\n    Field, J.V. (2013). *Kepler's Geometrical Cosmology*. Bloomsbury. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [9781472507037](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9781472507037 \"Special:BookSources\/9781472507037\")\n    \n    .\n17. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-17)**\n    Whitfield, John. [\"Journey to the 248th dimension\"](https:\/\/www.nature.com\/articles\/news070319-4). *Nature Online*. Retrieved 12 September 2025.\n18. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-18)**\n    Penrose, Roger (1974). \"The Rôle of Aesthetics in Pure and Applied Mathematical Research\". *Bulletin of the Institute of Mathematics and its Applications*. **10**: 226–271\\.\n19. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-19)**\n    [Chandrasekhar, Subrahmanyan](https:\/\/en.wikipedia.org\/wiki\/Subrahmanyan_Chandrasekhar \"Subrahmanyan Chandrasekhar\") (1987). *Truth and Beauty: Aesthetics and Motivation in Science*. Chicago, London: University of Chicago Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-226-10087-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-226-10087-6 \"Special:BookSources\/978-0-226-10087-6\")\n    \n    .\n20. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-20)**\n    Dirac, P.A.M. (1940). \"The Relation between Mathematics and Physics\". *Proceedings of the Royal Society of Edinburgh*. **59**: 122–129\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1017\/s0370164600012207](https:\/\/doi.org\/10.1017%2Fs0370164600012207).\n21. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-21)**\n    Penrose, Roger (2004). *The Road to Reality*. London: Jonathan Cape. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-224-04447-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-224-04447-9 \"Special:BookSources\/978-0-224-04447-9\")\n    \n    .\n22. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-22)**\n    McAllister, James W. (1996). [*Beauty & Revolution in Science*](https:\/\/hdl.handle.net\/1887\/10158). Ithaca: Cornell University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-8014-3240-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-8014-3240-8 \"Special:BookSources\/978-0-8014-3240-8\")\n    \n    .\n23. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-23)**\n    [Devlin, Keith](https:\/\/en.wikipedia.org\/wiki\/Keith_Devlin \"Keith Devlin\") (2000). \"Do Mathematicians Have Different Brains?\". [*The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip*](https:\/\/archive.org\/details\/mathgene00keit). [Basic Books](https:\/\/en.wikipedia.org\/wiki\/Basic_Books \"Basic Books\"). p. [140](https:\/\/archive.org\/details\/mathgene00keit\/page\/140). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-465-01619-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-465-01619-8 \"Special:BookSources\/978-0-465-01619-8\")\n    \n    . Retrieved 2008-08-22.\n24. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-24)**\n    Monastyrsky, Michael (2001). [\"Some Trends in Modern Mathematics and the Fields Medal\"](http:\/\/www.fields.utoronto.ca\/aboutus\/FieldsMedal_Monastyrsky.pdf) (PDF). *Can. Math. Soc. Notes*. **33** (2 and 3).\n25. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-25)**\n    McAllister, James W. (2005). \"Mathematical Beauty and the Evolution of the Standards of Mathematical Proof\". In Emmer, Michele (ed.). *The Visual Mind II*. MIT Press. pp. 15–34\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-262-05076-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-262-05076-0 \"Special:BookSources\/978-0-262-05076-0\")\n    \n    .\n26. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-26)**\n    [Loomis, Elisha Scott](https:\/\/en.wikipedia.org\/wiki\/Elisha_Scott_Loomis \"Elisha Scott Loomis\") (1968). *The Pythagorean Proposition*. Washington, DC: National Council of Teachers of Mathematics. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-873-53036-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-873-53036-1 \"Special:BookSources\/978-0-873-53036-1\")\n    \n    .\n27. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-27)**\n    Weisstein, Eric W. [\"Quadratic Reciprocity Theorem\"](http:\/\/mathworld.wolfram.com\/QuadraticReciprocityTheorem.html). *mathworld.wolfram.com*. Retrieved 2019-10-31.\n28. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-davis-experience_28-0)**\n    Davis, Philip J.; Hersh, Reuben (1981). *The Mathematical Experience*. Boston: Houghton Mifflin. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-395-32131-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-395-32131-7 \"Special:BookSources\/978-0-395-32131-7\")\n    \n    .\n29. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-29)**\n    Hoffman, Paul (1999). *The Man Who Loved Only Numbers*. London: Fourth Estate. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-85702-829-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-85702-829-4 \"Special:BookSources\/978-1-85702-829-4\")\n    \n    .\n30. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-30)**\n    [Aristotle](https:\/\/en.wikipedia.org\/wiki\/Aristotle \"Aristotle\") (1995). \"Metaphysics\". In Barnes, Jonathan (ed.). *The Complete Works of Aristotle*. Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-691-01650-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-691-01650-4 \"Special:BookSources\/978-0-691-01650-4\")\n    \n    .\n31. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-31)**\n    [Russell, Bertrand](https:\/\/en.wikipedia.org\/wiki\/Bertrand_Russell \"Bertrand Russell\") (1919). \"The Study of Mathematics\". [*Mysticism and Logic: And Other Essays*](https:\/\/archive.org\/details\/bub_gb_zwMQAAAAYAAJ). [Longman](https:\/\/en.wikipedia.org\/wiki\/Longman \"Longman\"). p. [60](https:\/\/archive.org\/details\/bub_gb_zwMQAAAAYAAJ\/page\/n68). Retrieved 2008-08-22. \"Mathematics rightly viewed possesses not only truth but supreme beauty a beauty cold and austere like that of sculpture without appeal to any part of our weaker nature without the gorgeous trappings Russell.\"\n32. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-32)**\n    [Harré, Rom](https:\/\/en.wikipedia.org\/wiki\/Rom_Harr%C3%A9 \"Rom Harré\") (1958). [\"Quasi-Aesthetic Appraisals\"](https:\/\/www.jstor.org\/stable\/3748562). *Philosophy*. **33**: 132–137\\.\n33. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-33)**\n    [Zangwill, Nick](https:\/\/en.wikipedia.org\/wiki\/Nick_Zangwill \"Nick Zangwill\") (2001). *The Metaphysics of Beauty*. Ithaca, London: Cornell University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-8014-3820-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-8014-3820-2 \"Special:BookSources\/978-0-8014-3820-2\")\n    \n    .\n34. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-34)** A. Moles: *Théorie de l'information et perception esthétique*, Paris, Denoël, 1973 ([Information Theory](https:\/\/en.wikipedia.org\/wiki\/Information_Theory \"Information Theory\") and aesthetical perception)\n35. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-35)**\n    F Nake (1974). Ästhetik als Informationsverarbeitung. ([Aesthetics](https:\/\/en.wikipedia.org\/wiki\/Aesthetics \"Aesthetics\") as information processing). Grundlagen und Anwendungen der Informatik im Bereich ästhetischer Produktion und Kritik. Springer, 1974,\n    \n    [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [3-211-81216-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/3-211-81216-4 \"Special:BookSources\/3-211-81216-4\")\n    \n    ,\n    \n    [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-3-211-81216-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-211-81216-7 \"Special:BookSources\/978-3-211-81216-7\")\n36. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-36)**\n    J. Schmidhuber. [Low-complexity art](https:\/\/en.wikipedia.org\/wiki\/Low-complexity_art \"Low-complexity art\"). [Leonardo](https:\/\/en.wikipedia.org\/wiki\/Leonardo_\\(journal\\) \"Leonardo (journal)\"), Journal of the International Society for the Arts, Sciences, and Technology ([Leonardo\/ISAST](https:\/\/en.wikipedia.org\/wiki\/Leonardo,_The_International_Society_of_the_Arts,_Sciences_and_Technology \"Leonardo, The International Society of the Arts, Sciences and Technology\")), 30(2):97–103, 1997.\n    \n    [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.2307\/1576418](https:\/\/doi.org\/10.2307%2F1576418). [JSTOR](https:\/\/en.wikipedia.org\/wiki\/JSTOR_\\(identifier\\) \"JSTOR (identifier)\") [1576418](https:\/\/www.jstor.org\/stable\/1576418).\n37. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-37)** J. Schmidhuber. Papers on the theory of beauty and [low-complexity art](https:\/\/en.wikipedia.org\/wiki\/Low-complexity_art \"Low-complexity art\") since 1994: <http:\/\/www.idsia.ch\/~juergen\/beauty.html>\n38. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-38)**\n    J. Schmidhuber. Simple Algorithmic Principles of Discovery, Subjective Beauty, Selective Attention, Curiosity & Creativity. Proc. 10th Intl. Conf. on Discovery Science (DS 2007) pp. 26–38, LNAI 4755, Springer, 2007. Also in Proc. 18th Intl. Conf. on Algorithmic Learning Theory (ALT 2007) p. 32, LNAI 4754, Springer, 2007. Joint invited lecture for DS 2007 and ALT 2007, Sendai, Japan, 2007.\n    \n    [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[0709\\.0674](https:\/\/arxiv.org\/abs\/0709.0674).\n39. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-39)** .J. Schmidhuber. Curious model-building control systems. International Joint Conference on Neural Networks, Singapore, vol 2, 1458–1463. IEEE press, 1991\n40. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-40)** Schmidhuber's theory of beauty and curiosity in a German TV show: <http:\/\/www.br-online.de\/bayerisches-fernsehen\/faszination-wissen\/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml> [Archived](https:\/\/web.archive.org\/web\/20080603221058\/http:\/\/www.br-online.de\/bayerisches-fernsehen\/faszination-wissen\/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml) June 3, 2008, at the [Wayback Machine](https:\/\/en.wikipedia.org\/wiki\/Wayback_Machine \"Wayback Machine\")\n41. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-41)**\n    Zeki, Semir; Romaya, John Paul; Benincasa, Dionigi M. T.; Atiyah, Michael F. (2014). [\"The experience of mathematical beauty and its neural correlates\"](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3923150). *Frontiers in Human Neuroscience*. **8**: 68. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3389\/fnhum.2014.00068](https:\/\/doi.org\/10.3389%2Ffnhum.2014.00068). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1662-5161](https:\/\/search.worldcat.org\/issn\/1662-5161). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [3923150](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3923150). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [24592230](https:\/\/pubmed.ncbi.nlm.nih.gov\/24592230).\n42. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-42)**\n    Zhang, Haoxuan; Zeki, Semir (May 2022). [\"Judgments of mathematical beauty are resistant to revision through external opinion\"](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC9790661). *PsyCh Journal*. **11** (5): 741–747\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1002\/pchj.556](https:\/\/doi.org\/10.1002%2Fpchj.556). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [2046-0252](https:\/\/search.worldcat.org\/issn\/2046-0252). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [9790661](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC9790661). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [35491015](https:\/\/pubmed.ncbi.nlm.nih.gov\/35491015).\n43. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-43)**  [Luque, Sergio. 2009. \"The Stochastic Synthesis of Iannis Xenakis.\" Leonardo Music Journal (19): 77–84](http:\/\/www.sergioluque.com\/stochastics)\n44. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-44)**\n    Norris, Chris (2001). [\"Hammer Of The Gods\"](https:\/\/web.archive.org\/web\/20111101195240\/http:\/\/toolshed.down.net\/articles\/text\/spinmag.jun.2001.html). Archived from [the original](http:\/\/toolshed.down.net\/articles\/text\/spinmag.jun.2001.html) on November 1, 2011. Retrieved April 25, 2007.\n45. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-45)**\n    Schell, Michael (December 11, 2018). [\"Elliott Carter (1908–2012): Legacy of a Centenarian\"](https:\/\/www.secondinversion.org\/2018\/12\/11\/elliott-carter-1908-2012-legacy-of-a-centenarian\/). *Second Inversion*. [Archived](https:\/\/web.archive.org\/web\/20181224054514\/https:\/\/www.secondinversion.org\/2018\/12\/11\/elliott-carter-1908-2012-legacy-of-a-centenarian\/) from the original on December 24, 2018. Retrieved December 11, 2018.\n46. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-46)** Hull, Thomas. \"Project Origami: Activities for Exploring Mathematics\". Taylor & Francis, 2006.\n47. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-47)** John Ernest's use of mathematics and especially group theory in his art works is analysed in *John Ernest, A Mathematical Artist* by Paul Ernest in *Philosophy of Mathematics Education Journal*, No. 24 Dec. 2009 (Special Issue on Mathematics and Art): <http:\/\/people.exeter.ac.uk\/PErnest\/pome24\/index.htm>\n48. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-48)**\n    Franco, Francesca (2017-10-05). [\"The Systems Group (Chapter 2)\"](https:\/\/books.google.com\/books?id=oJU4DwAAQBAJ&q=anthony+hill+and+peter+lowe&pg=PT61). *Generative Systems Art: The Work of Ernest Edmonds*. Routledge. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [9781317137436](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9781317137436 \"Special:BookSources\/9781317137436\")\n    \n    .\n\n## References\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=18 \"Edit section: References\")\\]\n\n- Cellucci, Carlo (2015), \"Mathematical beauty, understanding, and discovery\", *[Foundations of Science](https:\/\/en.wikipedia.org\/wiki\/Foundations_of_Science \"Foundations of Science\")*, **20** (4): 339–355, [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/s10699-014-9378-7](https:\/\/doi.org\/10.1007%2Fs10699-014-9378-7), [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [120068870](https:\/\/api.semanticscholar.org\/CorpusID:120068870)\n- [Hoffman, Paul](https:\/\/en.wikipedia.org\/wiki\/Paul_Hoffman_\\(science_writer\\) \"Paul Hoffman (science writer)\") (1992), *[The Man Who Loved Only Numbers](https:\/\/en.wikipedia.org\/wiki\/The_Man_Who_Loved_Only_Numbers \"The Man Who Loved Only Numbers\")*, Hyperion.\n- [Loomis, Elisha Scott](https:\/\/en.wikipedia.org\/wiki\/Elisha_Scott_Loomis \"Elisha Scott Loomis\") (1968), *The Pythagorean Proposition*, The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem.\n- [Pandey, S.K.](https:\/\/en.wikipedia.org\/w\/index.php?title=S.K._Pandey&action=edit&redlink=1 \"S.K. Pandey (page does not exist)\") . [*The Humming of Mathematics: Melody of Mathematics*](https:\/\/books.google.com\/books?id=BgYbzAEACAAJ). Independently Published, 2019.\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [1710134437](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/1710134437 \"Special:BookSources\/1710134437\")\n  .\n- Peitgen, H.-O., and Richter, P.H. (1986), *The Beauty of Fractals*, Springer-Verlag.\n- [Reber, R.](https:\/\/en.wikipedia.org\/wiki\/Rolf_Reber \"Rolf Reber\"); Brun, M.; Mitterndorfer, K. (2008). \"The use of heuristics in intuitive mathematical judgment\". *Psychonomic Bulletin & Review*. **15** (6): 1174–1178\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3758\/PBR.15.6.1174](https:\/\/doi.org\/10.3758%2FPBR.15.6.1174). [hdl](https:\/\/en.wikipedia.org\/wiki\/Hdl_\\(identifier\\) \"Hdl (identifier)\"):[1956\/2734](https:\/\/hdl.handle.net\/1956%2F2734). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [19001586](https:\/\/pubmed.ncbi.nlm.nih.gov\/19001586). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [5297500](https:\/\/api.semanticscholar.org\/CorpusID:5297500).\n- Strohmeier, John, and Westbrook, Peter (1999), *Divine Harmony, The Life and Teachings of Pythagoras*, Berkeley Hills Books, Berkeley, CA.\n- [Zeki, S.](https:\/\/en.wikipedia.org\/wiki\/Semir_Zeki \"Semir Zeki\"); Romaya, J. P.; Benincasa, D. M. T.; [Atiyah, M. F.](https:\/\/en.wikipedia.org\/wiki\/Michael_Atiyah \"Michael Atiyah\") (2014), \"The experience of mathematical beauty and its neural correlates\", *Frontiers in Human Neuroscience*, **8**: 68, [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3389\/fnhum.2014.00068](https:\/\/doi.org\/10.3389%2Ffnhum.2014.00068), [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [3923150](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3923150), [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [24592230](https:\/\/pubmed.ncbi.nlm.nih.gov\/24592230)\n\n## Further reading\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=19 \"Edit section: Further reading\")\\]\n\n- [Aigner, Martin](https:\/\/en.wikipedia.org\/wiki\/Martin_Aigner \"Martin Aigner\"); [Ziegler, Günter M.](https:\/\/en.wikipedia.org\/wiki\/G%C3%BCnter_M._Ziegler \"Günter M. Ziegler\") (2018). [*Proofs from THE BOOK*](https:\/\/en.wikipedia.org\/wiki\/Proofs_from_THE_BOOK \"Proofs from THE BOOK\") (6th ed.). Springer. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-3-662-57264-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-662-57264-1 \"Special:BookSources\/978-3-662-57264-1\")\n  \n  .\n- Cain, Alan J. (2024). [*Form & Number: A History of Mathematical Beauty*](https:\/\/archive.org\/details\/cain_formandnumber_ebook_large). Lisbon: Ebook.\n- [Hadamard, Jacques](https:\/\/en.wikipedia.org\/wiki\/Jacques_Hadamard \"Jacques Hadamard\") (1949). [*An Essay on the Psychology of Invention in the Mathematical Field*](https:\/\/archive.org\/details\/essayonpsycholog00hada) (2nd enlarged ed.). Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [0-486-20107-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-486-20107-4 \"Special:BookSources\/0-486-20107-4\")\n  \n  .\n  `{{cite book}}`: ISBN \/ Date incompatibility ([help](https:\/\/en.wikipedia.org\/wiki\/Help:CS1_errors#invalid_isbn_date \"Help:CS1 errors\"))\n- [Hardy, G.H.](https:\/\/en.wikipedia.org\/wiki\/G.H._Hardy \"G.H. Hardy\") (1967) \\[1st published 1940\\]. [*A Mathematician's Apology*](https:\/\/archive.org\/details\/hardy_annotated). Cambridge University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-1-107-60463-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-107-60463-6 \"Special:BookSources\/978-1-107-60463-6\")\n  \n  .\n- Huntley, H.E. (1970). [*The Divine Proportion: A Study in Mathematical Beauty*](https:\/\/archive.org\/details\/divineproportion0000hunt_o2w9). New York: Dover Publications. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-486-22254-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-486-22254-7 \"Special:BookSources\/978-0-486-22254-7\")\n  \n  .\n- Stewart, Ian (2007). *Why beauty is truth : a history of symmetry*. New York: Basic Books, a member of the Perseus Books Group. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-465-08236-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-465-08236-0 \"Special:BookSources\/978-0-465-08236-0\")\n  \n  . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [76481488](https:\/\/search.worldcat.org\/oclc\/76481488).\n\n## External links\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=20 \"Edit section: External links\")\\]\n\n- [Mathematics, Poetry and Beauty](https:\/\/web.archive.org\/web\/20151209130947\/http:\/\/raharoni.net.technion.ac.il\/mathematics-poetry-and-beauty\/)\n- [Is Mathematics Beautiful?](http:\/\/www.cut-the-knot.org\/manifesto\/beauty.shtml) cut-the-knot.org\n- [Justin Mullins.com](http:\/\/www.justinmullins.com\/)\n- [Edna St. Vincent Millay (poet): *Euclid alone has looked on beauty bare*](http:\/\/www.the-athenaeum.org\/poetry\/detail.php?id=80)\n- [Terence Tao](https:\/\/en.wikipedia.org\/wiki\/Terence_Tao \"Terence Tao\"), [*What is good mathematics?*](https:\/\/arxiv.org\/abs\/math\/0702396)\n- [Mathbeauty Blog](http:\/\/mathbeauty.wordpress.com\/)\n- The *[Aesthetic Appeal](https:\/\/archive.org\/details\/aestheticappeal)* collection at the [Internet Archive](https:\/\/en.wikipedia.org\/wiki\/Internet_Archive \"Internet Archive\")\n- [*A Mathematical Romance*](http:\/\/www.nybooks.com\/articles\/archives\/2013\/dec\/05\/mathematical-romance\/) [Jim Holt](https:\/\/en.wikipedia.org\/wiki\/Jim_Holt_\\(philosopher\\) \"Jim Holt (philosopher)\") December 5, 2013 issue of [The New York Review of Books](https:\/\/en.wikipedia.org\/wiki\/The_New_York_Review_of_Books \"The New York Review of Books\") review of *Love and Math: The Heart of Hidden Reality* by [Edward Frenkel](https:\/\/en.wikipedia.org\/wiki\/Edward_Frenkel \"Edward Frenkel\")\n\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:Aesthetics \"Template:Aesthetics\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:Aesthetics \"Template talk:Aesthetics\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:Aesthetics \"Special:EditPage\/Template:Aesthetics\")[Aesthetics](https:\/\/en.wikipedia.org\/wiki\/Aesthetics \"Aesthetics\") |   |\n|---|---|\n| Areas | [African](https:\/\/en.wikipedia.org\/wiki\/African_aesthetic \"African aesthetic\") [Ancient](https:\/\/en.wikipedia.org\/wiki\/Ancient_aesthetics \"Ancient aesthetics\") [Indian](https:\/\/en.wikipedia.org\/wiki\/Indian_aesthetics \"Indian aesthetics\") [Japanese](https:\/\/en.wikipedia.org\/wiki\/Japanese_aesthetics \"Japanese aesthetics\") [Mathematics](https:\/\/en.wikipedia.org\/wiki\/Mathematics_and_art \"Mathematics and art\") [Medieval](https:\/\/en.wikipedia.org\/wiki\/Medieval_aesthetics \"Medieval aesthetics\") [Music](https:\/\/en.wikipedia.org\/wiki\/Aesthetics_of_music \"Aesthetics of music\") [Nature](https:\/\/en.wikipedia.org\/wiki\/Aesthetics_of_nature \"Aesthetics of nature\") [Science](https:\/\/en.wikipedia.org\/wiki\/Aesthetics_of_science \"Aesthetics of science\") [Theology](https:\/\/en.wikipedia.org\/wiki\/Theological_aesthetics \"Theological aesthetics\") |\n| Schools | [Aestheticism](https:\/\/en.wikipedia.org\/wiki\/Aestheticism \"Aestheticism\") [Classicism](https:\/\/en.wikipedia.org\/wiki\/Classicism \"Classicism\") [Fascism](https:\/\/en.wikipedia.org\/wiki\/Fascism#Aesthetics \"Fascism\") [Feminism](https:\/\/en.wikipedia.org\/wiki\/Feminist_aesthetics \"Feminist aesthetics\") [Formalism](https:\/\/en.wikipedia.org\/wiki\/Formalism_\\(art\\) \"Formalism (art)\") [Historicism](https:\/\/en.wikipedia.org\/wiki\/Historicism_\\(art\\) \"Historicism (art)\") [Marxism](https:\/\/en.wikipedia.org\/wiki\/Marxist_aesthetics \"Marxist aesthetics\") [Modernism](https:\/\/en.wikipedia.org\/wiki\/Modernism \"Modernism\") [Postmodernism](https:\/\/en.wikipedia.org\/wiki\/Postmodernism \"Postmodernism\") [Psychoanalysis](https:\/\/en.wikipedia.org\/wiki\/Psychoanalytic_theory \"Psychoanalytic theory\") [Realism](https:\/\/en.wikipedia.org\/wiki\/Aesthetic_Realism \"Aesthetic Realism\") [Romanticism](https:\/\/en.wikipedia.org\/wiki\/Romanticism \"Romanticism\") [Symbolism](https:\/\/en.wikipedia.org\/wiki\/Symbolism_\\(movement\\) \"Symbolism (movement)\") [Theosophy](https:\/\/en.wikipedia.org\/wiki\/Theosophy_and_visual_arts \"Theosophy and visual arts\") *[more...](https:\/\/en.wikipedia.org\/wiki\/List_of_art_movements \"List of art movements\")* |\n| Philosophers | [Abhinavagupta](https:\/\/en.wikipedia.org\/wiki\/Abhinavagupta \"Abhinavagupta\") [Adorno](https:\/\/en.wikipedia.org\/wiki\/Theodor_W._Adorno \"Theodor W. Adorno\") [Alberti](https:\/\/en.wikipedia.org\/wiki\/Leon_Battista_Alberti \"Leon Battista Alberti\") [Akhundov](https:\/\/en.wikipedia.org\/wiki\/Mirza_Fatali_Akhundov \"Mirza Fatali Akhundov\") [Ibn Arabi](https:\/\/en.wikipedia.org\/wiki\/Ibn_Arabi \"Ibn Arabi\") [Aristotle](https:\/\/en.wikipedia.org\/wiki\/Aristotle \"Aristotle\") [Aquinas](https:\/\/en.wikipedia.org\/wiki\/Thomas_Aquinas \"Thomas Aquinas\") [Balázs](https:\/\/en.wikipedia.org\/wiki\/B%C3%A9la_Bal%C3%A1zs \"Béla Balázs\") [Balthasar](https:\/\/en.wikipedia.org\/wiki\/Hans_Urs_von_Balthasar \"Hans Urs von Balthasar\") [Bataille](https:\/\/en.wikipedia.org\/wiki\/Georges_Bataille \"Georges Bataille\") [Baudelaire](https:\/\/en.wikipedia.org\/wiki\/Charles_Baudelaire \"Charles Baudelaire\") [Baudrillard](https:\/\/en.wikipedia.org\/wiki\/Jean_Baudrillard \"Jean Baudrillard\") [Baumgarten](https:\/\/en.wikipedia.org\/wiki\/Alexander_Gottlieb_Baumgarten \"Alexander Gottlieb Baumgarten\") [Bell](https:\/\/en.wikipedia.org\/wiki\/Clive_Bell \"Clive Bell\") [Benjamin](https:\/\/en.wikipedia.org\/wiki\/Walter_Benjamin \"Walter Benjamin\") [Burke](https:\/\/en.wikipedia.org\/wiki\/Edmund_Burke \"Edmund Burke\") [Coleridge](https:\/\/en.wikipedia.org\/wiki\/Samuel_Taylor_Coleridge \"Samuel Taylor Coleridge\") [Collingwood](https:\/\/en.wikipedia.org\/wiki\/R._G._Collingwood \"R. G. Collingwood\") [Coomaraswamy](https:\/\/en.wikipedia.org\/wiki\/Ananda_Coomaraswamy \"Ananda Coomaraswamy\") [Danto](https:\/\/en.wikipedia.org\/wiki\/Arthur_Danto \"Arthur Danto\") [Deleuze](https:\/\/en.wikipedia.org\/wiki\/Gilles_Deleuze \"Gilles Deleuze\") [Dewey](https:\/\/en.wikipedia.org\/wiki\/John_Dewey \"John Dewey\") [Flaubert](https:\/\/en.wikipedia.org\/wiki\/Gustave_Flaubert \"Gustave Flaubert\") [Foucault](https:\/\/en.wikipedia.org\/wiki\/Michel_Foucault \"Michel Foucault\") [Fry](https:\/\/en.wikipedia.org\/wiki\/Roger_Fry \"Roger Fry\") [Goethe](https:\/\/en.wikipedia.org\/wiki\/Johann_Wolfgang_von_Goethe \"Johann Wolfgang von Goethe\") [Goodman](https:\/\/en.wikipedia.org\/wiki\/Nelson_Goodman \"Nelson Goodman\") [Gramsci](https:\/\/en.wikipedia.org\/wiki\/Antonio_Gramsci \"Antonio Gramsci\") [Greenberg](https:\/\/en.wikipedia.org\/wiki\/Clement_Greenberg \"Clement Greenberg\") [Hanslick](https:\/\/en.wikipedia.org\/wiki\/Eduard_Hanslick \"Eduard Hanslick\") [Hegel](https:\/\/en.wikipedia.org\/wiki\/Georg_Wilhelm_Friedrich_Hegel \"Georg Wilhelm Friedrich Hegel\") [Heidegger](https:\/\/en.wikipedia.org\/wiki\/Martin_Heidegger \"Martin Heidegger\") [Hume](https:\/\/en.wikipedia.org\/wiki\/David_Hume \"David Hume\") [Hutcheson](https:\/\/en.wikipedia.org\/wiki\/Francis_Hutcheson_\\(philosopher\\) \"Francis Hutcheson (philosopher)\") [Kant](https:\/\/en.wikipedia.org\/wiki\/Immanuel_Kant \"Immanuel Kant\") [Kierkegaard](https:\/\/en.wikipedia.org\/wiki\/S%C3%B8ren_Kierkegaard \"Søren Kierkegaard\") [Klee](https:\/\/en.wikipedia.org\/wiki\/Paul_Klee \"Paul Klee\") [Langer](https:\/\/en.wikipedia.org\/wiki\/Susanne_Langer \"Susanne Langer\") [Lipps](https:\/\/en.wikipedia.org\/wiki\/Theodor_Lipps \"Theodor Lipps\") [Liu](https:\/\/en.wikipedia.org\/wiki\/Liu_Xie \"Liu Xie\") [Lukács](https:\/\/en.wikipedia.org\/wiki\/Gy%C3%B6rgy_Luk%C3%A1cs \"György Lukács\") [Lyotard](https:\/\/en.wikipedia.org\/wiki\/Jean-Fran%C3%A7ois_Lyotard \"Jean-François Lyotard\") [de Man](https:\/\/en.wikipedia.org\/wiki\/Paul_de_Man \"Paul de Man\") [Marcuse](https:\/\/en.wikipedia.org\/wiki\/Herbert_Marcuse \"Herbert Marcuse\") [Maritain](https:\/\/en.wikipedia.org\/wiki\/Jacques_Maritain \"Jacques Maritain\") [Merleau-Ponty](https:\/\/en.wikipedia.org\/wiki\/Maurice_Merleau-Ponty \"Maurice Merleau-Ponty\") [Nietzsche](https:\/\/en.wikipedia.org\/wiki\/Friedrich_Nietzsche \"Friedrich Nietzsche\") [Ortega y Gasset](https:\/\/en.wikipedia.org\/wiki\/Jos%C3%A9_Ortega_y_Gasset \"José Ortega y Gasset\") [Orwell](https:\/\/en.wikipedia.org\/wiki\/George_Orwell \"George Orwell\") [Pater](https:\/\/en.wikipedia.org\/wiki\/Walter_Pater \"Walter Pater\") [Petrarch](https:\/\/en.wikipedia.org\/wiki\/Petrarch \"Petrarch\") [Plato](https:\/\/en.wikipedia.org\/wiki\/Plato \"Plato\") [Pythagoras](https:\/\/en.wikipedia.org\/wiki\/Pythagoras \"Pythagoras\") [Quintilian](https:\/\/en.wikipedia.org\/wiki\/Quintilian \"Quintilian\") [Rancière](https:\/\/en.wikipedia.org\/wiki\/Jacques_Ranci%C3%A8re \"Jacques Rancière\") [Rand](https:\/\/en.wikipedia.org\/wiki\/Ayn_Rand \"Ayn Rand\") [Richards](https:\/\/en.wikipedia.org\/wiki\/I._A._Richards \"I. A. Richards\") [Ruskin](https:\/\/en.wikipedia.org\/wiki\/John_Ruskin \"John Ruskin\") [Said](https:\/\/en.wikipedia.org\/wiki\/Edward_Said \"Edward Said\") [Santayana](https:\/\/en.wikipedia.org\/wiki\/George_Santayana \"George Santayana\") [Schiller](https:\/\/en.wikipedia.org\/wiki\/Friedrich_Schiller \"Friedrich Schiller\") [Schopenhauer](https:\/\/en.wikipedia.org\/wiki\/Arthur_Schopenhauer \"Arthur Schopenhauer\") [Scruton](https:\/\/en.wikipedia.org\/wiki\/Roger_Scruton \"Roger Scruton\") [Sontag](https:\/\/en.wikipedia.org\/wiki\/Susan_Sontag \"Susan Sontag\") [Tagore](https:\/\/en.wikipedia.org\/wiki\/Rabindranath_Tagore \"Rabindranath Tagore\") [Tanizaki](https:\/\/en.wikipedia.org\/wiki\/Jun%27ichir%C5%8D_Tanizaki \"Jun'ichirō Tanizaki\") [Tolkien](https:\/\/en.wikipedia.org\/wiki\/J._R._R._Tolkien \"J. R. R. Tolkien\") [Vasari](https:\/\/en.wikipedia.org\/wiki\/Giorgio_Vasari \"Giorgio Vasari\") [Wilde](https:\/\/en.wikipedia.org\/wiki\/Oscar_Wilde \"Oscar Wilde\") [Winckelmann](https:\/\/en.wikipedia.org\/wiki\/Johann_Joachim_Winckelmann \"Johann Joachim Winckelmann\") [Woolf](https:\/\/en.wikipedia.org\/wiki\/Virginia_Woolf \"Virginia Woolf\") [Zola](https:\/\/en.wikipedia.org\/wiki\/%C3%89mile_Zola \"Émile Zola\") *[more...](https:\/\/en.wikipedia.org\/wiki\/List_of_philosophers_of_art \"List of philosophers of art\")* |\n| Concepts | [Apollonian and Dionysian](https:\/\/en.wikipedia.org\/wiki\/Apollonian_and_Dionysian \"Apollonian and Dionysian\") [Appropriation](https:\/\/en.wikipedia.org\/wiki\/Appropriation_\\(art\\) \"Appropriation (art)\") [Art for art's sake](https:\/\/en.wikipedia.org\/wiki\/Art_for_art%27s_sake \"Art for art's sake\") [Art manifesto](https:\/\/en.wikipedia.org\/wiki\/Art_manifesto \"Art manifesto\") [Artistic merit](https:\/\/en.wikipedia.org\/wiki\/Artistic_merit \"Artistic merit\") [Authenticity](https:\/\/en.wikipedia.org\/wiki\/Authenticity_in_art \"Authenticity in art\") [Avant-garde](https:\/\/en.wikipedia.org\/wiki\/Avant-garde \"Avant-garde\") [Beauty](https:\/\/en.wikipedia.org\/wiki\/Beauty \"Beauty\") [Feminine](https:\/\/en.wikipedia.org\/wiki\/Feminine_beauty_ideal \"Feminine beauty ideal\") [Masculine](https:\/\/en.wikipedia.org\/wiki\/Masculine_beauty_ideal \"Masculine beauty ideal\") [Cool](https:\/\/en.wikipedia.org\/wiki\/Cool_\\(aesthetic\\) \"Cool (aesthetic)\") [Camp](https:\/\/en.wikipedia.org\/wiki\/Camp_\\(style\\) \"Camp (style)\") [Comedy](https:\/\/en.wikipedia.org\/wiki\/Comedy \"Comedy\") [Creativity](https:\/\/en.wikipedia.org\/wiki\/Creativity \"Creativity\") [Cuteness](https:\/\/en.wikipedia.org\/wiki\/Cuteness \"Cuteness\") [Depiction](https:\/\/en.wikipedia.org\/wiki\/Depiction \"Depiction\") [Disgust](https:\/\/en.wikipedia.org\/wiki\/Disgust \"Disgust\") [Ecstasy](https:\/\/en.wikipedia.org\/wiki\/Ecstasy_\\(philosophy\\) \"Ecstasy (philosophy)\") [Elegance](https:\/\/en.wikipedia.org\/wiki\/Elegance \"Elegance\") [Emotions](https:\/\/en.wikipedia.org\/wiki\/Aesthetic_emotions \"Aesthetic emotions\") [Entertainment](https:\/\/en.wikipedia.org\/wiki\/Entertainment \"Entertainment\") [Eroticism](https:\/\/en.wikipedia.org\/wiki\/Eroticism \"Eroticism\") [Exoticism](https:\/\/en.wikipedia.org\/wiki\/Exoticism \"Exoticism\") [Fashion](https:\/\/en.wikipedia.org\/wiki\/Fashion \"Fashion\") [Gaze](https:\/\/en.wikipedia.org\/wiki\/Gaze \"Gaze\") [Harmony](https:\/\/en.wikipedia.org\/wiki\/Harmony \"Harmony\") [Hegemony](https:\/\/en.wikipedia.org\/wiki\/Cultural_hegemony \"Cultural hegemony\") [Imperialism](https:\/\/en.wikipedia.org\/wiki\/Cultural_imperialism \"Cultural imperialism\") [Humour](https:\/\/en.wikipedia.org\/wiki\/Humour \"Humour\") [Iconography](https:\/\/en.wikipedia.org\/wiki\/Iconography \"Iconography\") [Aniconism](https:\/\/en.wikipedia.org\/wiki\/Aniconism \"Aniconism\") [Integrity](https:\/\/en.wikipedia.org\/wiki\/Artistic_integrity \"Artistic integrity\") [Interpretation](https:\/\/en.wikipedia.org\/wiki\/Aesthetic_interpretation \"Aesthetic interpretation\") [Judgment](https:\/\/en.wikipedia.org\/wiki\/Judgment \"Judgment\") *[Kama](https:\/\/en.wikipedia.org\/wiki\/Kama \"Kama\")* [Kitsch](https:\/\/en.wikipedia.org\/wiki\/Kitsch \"Kitsch\") [Life imitating art](https:\/\/en.wikipedia.org\/wiki\/Life_imitating_art \"Life imitating art\") [Magnificence](https:\/\/en.wikipedia.org\/wiki\/Magnificence_\\(history_of_ideas\\) \"Magnificence (history of ideas)\") [Mimesis](https:\/\/en.wikipedia.org\/wiki\/Mimesis \"Mimesis\") [Morality](https:\/\/en.wikipedia.org\/wiki\/Art_and_morality \"Art and morality\") [Perception](https:\/\/en.wikipedia.org\/wiki\/Perception \"Perception\") [Picturesque](https:\/\/en.wikipedia.org\/wiki\/Picturesque \"Picturesque\") [Quality](https:\/\/en.wikipedia.org\/wiki\/Quality_\\(philosophy\\) \"Quality (philosophy)\") *[Rasa](https:\/\/en.wikipedia.org\/wiki\/Rasa_\\(aesthetics\\) \"Rasa (aesthetics)\")* [Recreation](https:\/\/en.wikipedia.org\/wiki\/Recreation \"Recreation\") [Reverence](https:\/\/en.wikipedia.org\/wiki\/Reverence_\\(emotion\\) \"Reverence (emotion)\") [Satire](https:\/\/en.wikipedia.org\/wiki\/Satire \"Satire\") [Style](https:\/\/en.wikipedia.org\/wiki\/Style_\\(visual_arts\\) \"Style (visual arts)\") [Sublime](https:\/\/en.wikipedia.org\/wiki\/Sublime_\\(philosophy\\) \"Sublime (philosophy)\") [Taste](https:\/\/en.wikipedia.org\/wiki\/Taste_\\(sociology\\) \"Taste (sociology)\") [Tragedy](https:\/\/en.wikipedia.org\/wiki\/Tragedy \"Tragedy\") [Work of art](https:\/\/en.wikipedia.org\/wiki\/Work_of_art \"Work of art\") |\n| Works | *[Hippias Major](https:\/\/en.wikipedia.org\/wiki\/Hippias_Major \"Hippias Major\")* (c. 390 BCE) *[Poetics](https:\/\/en.wikipedia.org\/wiki\/Poetics_\\(Aristotle\\) \"Poetics (Aristotle)\")* (c. 335 BCE) *[Ars Poetica](https:\/\/en.wikipedia.org\/wiki\/Ars_Poetica_\\(Horace\\) \"Ars Poetica (Horace)\")* (c. 19 BCE) *[The Literary Mind and the Carving of Dragons](https:\/\/en.wikipedia.org\/wiki\/The_Literary_Mind_and_the_Carving_of_Dragons \"The Literary Mind and the Carving of Dragons\")* (c. 100) *[On the Sublime](https:\/\/en.wikipedia.org\/wiki\/On_the_Sublime \"On the Sublime\")* (c. 500) *[Asrar al-Balagha](https:\/\/en.wikipedia.org\/wiki\/Asrar_al-Balagha \"Asrar al-Balagha\")* (11th century) *[Mumyōzōshi](https:\/\/en.wikipedia.org\/wiki\/Mumy%C5%8Dz%C5%8Dshi \"Mumyōzōshi\")* (13th century) *[On the Sublime and Beautiful](https:\/\/en.wikipedia.org\/wiki\/On_the_Sublime_and_Beautiful \"On the Sublime and Beautiful\")* (1757) *[Lectures on Aesthetics](https:\/\/en.wikipedia.org\/wiki\/Lectures_on_Aesthetics \"Lectures on Aesthetics\")* (1835) \"[The Critic as Artist](https:\/\/en.wikipedia.org\/wiki\/The_Critic_as_Artist \"The Critic as Artist\")\" (1891) \"[A Room of One's Own](https:\/\/en.wikipedia.org\/wiki\/A_Room_of_One%27s_Own \"A Room of One's Own\")\" (1929) *[In Praise of Shadows](https:\/\/en.wikipedia.org\/wiki\/In_Praise_of_Shadows \"In Praise of Shadows\")* (1933) *[Art as Experience](https:\/\/en.wikipedia.org\/wiki\/Art_as_Experience \"Art as Experience\")* (1934) \"[The Work of Art in the Age of Mechanical Reproduction](https:\/\/en.wikipedia.org\/wiki\/The_Work_of_Art_in_the_Age_of_Mechanical_Reproduction \"The Work of Art in the Age of Mechanical Reproduction\")\" (1935) \"[Avant-Garde and Kitsch](https:\/\/en.wikipedia.org\/wiki\/Avant-Garde_and_Kitsch \"Avant-Garde and Kitsch\")\" (1939) *[Prison Notebooks](https:\/\/en.wikipedia.org\/wiki\/Prison_Notebooks \"Prison Notebooks\")* (Gramsci) 1947 *[Critical Essays](https:\/\/en.wikipedia.org\/wiki\/Critical_Essays_\\(Orwell\\) \"Critical Essays (Orwell)\")* (1946) *[Against Interpretation](https:\/\/en.wikipedia.org\/wiki\/Against_Interpretation \"Against Interpretation\")* (1966) \"[Notes on 'Camp'](https:\/\/en.wikipedia.org\/wiki\/Notes_on_%22Camp%22 \"Notes on \\\"Camp\\\"\")\" (1964) *[The Aesthetic Dimension](https:\/\/en.wikipedia.org\/wiki\/The_Aesthetic_Dimension \"The Aesthetic Dimension\")* (1977) *[Orientalism](https:\/\/en.wikipedia.org\/wiki\/Orientalism_\\(book\\) \"Orientalism (book)\")* (1978) *[Why Beauty Matters](https:\/\/en.wikipedia.org\/wiki\/Why_Beauty_Matters \"Why Beauty Matters\")* (2009) |\n| Related | [Applied aesthetics](https:\/\/en.wikipedia.org\/wiki\/Applied_aesthetics \"Applied aesthetics\") [Arts criticism](https:\/\/en.wikipedia.org\/wiki\/Arts_criticism \"Arts criticism\") [Art criticism](https:\/\/en.wikipedia.org\/wiki\/Art_criticism \"Art criticism\") [The arts and politics](https:\/\/en.wikipedia.org\/wiki\/The_arts_and_politics \"The arts and politics\") [Aestheticization of politics](https:\/\/en.wikipedia.org\/wiki\/Aestheticization_of_politics \"Aestheticization of politics\") [Artistic freedom](https:\/\/en.wikipedia.org\/wiki\/Artistic_freedom \"Artistic freedom\") [Axiology](https:\/\/en.wikipedia.org\/wiki\/Axiology \"Axiology\") [Economics of the arts and literature](https:\/\/en.wikipedia.org\/wiki\/Economics_of_the_arts_and_literature \"Economics of the arts and literature\") [Artistic patronage](https:\/\/en.wikipedia.org\/wiki\/Patronage#Arts \"Patronage\") [Cultural economics](https:\/\/en.wikipedia.org\/wiki\/Cultural_economics \"Cultural economics\") [Evolutionary aesthetics](https:\/\/en.wikipedia.org\/wiki\/Evolutionary_aesthetics \"Evolutionary aesthetics\") [Mathematical beauty]() [Neuroesthetics](https:\/\/en.wikipedia.org\/wiki\/Neuroesthetics \"Neuroesthetics\") [Patterns in nature](https:\/\/en.wikipedia.org\/wiki\/Patterns_in_nature \"Patterns in nature\") [Philosophy of design](https:\/\/en.wikipedia.org\/wiki\/Philosophy_of_design \"Philosophy of design\") [Philosophy of film](https:\/\/en.wikipedia.org\/wiki\/Philosophy_of_film \"Philosophy of film\") [Philosophy of language](https:\/\/en.wikipedia.org\/wiki\/Philosophy_of_language \"Philosophy of language\") [Semantics](https:\/\/en.wikipedia.org\/wiki\/Semantics \"Semantics\") [Philosophy of music](https:\/\/en.wikipedia.org\/wiki\/Philosophy_of_music \"Philosophy of music\") [Psychology of art](https:\/\/en.wikipedia.org\/wiki\/Psychology_of_art \"Psychology of art\") [Religious art](https:\/\/en.wikipedia.org\/wiki\/Religious_art \"Religious art\") [Theory of art](https:\/\/en.wikipedia.org\/wiki\/Theory_of_art \"Theory of art\") |\n| [Outline](https:\/\/en.wikipedia.org\/wiki\/Outline_of_aesthetics \"Outline of aesthetics\") [Category](https:\/\/en.wikipedia.org\/wiki\/Category:Aesthetics \"Category:Aesthetics\") ![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/cd\/Socrates.png\/20px-Socrates.png) [Philosophy portal](https:\/\/en.wikipedia.org\/wiki\/Portal:Philosophy \"Portal:Philosophy\") |   |\n\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:Mathematics_and_art \"Template:Mathematics and art\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:Mathematics_and_art \"Template talk:Mathematics and art\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:Mathematics_and_art \"Special:EditPage\/Template:Mathematics and art\")[Mathematics and art](https:\/\/en.wikipedia.org\/wiki\/Mathematics_and_art \"Mathematics and art\") |   |   |\n|---|---|---|\n| Concepts | [Algorithm](https:\/\/en.wikipedia.org\/wiki\/Algorithm \"Algorithm\") [Catenary](https:\/\/en.wikipedia.org\/wiki\/Catenary \"Catenary\") [Fractal](https:\/\/en.wikipedia.org\/wiki\/Fractal \"Fractal\") [Golden ratio](https:\/\/en.wikipedia.org\/wiki\/Golden_ratio \"Golden ratio\") [Hyperboloid structure](https:\/\/en.wikipedia.org\/wiki\/Hyperboloid_structure \"Hyperboloid structure\") [Minimal surface](https:\/\/en.wikipedia.org\/wiki\/Minimal_surface \"Minimal surface\") [Paraboloid](https:\/\/en.wikipedia.org\/wiki\/Paraboloid \"Paraboloid\") [Perspective](https:\/\/en.wikipedia.org\/wiki\/Perspective_\\(graphical\\) \"Perspective (graphical)\") [Camera lucida](https:\/\/en.wikipedia.org\/wiki\/Camera_lucida \"Camera lucida\") [Camera obscura](https:\/\/en.wikipedia.org\/wiki\/Camera_obscura \"Camera obscura\") [Plastic ratio](https:\/\/en.wikipedia.org\/wiki\/Plastic_ratio \"Plastic ratio\") [Projective geometry](https:\/\/en.wikipedia.org\/wiki\/Projective_geometry \"Projective geometry\") Proportion [Architecture](https:\/\/en.wikipedia.org\/wiki\/Proportion_\\(architecture\\) \"Proportion (architecture)\") [Human](https:\/\/en.wikipedia.org\/wiki\/Body_proportions \"Body proportions\") [Symmetry](https:\/\/en.wikipedia.org\/wiki\/Symmetry \"Symmetry\") [Tessellation](https:\/\/en.wikipedia.org\/wiki\/Tessellation \"Tessellation\") [Wallpaper group](https:\/\/en.wikipedia.org\/wiki\/Wallpaper_group \"Wallpaper group\") | [![Fibonacci word: detail of artwork by Samuel Monnier, 2009](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/9\/99\/FWF_Samuel_Monnier_%28vertical_detail%29.jpg\/120px-FWF_Samuel_Monnier_%28vertical_detail%29.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:FWF_Samuel_Monnier_\\(vertical_detail\\).jpg \"Fibonacci word: detail of artwork by Samuel Monnier, 2009\") |\n| Forms | [Algorithmic art](https:\/\/en.wikipedia.org\/wiki\/Algorithmic_art \"Algorithmic art\") [Anamorphic art](https:\/\/en.wikipedia.org\/wiki\/Anamorphosis \"Anamorphosis\") [Architecture](https:\/\/en.wikipedia.org\/wiki\/Mathematics_and_architecture \"Mathematics and architecture\") [Geodesic dome](https:\/\/en.wikipedia.org\/wiki\/Geodesic_dome \"Geodesic dome\") [Pyramid](https:\/\/en.wikipedia.org\/wiki\/Pyramid \"Pyramid\") [Vastu shastra](https:\/\/en.wikipedia.org\/wiki\/Vastu_shastra \"Vastu shastra\") [Computer art](https:\/\/en.wikipedia.org\/wiki\/Computer_art \"Computer art\") [Fiber arts](https:\/\/en.wikipedia.org\/wiki\/Mathematics_and_fiber_arts \"Mathematics and fiber arts\") [4D art](https:\/\/en.wikipedia.org\/wiki\/Fourth_dimension_in_art \"Fourth dimension in art\") [Fractal art](https:\/\/en.wikipedia.org\/wiki\/Fractal_art \"Fractal art\") [Islamic geometric patterns](https:\/\/en.wikipedia.org\/wiki\/Islamic_geometric_patterns \"Islamic geometric patterns\") [Girih](https:\/\/en.wikipedia.org\/wiki\/Girih \"Girih\") [Jali](https:\/\/en.wikipedia.org\/wiki\/Jali \"Jali\") [Muqarnas](https:\/\/en.wikipedia.org\/wiki\/Muqarnas \"Muqarnas\") [Zellij](https:\/\/en.wikipedia.org\/wiki\/Zellij \"Zellij\") [Knotting](https:\/\/en.wikipedia.org\/wiki\/Knot \"Knot\") [Celtic knot](https:\/\/en.wikipedia.org\/wiki\/Celtic_knot \"Celtic knot\") [Croatian interlace](https:\/\/en.wikipedia.org\/wiki\/Croatian_interlace \"Croatian interlace\") [Interlace](https:\/\/en.wikipedia.org\/wiki\/Interlace_\\(art\\) \"Interlace (art)\") [Music](https:\/\/en.wikipedia.org\/wiki\/Music_and_mathematics \"Music and mathematics\") [Origami](https:\/\/en.wikipedia.org\/wiki\/Origami \"Origami\") [Mathematics](https:\/\/en.wikipedia.org\/wiki\/Mathematics_of_paper_folding \"Mathematics of paper folding\") [Sculpture](https:\/\/en.wikipedia.org\/wiki\/Mathematical_sculpture \"Mathematical sculpture\") [String art](https:\/\/en.wikipedia.org\/wiki\/String_art \"String art\") [String figure](https:\/\/en.wikipedia.org\/wiki\/String_figure \"String figure\") [Tiling](https:\/\/en.wikipedia.org\/wiki\/Tessellation \"Tessellation\") |   |\n| Artworks | [List of works designed with the golden ratio](https:\/\/en.wikipedia.org\/wiki\/List_of_works_designed_with_the_golden_ratio \"List of works designed with the golden ratio\") *[Continuum](https:\/\/en.wikipedia.org\/wiki\/Continuum_\\(sculpture\\) \"Continuum (sculpture)\")* *[Mathemalchemy](https:\/\/en.wikipedia.org\/wiki\/Mathemalchemy \"Mathemalchemy\")* *[Mathematica: A World of Numbers... and Beyond](https:\/\/en.wikipedia.org\/wiki\/Mathematica:_A_World_of_Numbers..._and_Beyond \"Mathematica: A World of Numbers... and Beyond\")* *[Octacube](https:\/\/en.wikipedia.org\/wiki\/Octacube_\\(sculpture\\) \"Octacube (sculpture)\")* *[Pi](https:\/\/en.wikipedia.org\/wiki\/Pi_\\(art_project\\) \"Pi (art project)\")* *[Pi in the Sky](https:\/\/en.wikipedia.org\/wiki\/Pi_in_the_Sky \"Pi in the Sky\")* |   |\n| [Buildings](https:\/\/en.wikipedia.org\/wiki\/Mathematics_and_architecture \"Mathematics and architecture\") | [Cathedral of Saint Mary of the Assumption](https:\/\/en.wikipedia.org\/wiki\/Cathedral_of_Saint_Mary_of_the_Assumption_\\(San_Francisco\\) \"Cathedral of Saint Mary of the Assumption (San Francisco)\") [Hagia Sophia](https:\/\/en.wikipedia.org\/wiki\/Hagia_Sophia \"Hagia Sophia\") [Kresge Auditorium](https:\/\/en.wikipedia.org\/wiki\/Kresge_Auditorium \"Kresge Auditorium\") [Pantheon](https:\/\/en.wikipedia.org\/wiki\/Pantheon,_Rome \"Pantheon, Rome\") [Parthenon](https:\/\/en.wikipedia.org\/wiki\/Parthenon \"Parthenon\") [Pyramid of Khufu](https:\/\/en.wikipedia.org\/wiki\/Great_Pyramid_of_Giza \"Great Pyramid of Giza\") [Sagrada Família](https:\/\/en.wikipedia.org\/wiki\/Sagrada_Fam%C3%ADlia \"Sagrada Família\") [Sydney Opera House](https:\/\/en.wikipedia.org\/wiki\/Sydney_Opera_House \"Sydney Opera House\") [Taj Mahal](https:\/\/en.wikipedia.org\/wiki\/Taj_Mahal \"Taj Mahal\") |   |\n| [Artists](https:\/\/en.wikipedia.org\/wiki\/List_of_mathematical_artists \"List of mathematical artists\") |   |   |\n|   |   |   |\n| Renaissance | [Paolo Uccello](https:\/\/en.wikipedia.org\/wiki\/Paolo_Uccello \"Paolo Uccello\") [Piero della Francesca](https:\/\/en.wikipedia.org\/wiki\/Piero_della_Francesca \"Piero della Francesca\") [Leonardo da Vinci](https:\/\/en.wikipedia.org\/wiki\/Leonardo_da_Vinci \"Leonardo da Vinci\") *[Vitruvian Man](https:\/\/en.wikipedia.org\/wiki\/Vitruvian_Man \"Vitruvian Man\")* [Albrecht Dürer](https:\/\/en.wikipedia.org\/wiki\/Albrecht_D%C3%BCrer \"Albrecht Dürer\") [Parmigianino](https:\/\/en.wikipedia.org\/wiki\/Parmigianino \"Parmigianino\") *[Self-portrait in a Convex Mirror](https:\/\/en.wikipedia.org\/wiki\/Self-portrait_in_a_Convex_Mirror \"Self-portrait in a Convex Mirror\")* |   |\n| 19th–20th Century | [William Blake](https:\/\/en.wikipedia.org\/wiki\/William_Blake \"William Blake\") *[The Ancient of Days](https:\/\/en.wikipedia.org\/wiki\/The_Ancient_of_Days \"The Ancient of Days\")* *[Newton](https:\/\/en.wikipedia.org\/wiki\/Newton_\\(Blake\\) \"Newton (Blake)\")* [Jean Metzinger](https:\/\/en.wikipedia.org\/wiki\/Jean_Metzinger \"Jean Metzinger\") *[Danseuse au café](https:\/\/en.wikipedia.org\/wiki\/Dancer_in_a_Caf%C3%A9 \"Dancer in a Café\")* *[L'Oiseau bleu](https:\/\/en.wikipedia.org\/wiki\/L%27Oiseau_bleu_\\(Metzinger\\) \"L'Oiseau bleu (Metzinger)\")* [Giorgio de Chirico](https:\/\/en.wikipedia.org\/wiki\/Giorgio_de_Chirico \"Giorgio de Chirico\") [Man Ray](https:\/\/en.wikipedia.org\/wiki\/Man_Ray \"Man Ray\") [M. C. Escher](https:\/\/en.wikipedia.org\/wiki\/M._C._Escher \"M. C. Escher\") *[Circle Limit III](https:\/\/en.wikipedia.org\/wiki\/Circle_Limit_III \"Circle Limit III\")* *[Print Gallery](https:\/\/en.wikipedia.org\/wiki\/Print_Gallery_\\(M._C._Escher\\) \"Print Gallery (M. C. Escher)\")* *[Relativity](https:\/\/en.wikipedia.org\/wiki\/Relativity_\\(M._C._Escher\\) \"Relativity (M. C. Escher)\")* *[Reptiles](https:\/\/en.wikipedia.org\/wiki\/Reptiles_\\(M._C._Escher\\) \"Reptiles (M. C. Escher)\")* *[Waterfall](https:\/\/en.wikipedia.org\/wiki\/Waterfall_\\(M._C._Escher\\) \"Waterfall (M. C. Escher)\")* [René Magritte](https:\/\/en.wikipedia.org\/wiki\/Ren%C3%A9_Magritte \"René Magritte\") *[La condition humaine](https:\/\/en.wikipedia.org\/wiki\/The_Human_Condition_\\(Magritte\\) \"The Human Condition (Magritte)\")* [Salvador Dalí](https:\/\/en.wikipedia.org\/wiki\/Salvador_Dal%C3%AD \"Salvador Dalí\") *[Crucifixion](https:\/\/en.wikipedia.org\/wiki\/Crucifixion_\\(Corpus_Hypercubus\\) \"Crucifixion (Corpus Hypercubus)\")* *[The Swallow's Tail](https:\/\/en.wikipedia.org\/wiki\/The_Swallow%27s_Tail \"The Swallow's Tail\")* [Crockett Johnson](https:\/\/en.wikipedia.org\/wiki\/Crockett_Johnson \"Crockett Johnson\") |   |\n| Contemporary | [Max Bill](https:\/\/en.wikipedia.org\/wiki\/Max_Bill \"Max Bill\") [Martin](https:\/\/en.wikipedia.org\/wiki\/Martin_Demaine \"Martin Demaine\") and [Erik Demaine](https:\/\/en.wikipedia.org\/wiki\/Erik_Demaine \"Erik Demaine\") [Scott Draves](https:\/\/en.wikipedia.org\/wiki\/Scott_Draves \"Scott Draves\") [Jan Dibbets](https:\/\/en.wikipedia.org\/wiki\/Jan_Dibbets \"Jan Dibbets\") [John Ernest](https:\/\/en.wikipedia.org\/wiki\/John_Ernest \"John Ernest\") [Helaman Ferguson](https:\/\/en.wikipedia.org\/wiki\/Helaman_Ferguson \"Helaman Ferguson\") [Peter Forakis](https:\/\/en.wikipedia.org\/wiki\/Peter_Forakis \"Peter Forakis\") [Susan Goldstine](https:\/\/en.wikipedia.org\/wiki\/Susan_Goldstine \"Susan Goldstine\") [Bathsheba Grossman](https:\/\/en.wikipedia.org\/wiki\/Bathsheba_Grossman \"Bathsheba Grossman\") [George W. Hart](https:\/\/en.wikipedia.org\/wiki\/George_W._Hart \"George W. Hart\") [Desmond Paul Henry](https:\/\/en.wikipedia.org\/wiki\/Desmond_Paul_Henry \"Desmond Paul Henry\") [Anthony Hill](https:\/\/en.wikipedia.org\/wiki\/Anthony_Hill_\\(artist\\) \"Anthony Hill (artist)\") [Charles Jencks](https:\/\/en.wikipedia.org\/wiki\/Charles_Jencks \"Charles Jencks\") *[Garden of Cosmic Speculation](https:\/\/en.wikipedia.org\/wiki\/Garden_of_Cosmic_Speculation \"Garden of Cosmic Speculation\")* [Andy Lomas](https:\/\/en.wikipedia.org\/wiki\/Andy_Lomas \"Andy Lomas\") [Robert Longhurst](https:\/\/en.wikipedia.org\/wiki\/Robert_Longhurst \"Robert Longhurst\") [Jeanette McLeod](https:\/\/en.wikipedia.org\/wiki\/Jeanette_McLeod \"Jeanette McLeod\") [Hamid Naderi Yeganeh](https:\/\/en.wikipedia.org\/wiki\/Hamid_Naderi_Yeganeh \"Hamid Naderi Yeganeh\") [István Orosz](https:\/\/en.wikipedia.org\/wiki\/Istv%C3%A1n_Orosz \"István Orosz\") [Hinke Osinga](https:\/\/en.wikipedia.org\/wiki\/Hinke_Osinga \"Hinke Osinga\") [Antoine Pevsner](https:\/\/en.wikipedia.org\/wiki\/Antoine_Pevsner \"Antoine Pevsner\") [Tony Robbin](https:\/\/en.wikipedia.org\/wiki\/Tony_Robbin \"Tony Robbin\") [Alba Rojo Cama](https:\/\/en.wikipedia.org\/wiki\/Alba_Rojo_Cama \"Alba Rojo Cama\") [Reza Sarhangi](https:\/\/en.wikipedia.org\/wiki\/Reza_Sarhangi \"Reza Sarhangi\") [Oliver Sin](https:\/\/en.wikipedia.org\/wiki\/Oliver_Sin \"Oliver Sin\") [Hiroshi Sugimoto](https:\/\/en.wikipedia.org\/wiki\/Hiroshi_Sugimoto \"Hiroshi Sugimoto\") [Daina Taimiņa](https:\/\/en.wikipedia.org\/wiki\/Daina_Taimi%C5%86a \"Daina Taimiņa\") [Roman Verostko](https:\/\/en.wikipedia.org\/wiki\/Roman_Verostko \"Roman Verostko\") [Margaret Wertheim](https:\/\/en.wikipedia.org\/wiki\/Margaret_Wertheim \"Margaret Wertheim\") |   |\n| Theorists |   |   |\n|   |   |   |\n| Ancient | [Polykleitos](https:\/\/en.wikipedia.org\/wiki\/Polykleitos \"Polykleitos\") *Canon* [Vitruvius](https:\/\/en.wikipedia.org\/wiki\/Vitruvius \"Vitruvius\") *[De architectura](https:\/\/en.wikipedia.org\/wiki\/De_architectura \"De architectura\")* |   |\n| Renaissance | [Filippo Brunelleschi](https:\/\/en.wikipedia.org\/wiki\/Filippo_Brunelleschi \"Filippo Brunelleschi\") [Leon Battista Alberti](https:\/\/en.wikipedia.org\/wiki\/Leon_Battista_Alberti \"Leon Battista Alberti\") *[De pictura](https:\/\/en.wikipedia.org\/wiki\/De_pictura \"De pictura\")* *[De re aedificatoria](https:\/\/en.wikipedia.org\/wiki\/De_re_aedificatoria \"De re aedificatoria\")* [Piero della Francesca](https:\/\/en.wikipedia.org\/wiki\/Piero_della_Francesca \"Piero della Francesca\") *[De prospectiva pingendi](https:\/\/en.wikipedia.org\/wiki\/De_prospectiva_pingendi \"De prospectiva pingendi\")* [Luca Pacioli](https:\/\/en.wikipedia.org\/wiki\/Luca_Pacioli \"Luca Pacioli\") *[De divina proportione](https:\/\/en.wikipedia.org\/wiki\/Divina_proportione \"Divina proportione\")* [Leonardo da Vinci](https:\/\/en.wikipedia.org\/wiki\/Leonardo_da_Vinci \"Leonardo da Vinci\") *[A Treatise on Painting](https:\/\/en.wikipedia.org\/wiki\/A_Treatise_on_Painting \"A Treatise on Painting\")* [Albrecht Dürer](https:\/\/en.wikipedia.org\/wiki\/Albrecht_D%C3%BCrer \"Albrecht Dürer\") *Vier Bücher von Menschlicher Proportion* [Sebastiano Serlio](https:\/\/en.wikipedia.org\/wiki\/Sebastiano_Serlio \"Sebastiano Serlio\") *Regole generali d'architettura* [Andrea Palladio](https:\/\/en.wikipedia.org\/wiki\/Andrea_Palladio \"Andrea Palladio\") *[I quattro libri dell'architettura](https:\/\/en.wikipedia.org\/wiki\/I_quattro_libri_dell%27architettura \"I quattro libri dell'architettura\")* |   |\n| Romantic | [Samuel Colman](https:\/\/en.wikipedia.org\/wiki\/Samuel_Colman \"Samuel Colman\") *Nature's Harmonic Unity* [Frederik Macody Lund](https:\/\/en.wikipedia.org\/wiki\/Frederik_Macody_Lund \"Frederik Macody Lund\") *Ad Quadratum* [Jay Hambidge](https:\/\/en.wikipedia.org\/wiki\/Jay_Hambidge \"Jay Hambidge\") *The Greek Vase* |   |\n| Modern | [Owen Jones](https:\/\/en.wikipedia.org\/wiki\/Owen_Jones_\\(architect\\) \"Owen Jones (architect)\") *[The Grammar of Ornament](https:\/\/en.wikipedia.org\/wiki\/Owen_Jones_\\(architect\\)#The_Grammar_of_Ornament \"Owen Jones (architect)\")* [Ernest Hanbury Hankin](https:\/\/en.wikipedia.org\/wiki\/Ernest_Hanbury_Hankin \"Ernest Hanbury Hankin\") *The Drawing of Geometric Patterns in Saracenic Art* [G. H. Hardy](https:\/\/en.wikipedia.org\/wiki\/G._H._Hardy \"G. H. Hardy\") *[A Mathematician's Apology](https:\/\/en.wikipedia.org\/wiki\/A_Mathematician%27s_Apology \"A Mathematician's Apology\")* [George David Birkhoff](https:\/\/en.wikipedia.org\/wiki\/George_David_Birkhoff \"George David Birkhoff\") *Aesthetic Measure* [Douglas Hofstadter](https:\/\/en.wikipedia.org\/wiki\/Douglas_Hofstadter \"Douglas Hofstadter\") *[Gödel, Escher, Bach](https:\/\/en.wikipedia.org\/wiki\/G%C3%B6del,_Escher,_Bach \"Gödel, Escher, Bach\")* [Nikos Salingaros](https:\/\/en.wikipedia.org\/wiki\/Nikos_Salingaros \"Nikos Salingaros\") *The 'Life' of a Carpet* |   |\n| Publications | *[Journal of Mathematics and the Arts](https:\/\/en.wikipedia.org\/wiki\/Journal_of_Mathematics_and_the_Arts \"Journal of Mathematics and the Arts\")* *[Lumen Naturae](https:\/\/en.wikipedia.org\/wiki\/Lumen_Naturae \"Lumen Naturae\")* *[Making Mathematics with Needlework](https:\/\/en.wikipedia.org\/wiki\/Making_Mathematics_with_Needlework \"Making Mathematics with Needlework\")* *[Rhythm of Structure](https:\/\/en.wikipedia.org\/wiki\/Rhythm_of_Structure \"Rhythm of Structure\")* *[Viewpoints: Mathematical Perspective and Fractal Geometry in Art](https:\/\/en.wikipedia.org\/wiki\/Viewpoints:_Mathematical_Perspective_and_Fractal_Geometry_in_Art \"Viewpoints: Mathematical Perspective and Fractal Geometry in Art\")* |   |\n| Organizations | [Ars Mathematica](https:\/\/en.wikipedia.org\/wiki\/Ars_Mathematica_\\(organization\\) \"Ars Mathematica (organization)\") [The Bridges Organization](https:\/\/en.wikipedia.org\/wiki\/The_Bridges_Organization \"The Bridges Organization\") [European Society for Mathematics and the Arts](https:\/\/en.wikipedia.org\/wiki\/European_Society_for_Mathematics_and_the_Arts \"European Society for Mathematics and the Arts\") [Goudreau Museum of Mathematics in Art and Science](https:\/\/en.wikipedia.org\/wiki\/Goudreau_Museum_of_Mathematics_in_Art_and_Science \"Goudreau Museum of Mathematics in Art and Science\") [Institute For Figuring](https:\/\/en.wikipedia.org\/wiki\/Institute_For_Figuring \"Institute For Figuring\") [Mathemalchemy](https:\/\/en.wikipedia.org\/wiki\/Mathemalchemy \"Mathemalchemy\") [National Museum of Mathematics](https:\/\/en.wikipedia.org\/wiki\/National_Museum_of_Mathematics \"National Museum of Mathematics\") |   |\n| Related | [Droste effect](https:\/\/en.wikipedia.org\/wiki\/Droste_effect \"Droste effect\") [Mathematical beauty]() [Patterns in nature](https:\/\/en.wikipedia.org\/wiki\/Patterns_in_nature \"Patterns in nature\") [Sacred geometry](https:\/\/en.wikipedia.org\/wiki\/Sacred_geometry \"Sacred geometry\") |   |\n| ![](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/9\/96\/Symbol_category_class.svg\/20px-Symbol_category_class.svg.png) **[Category](https:\/\/en.wikipedia.org\/wiki\/Category:Mathematics_and_art \"Category:Mathematics and art\")** |   |   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By using this site, you agree to the [Terms of Use](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Terms_of_Use \"foundation:Special:MyLanguage\/Policy:Terms of Use\") and [Privacy Policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy \"foundation:Special:MyLanguage\/Policy:Privacy policy\"). Wikipedia® is a registered trademark of the [Wikimedia Foundation, Inc.](https:\/\/wikimediafoundation.org\/), a non-profit organization.\n\n- [Privacy policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy)\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About)\n- [Disclaimers](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:General_disclaimer)\n- [Contact Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us)\n- [Legal & safety contacts](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Legal:Wikimedia_Foundation_Legal_and_Safety_Contact_Information)\n- [Code of Conduct](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Universal_Code_of_Conduct)\n- [Developers](https:\/\/developer.wikimedia.org\/)\n- [Statistics](https:\/\/stats.wikimedia.org\/#\/en.wikipedia.org)\n- [Cookie statement](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Cookie_statement)\n- [Mobile view](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&mobileaction=toggle_view_mobile)\n\n- [![Wikimedia Foundation](https:\/\/en.wikipedia.org\/static\/images\/footer\/wikimedia.svg)](https:\/\/www.wikimedia.org\/)\n- [![Powered by MediaWiki](https:\/\/en.wikipedia.org\/w\/resources\/assets\/mediawiki_compact.svg)](https:\/\/www.mediawiki.org\/)\n\nSearch\n\nToggle the table of contents\n\nMathematical beauty\n\n24 languages\n\n[Add topic](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty)","attrs_readable_markdown":"**Mathematical beauty** is a type of [aesthetic](https:\/\/en.wikipedia.org\/wiki\/Aesthetics \"Aesthetics\") value that is experienced in doing or contemplating [mathematics](https:\/\/en.wikipedia.org\/wiki\/Mathematics \"Mathematics\"). The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such as [G.H. Hardy](https:\/\/en.wikipedia.org\/wiki\/G.H._Hardy \"G.H. Hardy\"), have characterized mathematics as an [art](https:\/\/en.wikipedia.org\/wiki\/Art \"Art\") form that seeks beauty.\n\nBeauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understanding [beauty](https:\/\/en.wikipedia.org\/wiki\/Beauty \"Beauty\") in general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstract *ideas* which can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than (for example) listening to music.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-1) Furthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors identify mathematical elegance with mathematical beauty; others distinguish elegance as a separate aesthetic value, or as being, for instance, limited to the form mathematical exposition.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-rota-phenomenology-2): 177–178  Beauty itself is often linked to, or thought to be dependent on, the abstractness, purity, simplicity, depth or order of mathematics.\n\n## Examples of beautiful mathematics\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=1 \"Edit section: Examples of beautiful mathematics\")\\]\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/3b\/EulerIdentity2.svg\/250px-EulerIdentity2.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:EulerIdentity2.svg)\n\nStarting at *e*0 = 1, travelling at the velocity *i* relative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an [Argand diagram](https:\/\/en.wikipedia.org\/wiki\/Argand_diagram \"Argand diagram\").)\n\n[Euler's identity](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity \"Euler's identity\") is often given as an example of a beautiful result:[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-3): 1–3 [\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-cain-form-4): 835–836 [\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-5)\n\n![{\\\\displaystyle \\\\displaystyle \\\\mathrm {e} ^{\\\\mathrm {i} \\\\pi }+1=0\\\\,.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fd5213bb3f233806ff3212142214fee997434a15)\n\nThis expression ties together arguably the five most important [mathematical constants](https:\/\/en.wikipedia.org\/wiki\/Mathematical_constant \"Mathematical constant\") (*e*, *i*, π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\"), which the physicist [Richard Feynman](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\") called \"our jewel\" and \"the most remarkable formula in mathematics\".[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-6)\n\nAnother example is [Fermat's theorem on sums of two squares](https:\/\/en.wikipedia.org\/wiki\/Fermat%27s_theorem_on_sums_of_two_squares \"Fermat's theorem on sums of two squares\"), which says that any [prime number](https:\/\/en.wikipedia.org\/wiki\/Prime_number \"Prime number\") such that ![{\\\\displaystyle p\\\\equiv 1{\\\\pmod {4}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/125d70c2c8848e96954eddfc7f9283ec4f676ed7) can be written as a sum of two square numbers (for example, ![{\\\\displaystyle 5=1^{2}+2^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c4b288581682d73e457bc6829e15be203280852a), ![{\\\\displaystyle 13=2^{2}+3^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8be5ea38d26942831bf630c465218ef5e3189181), ![{\\\\displaystyle 37=1^{2}+6^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0f81cd69710c7c05bdc903449dc7e834540c797a)), which both [G.H. Hardy](https:\/\/en.wikipedia.org\/wiki\/G.H._Hardy \"G.H. Hardy\")[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-hardy-apology-7): §12  and [E.T. Bell](https:\/\/en.wikipedia.org\/wiki\/E.T._Bell \"E.T. Bell\")[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-8): ch.4  thought was a beautiful result.\n\nIn a survey in which mathematicians were asked to evaluate 24 theorems for their beauty, the top-rated three theorems were:[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-9) Euler's equation; [Euler's polyhedron formula](https:\/\/en.wikipedia.org\/wiki\/Euler_characteristic \"Euler characteristic\"), which asserts that for a polyhedron with *V* vertices, *E* edges, and *F* faces, ![{\\\\displaystyle V-E+F=2}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/759601e482258ff7a359a7db381abf60372c5b06); and [Euclid's theorem](https:\/\/en.wikipedia.org\/wiki\/Euclid%27s_theorem \"Euclid's theorem\") that there are infinitely many prime numbers, which was also given by [Hardy](https:\/\/en.wikipedia.org\/wiki\/G.H._Hardy \"G.H. Hardy\") as an example of a beautiful theorem.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-hardy-apology-7): §12\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/9\/9a\/Pythagorean_proof_%281%29.svg\/250px-Pythagorean_proof_%281%29.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Pythagorean_proof_\\(1\\).svg)\n\nAn example of \"beauty in method\"—a simple and elegant visual descriptor of the [Pythagorean theorem](https:\/\/en.wikipedia.org\/wiki\/Pythagorean_theorem \"Pythagorean theorem\").\n\n[Cantor's diagonal argument](https:\/\/en.wikipedia.org\/wiki\/Cantor%27s_diagonal_argument \"Cantor's diagonal argument\"), which establishes that there are [infinite sets](https:\/\/en.wikipedia.org\/wiki\/Infinite_set \"Infinite set\") which cannot be put into [one-to-one correspondence](https:\/\/en.wikipedia.org\/wiki\/Bijection \"Bijection\") with the infinite set of [natural numbers](https:\/\/en.wikipedia.org\/wiki\/Natural_number \"Natural number\"), has been cited by both mathematicians[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-10) and philosophers[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-11) as an example of a beautiful proof.\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/e8\/Proofwithoutwords.svg\/250px-Proofwithoutwords.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Proofwithoutwords.svg)\n\nA proof without words for the sum of odd numbers theorem\n\nVisual proofs, such as the illustrated proof of the [Pythagorean theorem](https:\/\/en.wikipedia.org\/wiki\/Pythagorean_theorem \"Pythagorean theorem\"), and other [proofs without words](https:\/\/en.wikipedia.org\/wiki\/Proofs_without_words \"Proofs without words\") generally, such as the shown proof that the sum of all positive [odd numbers](https:\/\/en.wikipedia.org\/wiki\/Parity_\\(mathematics\\) \"Parity (mathematics)\") up to 2*n* − 1 is a [perfect square](https:\/\/en.wikipedia.org\/wiki\/Square_number \"Square number\"), have been thought beautiful.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-12)\n\nThe mathematician [Paul Erdős](https:\/\/en.wikipedia.org\/wiki\/Paul_Erd%C5%91s \"Paul Erdős\") spoke of *The Book*, an imaginary infinite book in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would proclaim it \"straight from The Book!\".[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-13): 35  His rhetorical device inspired the creation of [*Proofs from THE BOOK*](https:\/\/en.wikipedia.org\/wiki\/Proofs_from_THE_BOOK \"Proofs from THE BOOK\"), a collection of such proofs, including many suggested by Erdős himself.[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-14): v\n\nIn [Plato](https:\/\/en.wikipedia.org\/wiki\/Plato \"Plato\")'s [*Timaeus*](https:\/\/en.wikipedia.org\/wiki\/Timaeus_\\(dialogue\\) \"Timaeus (dialogue)\"), the five [regular convex polyhedra](https:\/\/en.wikipedia.org\/wiki\/Platonic_solids \"Platonic solids\"), called the *Platonic solids* for their role in this dialogue, are called the \"most beautiful\" (\"κάλλιστα\") bodies.[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-plato-timaeus-15): 53e  In the *Timaeus*, they are described as having been used by the [demiurge](https:\/\/en.wikipedia.org\/wiki\/Demiurge \"Demiurge\"), or creator-craftsman who builds the cosmos, for the four [classical elements](https:\/\/en.wikipedia.org\/wiki\/Classical_elements \"Classical elements\") plus the heavens, because of their beauty.[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-plato-timaeus-15): 54e–55e\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/19\/Kepler-solar-system-1.png\/250px-Kepler-solar-system-1.png)](https:\/\/en.wikipedia.org\/wiki\/File:Kepler-solar-system-1.png)\n\nKepler's Platonic solid model of the solar system\n\nIn his 1596 book [*Mysterium Cosmographicum*](https:\/\/en.wikipedia.org\/wiki\/Mysterium_Cosmographicum \"Mysterium Cosmographicum\"), [Johannes Kepler](https:\/\/en.wikipedia.org\/wiki\/Johannes_Kepler \"Johannes Kepler\") argued that the orbits of the then-known planets in the [Solar System](https:\/\/en.wikipedia.org\/wiki\/Solar_System \"Solar System\") have been arranged by [God](https:\/\/en.wikipedia.org\/wiki\/God \"God\") to correspond to a concentric arrangement of the five [Platonic solids](https:\/\/en.wikipedia.org\/wiki\/Platonic_solid \"Platonic solid\"), each orbit lying on the [circumsphere](https:\/\/en.wikipedia.org\/wiki\/Circumscribed_sphere \"Circumscribed sphere\") of one [polyhedron](https:\/\/en.wikipedia.org\/wiki\/Polyhedron \"Polyhedron\") and the [insphere](https:\/\/en.wikipedia.org\/wiki\/Inscribed_sphere \"Inscribed sphere\") of another. For Kepler, God had wanted to shape the universe according to the five regular solids because of their beauty, and this explained *why* there were six planets (according to the knowledge of the time).[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-16): ch.3 [\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-cain-form-4): 280--285\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/14\/E8Petrie.svg\/250px-E8Petrie.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:E8Petrie.svg)\n\nPetrie projection of ![{\\\\displaystyle E\\_{8}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/48479e96d90b4cfabc7784106cc3cfff907dda34)\n\nA more modern example is the exceptional [simple Lie group](https:\/\/en.wikipedia.org\/wiki\/Simple_Lie_group \"Simple Lie group\") ![{\\\\displaystyle E\\_{8}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/48479e96d90b4cfabc7784106cc3cfff907dda34), which has been called \"perhaps the most beautiful structure in all of mathematics\".[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-17)\n\n### Scientific theories\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=5 \"Edit section: Scientific theories\")\\]\n\nThe mathematical statements of scientific theories, especially in physics, are sometimes considered to be mathematically beautiful. For example, [Roger Penrose](https:\/\/en.wikipedia.org\/wiki\/Roger_Penrose \"Roger Penrose\") thought there was a \"special beauty\" in [Maxwell's equations](https:\/\/en.wikipedia.org\/wiki\/Maxwell%27s_equations \"Maxwell's equations\") of electromagnetism:[\\[18\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-18): 268  ![{\\\\displaystyle {\\\\begin{aligned}\\\\nabla \\\\cdot \\\\mathbf {E} \\\\,\\\\,\\\\,&={\\\\frac {\\\\rho }{\\\\varepsilon \\_{0}}}\\\\\\\\\\\\nabla \\\\cdot \\\\mathbf {B} \\\\,\\\\,\\\\,&=0\\\\\\\\\\\\nabla \\\\times \\\\mathbf {E} &=-{\\\\frac {\\\\partial \\\\mathbf {B} }{\\\\partial t}}\\\\\\\\\\\\nabla \\\\times \\\\mathbf {B} &=\\\\mu \\_{0}\\\\left(\\\\mathbf {J} +\\\\varepsilon \\_{0}{\\\\frac {\\\\partial \\\\mathbf {E} }{\\\\partial t}}\\\\right)\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1740d383a275f64105f457e209ff5c66eeeb21)\n\nEinstein's theory of [general relativity](https:\/\/en.wikipedia.org\/wiki\/General_relativity \"General relativity\") has been characterized as a work of art, and, among other aesthetic praise,[\\[19\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-19): 148  was described by [Paul Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\") as having \"great mathematical beauty\"[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-20): 123  and by Penrose as having \"supreme mathematical beauty\".[\\[21\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-21): 1038\n\n(There can be more to the beauty of a scientific theory than just its mathematical statement. For example, whether a theory is visualizable or deterministic might have an influence on whether it is seen as beautiful.[\\[22\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-22): 53 [\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-cain-form-4): 873–877 )\n\n## Properties of beautiful mathematics\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=6 \"Edit section: Properties of beautiful mathematics\")\\]\n\nMany mathematicians and philosophers who have written about mathematical beauty have tried to identify properties or criteria that are conducive to the perception of beauty in a piece of mathematics. It is debated whether beauty can be clarified or explained by such properties: [Paul Erdős](https:\/\/en.wikipedia.org\/wiki\/Paul_Erd%C5%91s \"Paul Erdős\") thought that it was no more possible to convince someone of the beauty of a piece of mathematics than to convince them of the beauty of [Beethoven's Ninth Symphony](https:\/\/en.wikipedia.org\/wiki\/Symphony_No._9_\\(Beethoven\\) \"Symphony No. 9 (Beethoven)\"), if they couldn't see it for themselves.[\\[23\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-23)\n\nIn his 1940 essay *[A Mathematician's Apology](https:\/\/en.wikipedia.org\/wiki\/A_Mathematician%27s_Apology \"A Mathematician's Apology\")*, [G. H. Hardy](https:\/\/en.wikipedia.org\/wiki\/G._H._Hardy \"G. H. Hardy\") said that a beautiful result, including its proof, possesses three \"purely aesthetic qualities\", namely \"inevitability\", \"unexpectedness\", and \"economy\". He particularly excluded enumeration of cases as \"one of the duller forms of mathematical argument\".[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-hardy-apology-7): §18\n\nIn 1997, [Gian-Carlo Rota](https:\/\/en.wikipedia.org\/wiki\/Gian-Carlo_Rota \"Gian-Carlo Rota\") disagreed with unexpectedness as a sufficient condition for beauty and proposed a counterexample:\n\n> A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago \\[from 1977\\] the proof of the existence of [non-equivalent differentiable structures](https:\/\/en.wikipedia.org\/wiki\/Exotic_sphere \"Exotic sphere\") on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or now.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-rota-phenomenology-2): 172\n\nIn contrast, Monastyrsky wrote in 2001:\n\n> It is very difficult to find an analogous invention in the past to [Milnor](https:\/\/en.wikipedia.org\/wiki\/Milnor \"Milnor\")'s beautiful construction of the different differential structures on the seven-dimensional sphere... The original proof of Milnor was not very constructive, but later E. Briscorn showed that these differential structures can be described in an extremely explicit and beautiful form.[\\[24\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-24): 44\n\nThis disagreement illustrates both the subjective nature of mathematical beauty, [like other forms of beauty in general](https:\/\/en.wikipedia.org\/wiki\/Beauty \"Beauty\"), and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them.\n\nBesides Hardy's properties of \"unexpectedness\", \"inevitability\", \"economy\", which he applied to proofs as well as results, mathematicians have customarily thought beautiful proofs that are short and simple.[\\[25\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-25): 22\n\nIn the search for an elegant proof, mathematicians often look for different independent ways to prove a result—as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly the [Pythagorean theorem](https:\/\/en.wikipedia.org\/wiki\/Pythagorean_theorem \"Pythagorean theorem\"), with hundreds of proofs having been published.[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-26) Another theorem that has been proved in many different ways is the theorem of [quadratic reciprocity](https:\/\/en.wikipedia.org\/wiki\/Quadratic_reciprocity \"Quadratic reciprocity\"). In fact, [Carl Friedrich Gauss](https:\/\/en.wikipedia.org\/wiki\/Carl_Friedrich_Gauss \"Carl Friedrich Gauss\") alone had eight different proofs of this theorem, six of which he published.[\\[27\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-27)\n\nIn contrast, results that are logically correct but involve laborious calculations or consideration of many cases, are not usually considered beautiful, and may be even referred to as *ugly* or *clumsy*. For example, [Kenneth Appel](https:\/\/en.wikipedia.org\/wiki\/Kenneth_Appel \"Kenneth Appel\") and [Wolfgang Haken](https:\/\/en.wikipedia.org\/wiki\/Wolfgang_Haken \"Wolfgang Haken\")'s proof of the [four color theorem](https:\/\/en.wikipedia.org\/wiki\/Four_color_theorem \"Four color theorem\") made use of computer checking of over a thousand cases. [Philip J. Davis](https:\/\/en.wikipedia.org\/wiki\/Philip_J._Davis \"Philip J. Davis\") and [Reuben Hersh](https:\/\/en.wikipedia.org\/wiki\/Reuben_Hersh \"Reuben Hersh\") said that when they first heard that about the proof, they hoped it contained a new insight \"whose beauty would transform my day\", and were disheartened when informed the proof was by case enumeration and computer verification.[\\[28\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-davis-experience-28): 384  [Paul Erdős](https:\/\/en.wikipedia.org\/wiki\/Paul_Erd%C5%91s \"Paul Erdős\") said it was \"not beautiful\" because it gave no insight into why the theorem was true.[\\[29\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-29): 44\n\n## Philosophical analysis\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=9 \"Edit section: Philosophical analysis\")\\]\n\n[Aristotle](https:\/\/en.wikipedia.org\/wiki\/Aristotle \"Aristotle\") thought that beauty was found especially in mathematics, writing in the [*Metaphysics*](https:\/\/en.wikipedia.org\/wiki\/Metaphysics_\\(Aristotle\\) \"Metaphysics (Aristotle)\") that\n\n> those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a very great deal about them; for if they do not expressly mention them, but prove attributes which are their results or their formulae, it is not true to say that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.[\\[30\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-30): 1078a32–35\n\nThe logician and philosopher [Bertrand Russell](https:\/\/en.wikipedia.org\/wiki\/Bertrand_Russell \"Bertrand Russell\") made a now-famous statement characterizing mathematical beauty in terms of purity and austerity:\n\n> Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.[\\[31\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-31)\n\nIn the twentieth century, some philosophers questioned whether there was genuinely beauty in mathematics. The philosopher of science [Rom Harré](https:\/\/en.wikipedia.org\/wiki\/Rom_Harr%C3%A9 \"Rom Harré\") argued that there were no true aesthetic appraisals of mathematics, but only *quasi-aesthetic* appraisals. Any mathematical success described by an aesthetic term was a second-order success besides understanding and correctness. In contrast, aesthetic appraisal of a work of art was first-order. Harré considered this to be the difference between a quasi-aesthetic and a genuinely aesthetic appraisal.[\\[32\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-32)\n\n[Nick Zangwill](https:\/\/en.wikipedia.org\/wiki\/Nick_Zangwill \"Nick Zangwill\") thought that there were no true aesthetic experiences of mathematics and that a proofs or theories could only be *metaphorically* beautiful. His argument had two bases. First, he thought that aesthetic properties depended on sensory properties, and so abstract entities could not have aesthetic properties. Second, he thought that proofs, theorems, theories, and so on had purposes such as demonstrating correctness or granting understanding, and that any praise of them reflected *only* how well they achieved their purpose.[\\[33\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-33): 140–142\n\n## Scientific analysis\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=10 \"Edit section: Scientific analysis\")\\]\n\n### Information-theory model\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=11 \"Edit section: Information-theory model\")\\]\n\nIn the 1970s, [Abraham Moles](https:\/\/en.wikipedia.org\/wiki\/Abraham_Moles \"Abraham Moles\") and [Frieder Nake](https:\/\/en.wikipedia.org\/wiki\/Frieder_Nake \"Frieder Nake\") analyzed links between beauty, [information processing](https:\/\/en.wikipedia.org\/wiki\/Data_processing \"Data processing\"), and [information theory](https:\/\/en.wikipedia.org\/wiki\/Information_theory \"Information theory\").[\\[34\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-34)[\\[35\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-35) In the 1990s, [Jürgen Schmidhuber](https:\/\/en.wikipedia.org\/wiki\/J%C3%BCrgen_Schmidhuber \"Jürgen Schmidhuber\") formulated a mathematical theory of observer-dependent subjective beauty based on [algorithmic information theory](https:\/\/en.wikipedia.org\/wiki\/Algorithmic_information_theory \"Algorithmic information theory\"): the most beautiful objects among subjectively comparable objects have short [algorithmic](https:\/\/en.wikipedia.org\/wiki\/Algorithmic_information_theory \"Algorithmic information theory\") descriptions (i.e., [Kolmogorov complexity](https:\/\/en.wikipedia.org\/wiki\/Kolmogorov_complexity \"Kolmogorov complexity\")) relative to what the observer already knows.[\\[36\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-36)[\\[37\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-37)[\\[38\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-38) Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the [first derivative](https:\/\/en.wikipedia.org\/wiki\/First_derivative \"First derivative\") of subjectively perceived beauty: the observer continually tries to improve the [predictability](https:\/\/en.wikipedia.org\/wiki\/Predictability \"Predictability\") and [compressibility](https:\/\/en.wikipedia.org\/wiki\/Data_compression \"Data compression\") of the observations by discovering regularities such as repetitions and [symmetries](https:\/\/en.wikipedia.org\/wiki\/Symmetries \"Symmetries\") and [fractal](https:\/\/en.wikipedia.org\/wiki\/Fractal \"Fractal\") [self-similarity](https:\/\/en.wikipedia.org\/wiki\/Self-similarity \"Self-similarity\"). Whenever the observer's learning process (possibly a predictive artificial [neural network](https:\/\/en.wikipedia.org\/wiki\/Neural_network \"Neural network\")) leads to improved data compression such that the observation sequence can be described by fewer [bits](https:\/\/en.wikipedia.org\/wiki\/Bit \"Bit\") than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.[\\[39\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-39)[\\[40\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-40)\n\nBrain imaging experiments conducted by [Semir Zeki](https:\/\/en.wikipedia.org\/wiki\/Semir_Zeki \"Semir Zeki\"), [Michael Atiyah](https:\/\/en.wikipedia.org\/wiki\/Michael_Atiyah \"Michael Atiyah\") and collaborators show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial [orbito-frontal cortex](https:\/\/en.wikipedia.org\/wiki\/Orbitofrontal_cortex \"Orbitofrontal cortex\") (mOFC) of the brain and that this activity is parametrically related to the declared intensity of beauty. The location of the activity is similar to the location of the activity that correlates with the experience of beauty from other sources, such as visual art or music.[\\[41\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-41) Moreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers.[\\[42\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-42)\n\n## Mathematical beauty and the arts\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Mathematical_beauty&action=edit&section=13 \"Edit section: Mathematical beauty and the arts\")\\]\n\nExamples of the use of mathematics in music include the [stochastic music](https:\/\/en.wikipedia.org\/wiki\/Stochastic_music \"Stochastic music\") of [Iannis Xenakis](https:\/\/en.wikipedia.org\/wiki\/Iannis_Xenakis \"Iannis Xenakis\"),[\\[43\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-43) the [Fibonacci sequence](https:\/\/en.wikipedia.org\/wiki\/Fibonacci_number \"Fibonacci number\") in [Tool](https:\/\/en.wikipedia.org\/wiki\/Tool_\\(band\\) \"Tool (band)\")'s [Lateralus](https:\/\/en.wikipedia.org\/wiki\/Lateralus_\\(song\\) \"Lateralus (song)\"),[\\[44\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-44) and the [Metric modulation](https:\/\/en.wikipedia.org\/wiki\/Metric_modulation \"Metric modulation\") of [Elliott Carter](https:\/\/en.wikipedia.org\/wiki\/Elliott_Carter \"Elliott Carter\").[\\[45\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-45)\n\nOther instances include the counterpoint of [Johann Sebastian Bach](https:\/\/en.wikipedia.org\/wiki\/Johann_Sebastian_Bach \"Johann Sebastian Bach\"), [polyrhythmic](https:\/\/en.wikipedia.org\/wiki\/Polyrhythm \"Polyrhythm\") structures (as in [Igor Stravinsky](https:\/\/en.wikipedia.org\/wiki\/Igor_Stravinsky \"Igor Stravinsky\")'s *[The Rite of Spring](https:\/\/en.wikipedia.org\/wiki\/The_Rite_of_Spring \"The Rite of Spring\")*), [permutation](https:\/\/en.wikipedia.org\/wiki\/Permutation \"Permutation\") theory in [serialism](https:\/\/en.wikipedia.org\/wiki\/Serialism \"Serialism\") beginning with [Arnold Schoenberg](https:\/\/en.wikipedia.org\/wiki\/Arnold_Schoenberg \"Arnold Schoenberg\"), the application of Shepard tones in [Karlheinz Stockhausen](https:\/\/en.wikipedia.org\/wiki\/Karlheinz_Stockhausen \"Karlheinz Stockhausen\")'s *[Hymnen](https:\/\/en.wikipedia.org\/wiki\/Hymnen \"Hymnen\")* and the application of [Group theory](https:\/\/en.wikipedia.org\/wiki\/Group_theory \"Group theory\") to transformations in music in the theoretical writings of [David Lewin](https:\/\/en.wikipedia.org\/wiki\/David_Lewin \"David Lewin\").\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/38\/Della_Pittura_Alberti_perspective_pillars_on_grid.jpg\/250px-Della_Pittura_Alberti_perspective_pillars_on_grid.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Della_Pittura_Alberti_perspective_pillars_on_grid.jpg)\n\nDiagram from [Leon Battista Alberti](https:\/\/en.wikipedia.org\/wiki\/Leon_Battista_Alberti \"Leon Battista Alberti\")'s 1435 *[Della Pittura](https:\/\/en.wikipedia.org\/wiki\/Della_Pittura \"Della Pittura\")*, with pillars in [perspective](https:\/\/en.wikipedia.org\/wiki\/Perspective_\\(graphical\\) \"Perspective (graphical)\") on a grid\n\nExamples of the use of mathematics in the visual arts include applications of [chaos theory](https:\/\/en.wikipedia.org\/wiki\/Chaos_theory \"Chaos theory\") and [fractal](https:\/\/en.wikipedia.org\/wiki\/Fractal \"Fractal\") geometry to [computer-generated art](https:\/\/en.wikipedia.org\/wiki\/Digital_art \"Digital art\"), symmetry studies of [Leonardo da Vinci](https:\/\/en.wikipedia.org\/wiki\/Leonardo_da_Vinci \"Leonardo da Vinci\"), [projective geometries](https:\/\/en.wikipedia.org\/wiki\/Projective_geometry \"Projective geometry\") in development of the [perspective](https:\/\/en.wikipedia.org\/wiki\/Perspective_\\(graphical\\) \"Perspective (graphical)\") theory of [Renaissance](https:\/\/en.wikipedia.org\/wiki\/Renaissance \"Renaissance\") art, [grids](https:\/\/en.wikipedia.org\/wiki\/Grid_\\(page_layout\\) \"Grid (page layout)\") in [Op art](https:\/\/en.wikipedia.org\/wiki\/Op_art \"Op art\"), optical geometry in the [camera obscura](https:\/\/en.wikipedia.org\/wiki\/Camera_obscura \"Camera obscura\") of [Giambattista della Porta](https:\/\/en.wikipedia.org\/wiki\/Giambattista_della_Porta \"Giambattista della Porta\"), and multiple perspective in analytic [cubism](https:\/\/en.wikipedia.org\/wiki\/Cubism \"Cubism\") and [futurism](https:\/\/en.wikipedia.org\/wiki\/Futurism \"Futurism\").\n\n[Sacred geometry](https:\/\/en.wikipedia.org\/wiki\/Sacred_geometry \"Sacred geometry\") is a field of its own, giving rise to countless art forms including some of the best known mystic symbols and religious motifs, and has a particularly rich history in [Islamic architecture](https:\/\/en.wikipedia.org\/wiki\/Islamic_architecture \"Islamic architecture\"). It also provides a means of meditation and comtemplation, for example the study of the [Kaballah](https:\/\/en.wikipedia.org\/wiki\/Kaballah \"Kaballah\") [Sefirot](https:\/\/en.wikipedia.org\/wiki\/Sefirot \"Sefirot\") (Tree Of Life) and [Metatron's Cube](https:\/\/en.wikipedia.org\/wiki\/Metatron%27s_Cube \"Metatron's Cube\"); and also the act of drawing itself.\n\nThe Dutch graphic designer [M. C. Escher](https:\/\/en.wikipedia.org\/wiki\/M._C._Escher \"M. C. Escher\") created mathematically inspired [woodcuts](https:\/\/en.wikipedia.org\/wiki\/Woodcut \"Woodcut\"), [lithographs](https:\/\/en.wikipedia.org\/wiki\/Lithograph \"Lithograph\"), and [mezzotints](https:\/\/en.wikipedia.org\/wiki\/Mezzotint \"Mezzotint\"). These feature impossible constructions, explorations of [infinity](https:\/\/en.wikipedia.org\/wiki\/Infinity \"Infinity\"), architecture, visual [paradoxes](https:\/\/en.wikipedia.org\/wiki\/Paradox \"Paradox\") and [tessellations](https:\/\/en.wikipedia.org\/wiki\/Tessellation \"Tessellation\").\n\nSome painters and sculptors create work distorted with the mathematical principles of [anamorphosis](https:\/\/en.wikipedia.org\/wiki\/Anamorphosis \"Anamorphosis\"), including South African sculptor [Jonty Hurwitz](https:\/\/en.wikipedia.org\/wiki\/Jonty_Hurwitz \"Jonty Hurwitz\").\n\n[Origami](https:\/\/en.wikipedia.org\/wiki\/Origami \"Origami\"), the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the [mathematics of paper folding](https:\/\/en.wikipedia.org\/wiki\/Mathematics_of_paper_folding \"Mathematics of paper folding\") by observing the [crease pattern](https:\/\/en.wikipedia.org\/wiki\/Crease_pattern \"Crease pattern\") on unfolded origami pieces.[\\[46\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-46)\n\nBritish constructionist artist [John Ernest](https:\/\/en.wikipedia.org\/wiki\/John_Ernest \"John Ernest\") created reliefs and paintings inspired by group theory.[\\[47\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-47) A number of other British artists of the constructionist and systems schools of thought also draw on mathematics models and structures as a source of inspiration, including [Anthony Hill](https:\/\/en.wikipedia.org\/wiki\/Anthony_Hill_\\(artist\\) \"Anthony Hill (artist)\") and [Peter Lowe](https:\/\/en.wikipedia.org\/wiki\/Peter_Lowe_\\(artist\\) \"Peter Lowe (artist)\").[\\[48\\]](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_note-48) Computer-generated art is based on mathematical [algorithms](https:\/\/en.wikipedia.org\/wiki\/Algorithm \"Algorithm\").\n\n- [Argument from beauty](https:\/\/en.wikipedia.org\/wiki\/Argument_from_beauty \"Argument from beauty\")\n- [Fluency heuristic](https:\/\/en.wikipedia.org\/wiki\/Fluency_heuristic \"Fluency heuristic\")\n- [Golden ratio](https:\/\/en.wikipedia.org\/wiki\/Golden_ratio \"Golden ratio\")\n- [Neuroesthetics](https:\/\/en.wikipedia.org\/wiki\/Neuroesthetics \"Neuroesthetics\")\n- [Philosophy of mathematics](https:\/\/en.wikipedia.org\/wiki\/Philosophy_of_mathematics \"Philosophy of mathematics\")\n- [Processing fluency theory of aesthetic pleasure](https:\/\/en.wikipedia.org\/wiki\/Processing_fluency_theory_of_aesthetic_pleasure \"Processing fluency theory of aesthetic pleasure\")\n- [Pythagoreanism](https:\/\/en.wikipedia.org\/wiki\/Pythagoreanism \"Pythagoreanism\")\n- [Sense of wonder](https:\/\/en.wikipedia.org\/wiki\/Sense_of_wonder \"Sense of wonder\")\n- [Theory of everything](https:\/\/en.wikipedia.org\/wiki\/Theory_of_everything \"Theory of everything\")\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-1)**\n   Phillips, George (2005). [\"Preface\"](https:\/\/books.google.com\/books?id=psFwdN6V6icC&q=there+is+nothing+in+the+world+of+mathematics+that+corresponds+to+an+audience+in+a+concert+hall,+where+the+passive+listen+to+the+active.+Happily,+mathematicians+are+all+doers,+not+spectators.&pg=PR7). *Mathematics Is Not a Spectator Sport*. [Springer Science+Business Media](https:\/\/en.wikipedia.org\/wiki\/Springer_Science%2BBusiness_Media \"Springer Science+Business Media\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0-387-25528-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-387-25528-1 \"Special:BookSources\/0-387-25528-1\")\n   \n   . Retrieved 2008-08-22. \"\"...there is nothing in the world of mathematics that corresponds to an audience in a concert hall, where the passive listen to the active. Happily, mathematicians are all *doers*, not spectators.\"\n2. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-rota-phenomenology_2-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-rota-phenomenology_2-1)\n   Rota, Gian-Carlo (May 1997). \"The Phenomenology of Mathematical Beauty\". *Synthese*. **111** (2): 171–182\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1023\/A:1004930722234](https:\/\/doi.org\/10.1023%2FA%3A1004930722234).\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-3)**\n   Wilson, Robin (2018). *Euler's Pioneering Equation: The most beautiful theorem in mathematics*. Oxford University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-19-879492-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-19-879492-9 \"Special:BookSources\/978-0-19-879492-9\")\n   \n   .\n4. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-cain-form_4-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-cain-form_4-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-cain-form_4-2)\n   Cain, Alan J. (2024). [*Form & Number: A History of Mathematical Beauty*](https:\/\/archive.org\/details\/cain_formandnumber_ebook_large). Lisbon: EBook.\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-5)**\n   Gallagher, James (13 February 2014). [\"Mathematics: Why the brain sees maths as beauty\"](https:\/\/www.bbc.co.uk\/news\/science-environment-26151062). *BBC News online*. Retrieved 13 February 2014.\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-6)**\n   Feynman, Richard P. (1977). [*The Feynman Lectures on Physics*](https:\/\/feynmanlectures.caltech.edu\/I_22.html#Ch22-S6-p4). Vol. I. Addison-Wesley. p. 22-16. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0-201-02010-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-201-02010-6 \"Special:BookSources\/0-201-02010-6\")\n   \n   .\n7. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-hardy-apology_7-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-hardy-apology_7-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-hardy-apology_7-2)\n   Hardy, G.H. (1967). [*A Mathematician's Apology*](https:\/\/archive.org\/details\/hardy_annotated). Cambridge University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-1-107-60463-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-107-60463-6 \"Special:BookSources\/978-1-107-60463-6\")\n   \n   .\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-8)**\n   Bell, E.T. (1937). *Men of Mathematics*. Simon & Schuster.\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-9)**\n   Wells, David (June 1990). \"Are these the most beautiful?\". *The Mathematical Intelligencer*. **12** (3): 37–41\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/BF03024015](https:\/\/doi.org\/10.1007%2FBF03024015).\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-10)**\n    Paulos, John Allen (1991). *Beyond Numeracy*. New York: Alfred A. Knopf. pp. 125–127\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-394-586409](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-394-586409 \"Special:BookSources\/0-394-586409\")\n    \n    .\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-11)**\n    Dutilh Novaes, Catarina (2019). \"The Beauty (?) of Mathematical Proofs\". In Aberdein, Andrew; Inglis, Matthew (eds.). *Advances in Experimental Philosophy of Logic and Mathematics*. Bloomsbury Academic. pp. 69–71\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-350-03901-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-350-03901-8 \"Special:BookSources\/978-1-350-03901-8\")\n    \n    .\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-12)**\n    Polster, Burkert (2004). *Q.E.D.: Beauty in Mathematical Proof*. Walker & Company. pp. 32–33\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-8027-1431-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-8027-1431-2 \"Special:BookSources\/978-0-8027-1431-2\")\n    \n    .\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-13)**\n    Schechter, Bruce (2000). *My Brain is Open: The Mathematical Journeys of Paul Erdős*. New York: [Simon & Schuster](https:\/\/en.wikipedia.org\/wiki\/Simon_%26_Schuster \"Simon & Schuster\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-684-85980-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-684-85980-7 \"Special:BookSources\/0-684-85980-7\")\n    \n    .\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-14)**\n    [Aigner, Martin](https:\/\/en.wikipedia.org\/wiki\/Martin_Aigner \"Martin Aigner\"); [Ziegler, Günter M.](https:\/\/en.wikipedia.org\/wiki\/G%C3%BCnter_M._Ziegler \"Günter M. Ziegler\") (2018). [*Proofs from THE BOOK*](https:\/\/en.wikipedia.org\/wiki\/Proofs_from_THE_BOOK \"Proofs from THE BOOK\") (6th ed.). Springer. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-3-662-57264-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-662-57264-1 \"Special:BookSources\/978-3-662-57264-1\")\n    \n    .\n15. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-plato-timaeus_15-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-plato-timaeus_15-1)\n    Plato (1929). [*Timaeus*](https:\/\/www.loebclassics.com\/view\/LCL234\/1929\/volume.xml). Cambridge, MA: Harvard University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-674-99257-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-674-99257-3 \"Special:BookSources\/978-0-674-99257-3\")\n    \n    .\n16. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-16)**\n    Field, J.V. (2013). *Kepler's Geometrical Cosmology*. Bloomsbury. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [9781472507037](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9781472507037 \"Special:BookSources\/9781472507037\")\n    \n    .\n17. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-17)**\n    Whitfield, John. [\"Journey to the 248th dimension\"](https:\/\/www.nature.com\/articles\/news070319-4). *Nature Online*. Retrieved 12 September 2025.\n18. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-18)**\n    Penrose, Roger (1974). \"The Rôle of Aesthetics in Pure and Applied Mathematical Research\". *Bulletin of the Institute of Mathematics and its Applications*. **10**: 226–271\\.\n19. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-19)**\n    [Chandrasekhar, Subrahmanyan](https:\/\/en.wikipedia.org\/wiki\/Subrahmanyan_Chandrasekhar \"Subrahmanyan Chandrasekhar\") (1987). *Truth and Beauty: Aesthetics and Motivation in Science*. Chicago, London: University of Chicago Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-226-10087-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-226-10087-6 \"Special:BookSources\/978-0-226-10087-6\")\n    \n    .\n20. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-20)**\n    Dirac, P.A.M. (1940). \"The Relation between Mathematics and Physics\". *Proceedings of the Royal Society of Edinburgh*. **59**: 122–129\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1017\/s0370164600012207](https:\/\/doi.org\/10.1017%2Fs0370164600012207).\n21. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-21)**\n    Penrose, Roger (2004). *The Road to Reality*. London: Jonathan Cape. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-224-04447-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-224-04447-9 \"Special:BookSources\/978-0-224-04447-9\")\n    \n    .\n22. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-22)**\n    McAllister, James W. (1996). [*Beauty & Revolution in Science*](https:\/\/hdl.handle.net\/1887\/10158). Ithaca: Cornell University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-8014-3240-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-8014-3240-8 \"Special:BookSources\/978-0-8014-3240-8\")\n    \n    .\n23. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-23)**\n    [Devlin, Keith](https:\/\/en.wikipedia.org\/wiki\/Keith_Devlin \"Keith Devlin\") (2000). \"Do Mathematicians Have Different Brains?\". [*The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip*](https:\/\/archive.org\/details\/mathgene00keit). [Basic Books](https:\/\/en.wikipedia.org\/wiki\/Basic_Books \"Basic Books\"). p. [140](https:\/\/archive.org\/details\/mathgene00keit\/page\/140). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-465-01619-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-465-01619-8 \"Special:BookSources\/978-0-465-01619-8\")\n    \n    . Retrieved 2008-08-22.\n24. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-24)**\n    Monastyrsky, Michael (2001). [\"Some Trends in Modern Mathematics and the Fields Medal\"](http:\/\/www.fields.utoronto.ca\/aboutus\/FieldsMedal_Monastyrsky.pdf) (PDF). *Can. Math. Soc. Notes*. **33** (2 and 3).\n25. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-25)**\n    McAllister, James W. (2005). \"Mathematical Beauty and the Evolution of the Standards of Mathematical Proof\". In Emmer, Michele (ed.). *The Visual Mind II*. MIT Press. pp. 15–34\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-262-05076-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-262-05076-0 \"Special:BookSources\/978-0-262-05076-0\")\n    \n    .\n26. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-26)**\n    [Loomis, Elisha Scott](https:\/\/en.wikipedia.org\/wiki\/Elisha_Scott_Loomis \"Elisha Scott Loomis\") (1968). *The Pythagorean Proposition*. Washington, DC: National Council of Teachers of Mathematics. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-873-53036-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-873-53036-1 \"Special:BookSources\/978-0-873-53036-1\")\n    \n    .\n27. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-27)**\n    Weisstein, Eric W. [\"Quadratic Reciprocity Theorem\"](http:\/\/mathworld.wolfram.com\/QuadraticReciprocityTheorem.html). *mathworld.wolfram.com*. Retrieved 2019-10-31.\n28. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-davis-experience_28-0)**\n    Davis, Philip J.; Hersh, Reuben (1981). *The Mathematical Experience*. Boston: Houghton Mifflin. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-395-32131-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-395-32131-7 \"Special:BookSources\/978-0-395-32131-7\")\n    \n    .\n29. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-29)**\n    Hoffman, Paul (1999). *The Man Who Loved Only Numbers*. London: Fourth Estate. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-85702-829-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-85702-829-4 \"Special:BookSources\/978-1-85702-829-4\")\n    \n    .\n30. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-30)**\n    [Aristotle](https:\/\/en.wikipedia.org\/wiki\/Aristotle \"Aristotle\") (1995). \"Metaphysics\". In Barnes, Jonathan (ed.). *The Complete Works of Aristotle*. Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-691-01650-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-691-01650-4 \"Special:BookSources\/978-0-691-01650-4\")\n    \n    .\n31. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-31)**\n    [Russell, Bertrand](https:\/\/en.wikipedia.org\/wiki\/Bertrand_Russell \"Bertrand Russell\") (1919). \"The Study of Mathematics\". [*Mysticism and Logic: And Other Essays*](https:\/\/archive.org\/details\/bub_gb_zwMQAAAAYAAJ). [Longman](https:\/\/en.wikipedia.org\/wiki\/Longman \"Longman\"). p. [60](https:\/\/archive.org\/details\/bub_gb_zwMQAAAAYAAJ\/page\/n68). Retrieved 2008-08-22. \"Mathematics rightly viewed possesses not only truth but supreme beauty a beauty cold and austere like that of sculpture without appeal to any part of our weaker nature without the gorgeous trappings Russell.\"\n32. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-32)**\n    [Harré, Rom](https:\/\/en.wikipedia.org\/wiki\/Rom_Harr%C3%A9 \"Rom Harré\") (1958). [\"Quasi-Aesthetic Appraisals\"](https:\/\/www.jstor.org\/stable\/3748562). *Philosophy*. **33**: 132–137\\.\n33. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-33)**\n    [Zangwill, Nick](https:\/\/en.wikipedia.org\/wiki\/Nick_Zangwill \"Nick Zangwill\") (2001). *The Metaphysics of Beauty*. Ithaca, London: Cornell University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-8014-3820-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-8014-3820-2 \"Special:BookSources\/978-0-8014-3820-2\")\n    \n    .\n34. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-34)** A. Moles: *Théorie de l'information et perception esthétique*, Paris, Denoël, 1973 ([Information Theory](https:\/\/en.wikipedia.org\/wiki\/Information_Theory \"Information Theory\") and aesthetical perception)\n35. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-35)**\n    F Nake (1974). Ästhetik als Informationsverarbeitung. ([Aesthetics](https:\/\/en.wikipedia.org\/wiki\/Aesthetics \"Aesthetics\") as information processing). Grundlagen und Anwendungen der Informatik im Bereich ästhetischer Produktion und Kritik. Springer, 1974, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [3-211-81216-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/3-211-81216-4 \"Special:BookSources\/3-211-81216-4\")\n    \n    , [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-3-211-81216-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-211-81216-7 \"Special:BookSources\/978-3-211-81216-7\")\n36. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-36)** J. Schmidhuber. [Low-complexity art](https:\/\/en.wikipedia.org\/wiki\/Low-complexity_art \"Low-complexity art\"). [Leonardo](https:\/\/en.wikipedia.org\/wiki\/Leonardo_\\(journal\\) \"Leonardo (journal)\"), Journal of the International Society for the Arts, Sciences, and Technology ([Leonardo\/ISAST](https:\/\/en.wikipedia.org\/wiki\/Leonardo,_The_International_Society_of_the_Arts,_Sciences_and_Technology \"Leonardo, The International Society of the Arts, Sciences and Technology\")), 30(2):97–103, 1997. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.2307\/1576418](https:\/\/doi.org\/10.2307%2F1576418). [JSTOR](https:\/\/en.wikipedia.org\/wiki\/JSTOR_\\(identifier\\) \"JSTOR (identifier)\") [1576418](https:\/\/www.jstor.org\/stable\/1576418).\n37. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-37)** J. Schmidhuber. Papers on the theory of beauty and [low-complexity art](https:\/\/en.wikipedia.org\/wiki\/Low-complexity_art \"Low-complexity art\") since 1994: <http:\/\/www.idsia.ch\/~juergen\/beauty.html>\n38. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-38)** J. Schmidhuber. Simple Algorithmic Principles of Discovery, Subjective Beauty, Selective Attention, Curiosity & Creativity. Proc. 10th Intl. Conf. on Discovery Science (DS 2007) pp. 26–38, LNAI 4755, Springer, 2007. Also in Proc. 18th Intl. Conf. on Algorithmic Learning Theory (ALT 2007) p. 32, LNAI 4754, Springer, 2007. Joint invited lecture for DS 2007 and ALT 2007, Sendai, Japan, 2007. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[0709\\.0674](https:\/\/arxiv.org\/abs\/0709.0674).\n39. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-39)** .J. Schmidhuber. Curious model-building control systems. International Joint Conference on Neural Networks, Singapore, vol 2, 1458–1463. IEEE press, 1991\n40. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-40)** Schmidhuber's theory of beauty and curiosity in a German TV show: <http:\/\/www.br-online.de\/bayerisches-fernsehen\/faszination-wissen\/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml> [Archived](https:\/\/web.archive.org\/web\/20080603221058\/http:\/\/www.br-online.de\/bayerisches-fernsehen\/faszination-wissen\/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml) June 3, 2008, at the [Wayback Machine](https:\/\/en.wikipedia.org\/wiki\/Wayback_Machine \"Wayback Machine\")\n41. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-41)**\n    Zeki, Semir; Romaya, John Paul; Benincasa, Dionigi M. T.; Atiyah, Michael F. (2014). [\"The experience of mathematical beauty and its neural correlates\"](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3923150). *Frontiers in Human Neuroscience*. **8**: 68. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3389\/fnhum.2014.00068](https:\/\/doi.org\/10.3389%2Ffnhum.2014.00068). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1662-5161](https:\/\/search.worldcat.org\/issn\/1662-5161). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [3923150](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3923150). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [24592230](https:\/\/pubmed.ncbi.nlm.nih.gov\/24592230).\n42. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-42)**\n    Zhang, Haoxuan; Zeki, Semir (May 2022). [\"Judgments of mathematical beauty are resistant to revision through external opinion\"](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC9790661). *PsyCh Journal*. **11** (5): 741–747\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1002\/pchj.556](https:\/\/doi.org\/10.1002%2Fpchj.556). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [2046-0252](https:\/\/search.worldcat.org\/issn\/2046-0252). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [9790661](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC9790661). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [35491015](https:\/\/pubmed.ncbi.nlm.nih.gov\/35491015).\n43. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-43)**  [Luque, Sergio. 2009. \"The Stochastic Synthesis of Iannis Xenakis.\" Leonardo Music Journal (19): 77–84](http:\/\/www.sergioluque.com\/stochastics)\n44. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-44)**\n    Norris, Chris (2001). [\"Hammer Of The Gods\"](https:\/\/web.archive.org\/web\/20111101195240\/http:\/\/toolshed.down.net\/articles\/text\/spinmag.jun.2001.html). Archived from [the original](http:\/\/toolshed.down.net\/articles\/text\/spinmag.jun.2001.html) on November 1, 2011. Retrieved April 25, 2007.\n45. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-45)**\n    Schell, Michael (December 11, 2018). [\"Elliott Carter (1908–2012): Legacy of a Centenarian\"](https:\/\/www.secondinversion.org\/2018\/12\/11\/elliott-carter-1908-2012-legacy-of-a-centenarian\/). *Second Inversion*. [Archived](https:\/\/web.archive.org\/web\/20181224054514\/https:\/\/www.secondinversion.org\/2018\/12\/11\/elliott-carter-1908-2012-legacy-of-a-centenarian\/) from the original on December 24, 2018. Retrieved December 11, 2018.\n46. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-46)** Hull, Thomas. \"Project Origami: Activities for Exploring Mathematics\". Taylor & Francis, 2006.\n47. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-47)** John Ernest's use of mathematics and especially group theory in his art works is analysed in *John Ernest, A Mathematical Artist* by Paul Ernest in *Philosophy of Mathematics Education Journal*, No. 24 Dec. 2009 (Special Issue on Mathematics and Art): <http:\/\/people.exeter.ac.uk\/PErnest\/pome24\/index.htm>\n48. **[^](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty#cite_ref-48)**\n    Franco, Francesca (2017-10-05). [\"The Systems Group (Chapter 2)\"](https:\/\/books.google.com\/books?id=oJU4DwAAQBAJ&q=anthony+hill+and+peter+lowe&pg=PT61). *Generative Systems Art: The Work of Ernest Edmonds*. Routledge. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [9781317137436](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9781317137436 \"Special:BookSources\/9781317137436\")\n    \n    .\n\n- Cellucci, Carlo (2015), \"Mathematical beauty, understanding, and discovery\", *[Foundations of Science](https:\/\/en.wikipedia.org\/wiki\/Foundations_of_Science \"Foundations of Science\")*, **20** (4): 339–355, [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/s10699-014-9378-7](https:\/\/doi.org\/10.1007%2Fs10699-014-9378-7), [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [120068870](https:\/\/api.semanticscholar.org\/CorpusID:120068870)\n- [Hoffman, Paul](https:\/\/en.wikipedia.org\/wiki\/Paul_Hoffman_\\(science_writer\\) \"Paul Hoffman (science writer)\") (1992), *[The Man Who Loved Only Numbers](https:\/\/en.wikipedia.org\/wiki\/The_Man_Who_Loved_Only_Numbers \"The Man Who Loved Only Numbers\")*, Hyperion.\n- [Loomis, Elisha Scott](https:\/\/en.wikipedia.org\/wiki\/Elisha_Scott_Loomis \"Elisha Scott Loomis\") (1968), *The Pythagorean Proposition*, The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem.\n- [Pandey, S.K.](https:\/\/en.wikipedia.org\/w\/index.php?title=S.K._Pandey&action=edit&redlink=1 \"S.K. Pandey (page does not exist)\") . [*The Humming of Mathematics: Melody of Mathematics*](https:\/\/books.google.com\/books?id=BgYbzAEACAAJ). Independently Published, 2019. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [1710134437](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/1710134437 \"Special:BookSources\/1710134437\")\n  .\n- Peitgen, H.-O., and Richter, P.H. (1986), *The Beauty of Fractals*, Springer-Verlag.\n- [Reber, R.](https:\/\/en.wikipedia.org\/wiki\/Rolf_Reber \"Rolf Reber\"); Brun, M.; Mitterndorfer, K. (2008). \"The use of heuristics in intuitive mathematical judgment\". *Psychonomic Bulletin & Review*. **15** (6): 1174–1178\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3758\/PBR.15.6.1174](https:\/\/doi.org\/10.3758%2FPBR.15.6.1174). [hdl](https:\/\/en.wikipedia.org\/wiki\/Hdl_\\(identifier\\) \"Hdl (identifier)\"):[1956\/2734](https:\/\/hdl.handle.net\/1956%2F2734). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [19001586](https:\/\/pubmed.ncbi.nlm.nih.gov\/19001586). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [5297500](https:\/\/api.semanticscholar.org\/CorpusID:5297500).\n- Strohmeier, John, and Westbrook, Peter (1999), *Divine Harmony, The Life and Teachings of Pythagoras*, Berkeley Hills Books, Berkeley, CA.\n- [Zeki, S.](https:\/\/en.wikipedia.org\/wiki\/Semir_Zeki \"Semir Zeki\"); Romaya, J. P.; Benincasa, D. M. T.; [Atiyah, M. F.](https:\/\/en.wikipedia.org\/wiki\/Michael_Atiyah \"Michael Atiyah\") (2014), \"The experience of mathematical beauty and its neural correlates\", *Frontiers in Human Neuroscience*, **8**: 68, [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3389\/fnhum.2014.00068](https:\/\/doi.org\/10.3389%2Ffnhum.2014.00068), [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [3923150](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3923150), [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [24592230](https:\/\/pubmed.ncbi.nlm.nih.gov\/24592230)\n\n- [Aigner, Martin](https:\/\/en.wikipedia.org\/wiki\/Martin_Aigner \"Martin Aigner\"); [Ziegler, Günter M.](https:\/\/en.wikipedia.org\/wiki\/G%C3%BCnter_M._Ziegler \"Günter M. Ziegler\") (2018). [*Proofs from THE BOOK*](https:\/\/en.wikipedia.org\/wiki\/Proofs_from_THE_BOOK \"Proofs from THE BOOK\") (6th ed.). Springer. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-3-662-57264-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-662-57264-1 \"Special:BookSources\/978-3-662-57264-1\")\n  \n  .\n- Cain, Alan J. (2024). [*Form & Number: A History of Mathematical Beauty*](https:\/\/archive.org\/details\/cain_formandnumber_ebook_large). Lisbon: Ebook.\n- [Hadamard, Jacques](https:\/\/en.wikipedia.org\/wiki\/Jacques_Hadamard \"Jacques Hadamard\") (1949). [*An Essay on the Psychology of Invention in the Mathematical Field*](https:\/\/archive.org\/details\/essayonpsycholog00hada) (2nd enlarged ed.). Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [0-486-20107-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-486-20107-4 \"Special:BookSources\/0-486-20107-4\")\n  \n  .\n- [Hardy, G.H.](https:\/\/en.wikipedia.org\/wiki\/G.H._Hardy \"G.H. Hardy\") (1967) \\[1st published 1940\\]. [*A Mathematician's Apology*](https:\/\/archive.org\/details\/hardy_annotated). Cambridge University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-1-107-60463-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-107-60463-6 \"Special:BookSources\/978-1-107-60463-6\")\n  \n  .\n- Huntley, H.E. (1970). [*The Divine Proportion: A Study in Mathematical Beauty*](https:\/\/archive.org\/details\/divineproportion0000hunt_o2w9). New York: Dover Publications. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-486-22254-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-486-22254-7 \"Special:BookSources\/978-0-486-22254-7\")\n  \n  .\n- Stewart, Ian (2007). *Why beauty is truth : a history of symmetry*. New York: Basic Books, a member of the Perseus Books Group. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-465-08236-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-465-08236-0 \"Special:BookSources\/978-0-465-08236-0\")\n  \n  . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [76481488](https:\/\/search.worldcat.org\/oclc\/76481488).\n\n- [Mathematics, Poetry and Beauty](https:\/\/web.archive.org\/web\/20151209130947\/http:\/\/raharoni.net.technion.ac.il\/mathematics-poetry-and-beauty\/)\n- [Is Mathematics Beautiful?](http:\/\/www.cut-the-knot.org\/manifesto\/beauty.shtml) cut-the-knot.org\n- [Justin Mullins.com](http:\/\/www.justinmullins.com\/)\n- [Edna St. Vincent Millay (poet): *Euclid alone has looked on beauty bare*](http:\/\/www.the-athenaeum.org\/poetry\/detail.php?id=80)\n- [Terence Tao](https:\/\/en.wikipedia.org\/wiki\/Terence_Tao \"Terence Tao\"), [*What is good mathematics?*](https:\/\/arxiv.org\/abs\/math\/0702396)\n- [Mathbeauty Blog](http:\/\/mathbeauty.wordpress.com\/)\n- The *[Aesthetic Appeal](https:\/\/archive.org\/details\/aestheticappeal)* collection at the [Internet Archive](https:\/\/en.wikipedia.org\/wiki\/Internet_Archive \"Internet Archive\")\n- [*A Mathematical Romance*](http:\/\/www.nybooks.com\/articles\/archives\/2013\/dec\/05\/mathematical-romance\/) [Jim Holt](https:\/\/en.wikipedia.org\/wiki\/Jim_Holt_\\(philosopher\\) \"Jim Holt (philosopher)\") December 5, 2013 issue of [The New York Review of Books](https:\/\/en.wikipedia.org\/wiki\/The_New_York_Review_of_Books \"The New York Review of Books\") review of *Love and Math: The Heart of Hidden Reality* by [Edward Frenkel](https:\/\/en.wikipedia.org\/wiki\/Edward_Frenkel \"Edward Frenkel\")","meta_canonical":null}

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Boilerpipe Text	Mathematical beauty is a type of aesthetic value that is experienced in doing or contemplating mathematics . The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such as G.H. Hardy , have characterized mathematics as an art form that seeks beauty. Beauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understanding beauty in general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstract ideas which can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than (for example) listening to music. [ 1 ] Furthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors identify mathematical elegance with mathematical beauty; others distinguish elegance as a separate aesthetic value, or as being, for instance, limited to the form mathematical exposition. [ 2 ] : 177–178 Beauty itself is often linked to, or thought to be dependent on, the abstractness, purity, simplicity, depth or order of mathematics. Examples of beautiful mathematics [ edit ] Starting at e 0 = 1, travelling at the velocity i relative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an Argand diagram .) Euler's identity is often given as an example of a beautiful result: [ 3 ] : 1–3 [ 4 ] : 835–836 [ 5 ] This expression ties together arguably the five most important mathematical constants ( e , i , π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of Euler's formula , which the physicist Richard Feynman called "our jewel" and "the most remarkable formula in mathematics". [ 6 ] Another example is Fermat's theorem on sums of two squares , which says that any prime number such that can be written as a sum of two square numbers (for example, , , ), which both G.H. Hardy [ 7 ] : §12 and E.T. Bell [ 8 ] : ch.4 thought was a beautiful result. In a survey in which mathematicians were asked to evaluate 24 theorems for their beauty, the top-rated three theorems were: [ 9 ] Euler's equation; Euler's polyhedron formula , which asserts that for a polyhedron with V vertices, E edges, and F faces, ; and Euclid's theorem that there are infinitely many prime numbers, which was also given by Hardy as an example of a beautiful theorem. [ 7 ] : §12 An example of "beauty in method"—a simple and elegant visual descriptor of the Pythagorean theorem . Cantor's diagonal argument , which establishes that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers , has been cited by both mathematicians [ 10 ] and philosophers [ 11 ] as an example of a beautiful proof. A proof without words for the sum of odd numbers theorem Visual proofs, such as the illustrated proof of the Pythagorean theorem , and other proofs without words generally, such as the shown proof that the sum of all positive odd numbers up to 2 n − 1 is a perfect square , have been thought beautiful. [ 12 ] The mathematician Paul Erdős spoke of The Book , an imaginary infinite book in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would proclaim it "straight from The Book!". [ 13 ] : 35 His rhetorical device inspired the creation of Proofs from THE BOOK , a collection of such proofs, including many suggested by Erdős himself. [ 14 ] : v In Plato 's Timaeus , the five regular convex polyhedra , called the Platonic solids for their role in this dialogue, are called the "most beautiful" ("κάλλιστα") bodies. [ 15 ] : 53e In the Timaeus , they are described as having been used by the demiurge , or creator-craftsman who builds the cosmos, for the four classical elements plus the heavens, because of their beauty. [ 15 ] : 54e–55e Kepler's Platonic solid model of the solar system In his 1596 book Mysterium Cosmographicum , Johannes Kepler argued that the orbits of the then-known planets in the Solar System have been arranged by God to correspond to a concentric arrangement of the five Platonic solids , each orbit lying on the circumsphere of one polyhedron and the insphere of another. For Kepler, God had wanted to shape the universe according to the five regular solids because of their beauty, and this explained why there were six planets (according to the knowledge of the time). [ 16 ] : ch.3 [ 4 ] : 280--285 Petrie projection of A more modern example is the exceptional simple Lie group , which has been called "perhaps the most beautiful structure in all of mathematics". [ 17 ] Scientific theories [ edit ] The mathematical statements of scientific theories, especially in physics, are sometimes considered to be mathematically beautiful. For example, Roger Penrose thought there was a "special beauty" in Maxwell's equations of electromagnetism: [ 18 ] : 268 Einstein's theory of general relativity has been characterized as a work of art, and, among other aesthetic praise, [ 19 ] : 148 was described by Paul Dirac as having "great mathematical beauty" [ 20 ] : 123 and by Penrose as having "supreme mathematical beauty". [ 21 ] : 1038 (There can be more to the beauty of a scientific theory than just its mathematical statement. For example, whether a theory is visualizable or deterministic might have an influence on whether it is seen as beautiful. [ 22 ] : 53 [ 4 ] : 873–877 ) Properties of beautiful mathematics [ edit ] Many mathematicians and philosophers who have written about mathematical beauty have tried to identify properties or criteria that are conducive to the perception of beauty in a piece of mathematics. It is debated whether beauty can be clarified or explained by such properties: Paul Erdős thought that it was no more possible to convince someone of the beauty of a piece of mathematics than to convince them of the beauty of Beethoven's Ninth Symphony , if they couldn't see it for themselves. [ 23 ] In his 1940 essay A Mathematician's Apology , G. H. Hardy said that a beautiful result, including its proof, possesses three "purely aesthetic qualities", namely "inevitability", "unexpectedness", and "economy". He particularly excluded enumeration of cases as "one of the duller forms of mathematical argument". [ 7 ] : §18 In 1997, Gian-Carlo Rota disagreed with unexpectedness as a sufficient condition for beauty and proposed a counterexample: A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago [from 1977] the proof of the existence of non-equivalent differentiable structures on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or now. [ 2 ] : 172 In contrast, Monastyrsky wrote in 2001: It is very difficult to find an analogous invention in the past to Milnor 's beautiful construction of the different differential structures on the seven-dimensional sphere... The original proof of Milnor was not very constructive, but later E. Briscorn showed that these differential structures can be described in an extremely explicit and beautiful form. [ 24 ] : 44 This disagreement illustrates both the subjective nature of mathematical beauty, like other forms of beauty in general , and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them. Besides Hardy's properties of "unexpectedness", "inevitability", "economy", which he applied to proofs as well as results, mathematicians have customarily thought beautiful proofs that are short and simple. [ 25 ] : 22 In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem , with hundreds of proofs having been published. [ 26 ] Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity . In fact, Carl Friedrich Gauss alone had eight different proofs of this theorem, six of which he published. [ 27 ] In contrast, results that are logically correct but involve laborious calculations or consideration of many cases, are not usually considered beautiful, and may be even referred to as ugly or clumsy . For example, Kenneth Appel and Wolfgang Haken 's proof of the four color theorem made use of computer checking of over a thousand cases. Philip J. Davis and Reuben Hersh said that when they first heard that about the proof, they hoped it contained a new insight "whose beauty would transform my day", and were disheartened when informed the proof was by case enumeration and computer verification. [ 28 ] : 384 Paul Erdős said it was "not beautiful" because it gave no insight into why the theorem was true. [ 29 ] : 44 Philosophical analysis [ edit ] Aristotle thought that beauty was found especially in mathematics, writing in the Metaphysics that those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a very great deal about them; for if they do not expressly mention them, but prove attributes which are their results or their formulae, it is not true to say that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree. [ 30 ] : 1078a32–35 The logician and philosopher Bertrand Russell made a now-famous statement characterizing mathematical beauty in terms of purity and austerity: Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. [ 31 ] In the twentieth century, some philosophers questioned whether there was genuinely beauty in mathematics. The philosopher of science Rom Harré argued that there were no true aesthetic appraisals of mathematics, but only quasi-aesthetic appraisals. Any mathematical success described by an aesthetic term was a second-order success besides understanding and correctness. In contrast, aesthetic appraisal of a work of art was first-order. Harré considered this to be the difference between a quasi-aesthetic and a genuinely aesthetic appraisal. [ 32 ] Nick Zangwill thought that there were no true aesthetic experiences of mathematics and that a proofs or theories could only be metaphorically beautiful. His argument had two bases. First, he thought that aesthetic properties depended on sensory properties, and so abstract entities could not have aesthetic properties. Second, he thought that proofs, theorems, theories, and so on had purposes such as demonstrating correctness or granting understanding, and that any praise of them reflected only how well they achieved their purpose. [ 33 ] : 140–142 Scientific analysis [ edit ] Information-theory model [ edit ] In the 1970s, Abraham Moles and Frieder Nake analyzed links between beauty, information processing , and information theory . [ 34 ] [ 35 ] In the 1990s, Jürgen Schmidhuber formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory : the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e., Kolmogorov complexity ) relative to what the observer already knows. [ 36 ] [ 37 ] [ 38 ] Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the first derivative of subjectively perceived beauty: the observer continually tries to improve the predictability and compressibility of the observations by discovering regularities such as repetitions and symmetries and fractal self-similarity . Whenever the observer's learning process (possibly a predictive artificial neural network ) leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward. [ 39 ] [ 40 ] Brain imaging experiments conducted by Semir Zeki , Michael Atiyah and collaborators show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial orbito-frontal cortex (mOFC) of the brain and that this activity is parametrically related to the declared intensity of beauty. The location of the activity is similar to the location of the activity that correlates with the experience of beauty from other sources, such as visual art or music. [ 41 ] Moreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers. [ 42 ] Mathematical beauty and the arts [ edit ] Examples of the use of mathematics in music include the stochastic music of Iannis Xenakis , [ 43 ] the Fibonacci sequence in Tool 's Lateralus , [ 44 ] and the Metric modulation of Elliott Carter . [ 45 ] Other instances include the counterpoint of Johann Sebastian Bach , polyrhythmic structures (as in Igor Stravinsky 's The Rite of Spring ), permutation theory in serialism beginning with Arnold Schoenberg , the application of Shepard tones in Karlheinz Stockhausen 's Hymnen and the application of Group theory to transformations in music in the theoretical writings of David Lewin . Diagram from Leon Battista Alberti 's 1435 Della Pittura , with pillars in perspective on a grid Examples of the use of mathematics in the visual arts include applications of chaos theory and fractal geometry to computer-generated art , symmetry studies of Leonardo da Vinci , projective geometries in development of the perspective theory of Renaissance art, grids in Op art , optical geometry in the camera obscura of Giambattista della Porta , and multiple perspective in analytic cubism and futurism . Sacred geometry is a field of its own, giving rise to countless art forms including some of the best known mystic symbols and religious motifs, and has a particularly rich history in Islamic architecture . It also provides a means of meditation and comtemplation, for example the study of the Kaballah Sefirot (Tree Of Life) and Metatron's Cube ; and also the act of drawing itself. The Dutch graphic designer M. C. Escher created mathematically inspired woodcuts , lithographs , and mezzotints . These feature impossible constructions, explorations of infinity , architecture, visual paradoxes and tessellations . Some painters and sculptors create work distorted with the mathematical principles of anamorphosis , including South African sculptor Jonty Hurwitz . Origami , the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the mathematics of paper folding by observing the crease pattern on unfolded origami pieces. [ 46 ] British constructionist artist John Ernest created reliefs and paintings inspired by group theory. [ 47 ] A number of other British artists of the constructionist and systems schools of thought also draw on mathematics models and structures as a source of inspiration, including Anthony Hill and Peter Lowe . [ 48 ] Computer-generated art is based on mathematical algorithms . Argument from beauty Fluency heuristic Golden ratio Neuroesthetics Philosophy of mathematics Processing fluency theory of aesthetic pleasure Pythagoreanism Sense of wonder Theory of everything ^ Phillips, George (2005). "Preface" . Mathematics Is Not a Spectator Sport . Springer Science+Business Media . ISBN 0-387-25528-1 . Retrieved 2008-08-22 . "...there is nothing in the world of mathematics that corresponds to an audience in a concert hall, where the passive listen to the active. Happily, mathematicians are all doers , not spectators. ^ a b Rota, Gian-Carlo (May 1997). "The Phenomenology of Mathematical Beauty". Synthese . 111 (2): 171– 182. doi : 10.1023/A:1004930722234 . ^ Wilson, Robin (2018). Euler's Pioneering Equation: The most beautiful theorem in mathematics . Oxford University Press. ISBN 978-0-19-879492-9 . ^ a b c Cain, Alan J. (2024). Form & Number: A History of Mathematical Beauty . Lisbon: EBook. ^ Gallagher, James (13 February 2014). "Mathematics: Why the brain sees maths as beauty" . BBC News online . Retrieved 13 February 2014 . ^ Feynman, Richard P. (1977). The Feynman Lectures on Physics . Vol. I. Addison-Wesley. p. 22-16. ISBN 0-201-02010-6 . ^ a b c Hardy, G.H. (1967). A Mathematician's Apology . Cambridge University Press. ISBN 978-1-107-60463-6 . ^ Bell, E.T. (1937). Men of Mathematics . Simon & Schuster. ^ Wells, David (June 1990). "Are these the most beautiful?". The Mathematical Intelligencer . 12 (3): 37– 41. doi : 10.1007/BF03024015 . ^ Paulos, John Allen (1991). Beyond Numeracy . New York: Alfred A. Knopf. pp. 125– 127. ISBN 0-394-586409 . ^ Dutilh Novaes, Catarina (2019). "The Beauty (?) of Mathematical Proofs". In Aberdein, Andrew; Inglis, Matthew (eds.). Advances in Experimental Philosophy of Logic and Mathematics . Bloomsbury Academic. pp. 69– 71. ISBN 978-1-350-03901-8 . ^ Polster, Burkert (2004). Q.E.D.: Beauty in Mathematical Proof . Walker & Company. pp. 32– 33. ISBN 978-0-8027-1431-2 . ^ Schechter, Bruce (2000). My Brain is Open: The Mathematical Journeys of Paul Erdős . New York: Simon & Schuster . ISBN 0-684-85980-7 . ^ Aigner, Martin ; Ziegler, Günter M. (2018). Proofs from THE BOOK (6th ed.). Springer. ISBN 978-3-662-57264-1 . ^ a b Plato (1929). Timaeus . Cambridge, MA: Harvard University Press. ISBN 978-0-674-99257-3 . ^ Field, J.V. (2013). Kepler's Geometrical Cosmology . Bloomsbury. ISBN 9781472507037 . ^ Whitfield, John. "Journey to the 248th dimension" . Nature Online . Retrieved 12 September 2025 . ^ Penrose, Roger (1974). "The Rôle of Aesthetics in Pure and Applied Mathematical Research". Bulletin of the Institute of Mathematics and its Applications . 10 : 226– 271. ^ Chandrasekhar, Subrahmanyan (1987). Truth and Beauty: Aesthetics and Motivation in Science . Chicago, London: University of Chicago Press. ISBN 978-0-226-10087-6 . ^ Dirac, P.A.M. (1940). "The Relation between Mathematics and Physics". Proceedings of the Royal Society of Edinburgh . 59 : 122– 129. doi : 10.1017/s0370164600012207 . ^ Penrose, Roger (2004). The Road to Reality . London: Jonathan Cape. ISBN 978-0-224-04447-9 . ^ McAllister, James W. (1996). Beauty & Revolution in Science . Ithaca: Cornell University Press. ISBN 978-0-8014-3240-8 . ^ Devlin, Keith (2000). "Do Mathematicians Have Different Brains?". The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip . Basic Books . p. 140 . ISBN 978-0-465-01619-8 . Retrieved 2008-08-22 . ^ Monastyrsky, Michael (2001). "Some Trends in Modern Mathematics and the Fields Medal" (PDF) . Can. Math. Soc. Notes . 33 (2 and 3). ^ McAllister, James W. (2005). "Mathematical Beauty and the Evolution of the Standards of Mathematical Proof". In Emmer, Michele (ed.). The Visual Mind II . MIT Press. pp. 15– 34. ISBN 978-0-262-05076-0 . ^ Loomis, Elisha Scott (1968). The Pythagorean Proposition . Washington, DC: National Council of Teachers of Mathematics. ISBN 978-0-873-53036-1 . ^ Weisstein, Eric W. "Quadratic Reciprocity Theorem" . mathworld.wolfram.com . Retrieved 2019-10-31 . ^ Davis, Philip J.; Hersh, Reuben (1981). The Mathematical Experience . Boston: Houghton Mifflin. ISBN 978-0-395-32131-7 . ^ Hoffman, Paul (1999). The Man Who Loved Only Numbers . London: Fourth Estate. ISBN 978-1-85702-829-4 . ^ Aristotle (1995). "Metaphysics". In Barnes, Jonathan (ed.). The Complete Works of Aristotle . Princeton University Press. ISBN 978-0-691-01650-4 . ^ Russell, Bertrand (1919). "The Study of Mathematics". Mysticism and Logic: And Other Essays . Longman . p. 60 . Retrieved 2008-08-22 . Mathematics rightly viewed possesses not only truth but supreme beauty a beauty cold and austere like that of sculpture without appeal to any part of our weaker nature without the gorgeous trappings Russell. ^ Harré, Rom (1958). "Quasi-Aesthetic Appraisals" . Philosophy . 33 : 132– 137. ^ Zangwill, Nick (2001). The Metaphysics of Beauty . Ithaca, London: Cornell University Press. ISBN 978-0-8014-3820-2 . ^ A. Moles: Théorie de l'information et perception esthétique , Paris, Denoël, 1973 ( Information Theory and aesthetical perception) ^ F Nake (1974). Ästhetik als Informationsverarbeitung. ( Aesthetics as information processing). Grundlagen und Anwendungen der Informatik im Bereich ästhetischer Produktion und Kritik. Springer, 1974, ISBN 3-211-81216-4 , ISBN 978-3-211-81216-7 ^ J. Schmidhuber. Low-complexity art . Leonardo , Journal of the International Society for the Arts, Sciences, and Technology ( Leonardo/ISAST ), 30(2):97–103, 1997. doi : 10.2307/1576418 . JSTOR 1576418 . ^ J. Schmidhuber. Papers on the theory of beauty and low-complexity art since 1994: http://www.idsia.ch/~juergen/beauty.html ^ J. Schmidhuber. Simple Algorithmic Principles of Discovery, Subjective Beauty, Selective Attention, Curiosity & Creativity. Proc. 10th Intl. Conf. on Discovery Science (DS 2007) pp. 26–38, LNAI 4755, Springer, 2007. Also in Proc. 18th Intl. Conf. on Algorithmic Learning Theory (ALT 2007) p. 32, LNAI 4754, Springer, 2007. Joint invited lecture for DS 2007 and ALT 2007, Sendai, Japan, 2007. arXiv : 0709.0674 . ^ .J. Schmidhuber. Curious model-building control systems. International Joint Conference on Neural Networks, Singapore, vol 2, 1458–1463. IEEE press, 1991 ^ Schmidhuber's theory of beauty and curiosity in a German TV show: http://www.br-online.de/bayerisches-fernsehen/faszination-wissen/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml Archived June 3, 2008, at the Wayback Machine ^ Zeki, Semir; Romaya, John Paul; Benincasa, Dionigi M. T.; Atiyah, Michael F. (2014). "The experience of mathematical beauty and its neural correlates" . Frontiers in Human Neuroscience . 8 : 68. doi : 10.3389/fnhum.2014.00068 . ISSN 1662-5161 . PMC 3923150 . PMID 24592230 . ^ Zhang, Haoxuan; Zeki, Semir (May 2022). "Judgments of mathematical beauty are resistant to revision through external opinion" . PsyCh Journal . 11 (5): 741– 747. doi : 10.1002/pchj.556 . ISSN 2046-0252 . PMC 9790661 . PMID 35491015 . ^ Luque, Sergio. 2009. "The Stochastic Synthesis of Iannis Xenakis." Leonardo Music Journal (19): 77–84 ^ Norris, Chris (2001). "Hammer Of The Gods" . Archived from the original on November 1, 2011 . Retrieved April 25, 2007 . ^ Schell, Michael (December 11, 2018). "Elliott Carter (1908–2012): Legacy of a Centenarian" . Second Inversion . Archived from the original on December 24, 2018 . Retrieved December 11, 2018 . ^ Hull, Thomas. "Project Origami: Activities for Exploring Mathematics". Taylor & Francis, 2006. ^ John Ernest's use of mathematics and especially group theory in his art works is analysed in John Ernest, A Mathematical Artist by Paul Ernest in Philosophy of Mathematics Education Journal , No. 24 Dec. 2009 (Special Issue on Mathematics and Art): http://people.exeter.ac.uk/PErnest/pome24/index.htm ^ Franco, Francesca (2017-10-05). "The Systems Group (Chapter 2)" . Generative Systems Art: The Work of Ernest Edmonds . Routledge. ISBN 9781317137436 . Cellucci, Carlo (2015), "Mathematical beauty, understanding, and discovery", Foundations of Science , 20 (4): 339– 355, doi : 10.1007/s10699-014-9378-7 , S2CID 120068870 Hoffman, Paul (1992), The Man Who Loved Only Numbers , Hyperion. Loomis, Elisha Scott (1968), The Pythagorean Proposition , The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem. Pandey, S.K. . The Humming of Mathematics: Melody of Mathematics . Independently Published, 2019. ISBN 1710134437 . Peitgen, H.-O., and Richter, P.H. (1986), The Beauty of Fractals , Springer-Verlag. Reber, R. ; Brun, M.; Mitterndorfer, K. (2008). "The use of heuristics in intuitive mathematical judgment". Psychonomic Bulletin & Review . 15 (6): 1174– 1178. doi : 10.3758/PBR.15.6.1174 . hdl : 1956/2734 . PMID 19001586 . S2CID 5297500 . Strohmeier, John, and Westbrook, Peter (1999), Divine Harmony, The Life and Teachings of Pythagoras , Berkeley Hills Books, Berkeley, CA. Zeki, S. ; Romaya, J. P.; Benincasa, D. M. T.; Atiyah, M. F. (2014), "The experience of mathematical beauty and its neural correlates", Frontiers in Human Neuroscience , 8 : 68, doi : 10.3389/fnhum.2014.00068 , PMC 3923150 , PMID 24592230 Aigner, Martin ; Ziegler, Günter M. (2018). Proofs from THE BOOK (6th ed.). Springer. ISBN 978-3-662-57264-1 . Cain, Alan J. (2024). Form & Number: A History of Mathematical Beauty . Lisbon: Ebook. Hadamard, Jacques (1949). An Essay on the Psychology of Invention in the Mathematical Field (2nd enlarged ed.). Princeton University Press. ISBN 0-486-20107-4 . Hardy, G.H. (1967) [1st published 1940]. A Mathematician's Apology . Cambridge University Press. ISBN 978-1-107-60463-6 . Huntley, H.E. (1970). The Divine Proportion: A Study in Mathematical Beauty . New York: Dover Publications. ISBN 978-0-486-22254-7 . Stewart, Ian (2007). Why beauty is truth : a history of symmetry . New York: Basic Books, a member of the Perseus Books Group. ISBN 978-0-465-08236-0 . OCLC 76481488 . Mathematics, Poetry and Beauty Is Mathematics Beautiful? cut-the-knot.org Justin Mullins.com Edna St. Vincent Millay (poet): Euclid alone has looked on beauty bare Terence Tao , What is good mathematics? Mathbeauty Blog The Aesthetic Appeal collection at the Internet Archive A Mathematical Romance Jim Holt December 5, 2013 issue of The New York Review of Books review of Love and Math: The Heart of Hidden Reality by Edward Frenkel
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[o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Mathematical_beauty) - [1 Examples of beautiful mathematics](https://en.wikipedia.org/wiki/Mathematical_beauty#Examples_of_beautiful_mathematics) Toggle Examples of beautiful mathematics subsection - [1\.1 Results](https://en.wikipedia.org/wiki/Mathematical_beauty#Results) - [1\.2 Proofs](https://en.wikipedia.org/wiki/Mathematical_beauty#Proofs) - [1\.3 Objects](https://en.wikipedia.org/wiki/Mathematical_beauty#Objects) - [1\.4 Scientific theories](https://en.wikipedia.org/wiki/Mathematical_beauty#Scientific_theories) - [2 Properties of beautiful mathematics](https://en.wikipedia.org/wiki/Mathematical_beauty#Properties_of_beautiful_mathematics) Toggle Properties of beautiful mathematics subsection - [2\.1 Results](https://en.wikipedia.org/wiki/Mathematical_beauty#Results_2) - [2\.2 Proofs](https://en.wikipedia.org/wiki/Mathematical_beauty#Proofs_2) - [3 Philosophical analysis](https://en.wikipedia.org/wiki/Mathematical_beauty#Philosophical_analysis) - [4 Scientific analysis](https://en.wikipedia.org/wiki/Mathematical_beauty#Scientific_analysis) Toggle Scientific analysis subsection - [4\.1 Information-theory model](https://en.wikipedia.org/wiki/Mathematical_beauty#Information-theory_model) - [4\.2 Neural correlates](https://en.wikipedia.org/wiki/Mathematical_beauty#Neural_correlates) - [5 Mathematical beauty and the arts](https://en.wikipedia.org/wiki/Mathematical_beauty#Mathematical_beauty_and_the_arts) Toggle Mathematical beauty and the arts subsection - [5\.1 Music](https://en.wikipedia.org/wiki/Mathematical_beauty#Music) - [5\.2 Visual arts](https://en.wikipedia.org/wiki/Mathematical_beauty#Visual_arts) - [6 See also](https://en.wikipedia.org/wiki/Mathematical_beauty#See_also) - [7 Notes](https://en.wikipedia.org/wiki/Mathematical_beauty#Notes) - [8 References](https://en.wikipedia.org/wiki/Mathematical_beauty#References) - [9 Further reading](https://en.wikipedia.org/wiki/Mathematical_beauty#Further_reading) - [10 External links](https://en.wikipedia.org/wiki/Mathematical_beauty#External_links) Toggle the table of contents # Mathematical beauty 24 languages - [العربية](https://ar.wikipedia.org/wiki/%D8%AC%D9%85%D8%A7%D9%84_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA%D9%8A "جمال رياضياتي – Arabic") - [বাংলা](https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%B8%E0%A7%8C%E0%A6%A8%E0%A7%8D%E0%A6%A6%E0%A6%B0%E0%A7%8D%E0%A6%AF "গাণিতিক সৌন্দর্য – Bangla") - [Català](https://ca.wikipedia.org/wiki/Bellesa_matem%C3%A0tica "Bellesa matemàtica – Catalan") - [Dansk](https://da.wikipedia.org/wiki/Matematisk_sk%C3%B8nhed "Matematisk skønhed – Danish") - [Español](https://es.wikipedia.org/wiki/Belleza_matem%C3%A1tica "Belleza matemática – Spanish") - [فارسی](https://fa.wikipedia.org/wiki/%D8%B2%DB%8C%D8%A8%D8%A7%DB%8C%DB%8C_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C "زیبایی ریاضی – Persian") - [Suomi](https://fi.wikipedia.org/wiki/Matematiikan_kauneus "Matematiikan kauneus – Finnish") - [Français](https://fr.wikipedia.org/wiki/Beaut%C3%A9_math%C3%A9matique "Beauté mathématique – French") - [עברית](https://he.wikipedia.org/wiki/%D7%99%D7%95%D7%A4%D7%99_%D7%9E%D7%AA%D7%9E%D7%98%D7%99 "יופי מתמטי – Hebrew") - [हिन्दी](https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A5%8C%E0%A4%A8%E0%A5%8D%E0%A4%A6%E0%A4%B0%E0%A5%8D%E0%A4%AF "गणितीय सौन्दर्य – Hindi") - [Bahasa Indonesia](https://id.wikipedia.org/wiki/Keindahan_matematis "Keindahan matematis – Indonesian") - [Italiano](https://it.wikipedia.org/wiki/Bellezza_matematica "Bellezza matematica – Italian") - [日本語](https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E7%9A%84%E3%81%AA%E7%BE%8E "数学的な美 – Japanese") - [한국어](https://ko.wikipedia.org/wiki/%EC%88%98%ED%95%99%EC%A0%81_%EB%AF%B8 "수학적 미 – Korean") - [Lietuvių](https://lt.wikipedia.org/wiki/Matematinis_gro%C5%BEis "Matematinis grožis – Lithuanian") - [မြန်မာဘာသာ](https://my.wikipedia.org/wiki/%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC%E1%80%86%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%9B%E1%80%AC%E1%80%A1%E1%80%9C%E1%80%BE%E1%80%A1%E1%80%95 "သင်္ချာဆိုင်ရာအလှအပ – Burmese") - [Polski](https://pl.wikipedia.org/wiki/Matematyka_a_estetyka "Matematyka a estetyka – Polish") - [Português](https://pt.wikipedia.org/wiki/Beleza_da_matem%C3%A1tica "Beleza da matemática – Portuguese") - [Русский](https://ru.wikipedia.org/wiki/%D0%9A%D1%80%D0%B0%D1%81%D0%BE%D1%82%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B8 "Красота математики – Russian") - [Српски / srpski](https://sr.wikipedia.org/wiki/%D0%9B%D0%B5%D0%BF%D0%BE%D1%82%D0%B0_%D1%83_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%86%D0%B8 "Лепота у математици – Serbian") - [Українська](https://uk.wikipedia.org/wiki/%D0%9A%D1%80%D0%B0%D1%81%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B8 "Краса математики – Ukrainian") - [Tiếng Việt](https://vi.wikipedia.org/wiki/V%E1%BA%BB_%C4%91%E1%BA%B9p_c%E1%BB%A7a_to%C3%A1n_h%E1%BB%8Dc "Vẻ đẹp của toán học – Vietnamese") - [粵語](https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E4%B9%8B%E7%BE%8E "數學之美 – Cantonese") - [中文](https://zh.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E4%B9%8B%E7%BE%8E "數學之美 – Chinese") [Edit links](https://www.wikidata.org/wiki/Special:EntityPage/Q2248521#sitelinks-wikipedia "Edit interlanguage links") - 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[Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q2248521 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia Aesthetic value of mathematics Mathematical beauty is a type of [aesthetic](https://en.wikipedia.org/wiki/Aesthetics "Aesthetics") value that is experienced in doing or contemplating [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"). The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such as [G.H. Hardy](https://en.wikipedia.org/wiki/G.H._Hardy "G.H. Hardy"), have characterized mathematics as an [art](https://en.wikipedia.org/wiki/Art "Art") form that seeks beauty. Beauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understanding [beauty](https://en.wikipedia.org/wiki/Beauty "Beauty") in general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstract ideas which can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than (for example) listening to music.[\[1\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-1) Furthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors identify mathematical elegance with mathematical beauty; others distinguish elegance as a separate aesthetic value, or as being, for instance, limited to the form mathematical exposition.[\[2\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-rota-phenomenology-2): 177–178 Beauty itself is often linked to, or thought to be dependent on, the abstractness, purity, simplicity, depth or order of mathematics. ## Examples of beautiful mathematics \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=1 "Edit section: Examples of beautiful mathematics")\] ### Results \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=2 "Edit section: Results")\] [![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/3b/EulerIdentity2.svg/250px-EulerIdentity2.svg.png)](https://en.wikipedia.org/wiki/File:EulerIdentity2.svg) Starting at e0 = 1, travelling at the velocity i relative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an [Argand diagram](https://en.wikipedia.org/wiki/Argand_diagram "Argand diagram").) [Euler's identity](https://en.wikipedia.org/wiki/Euler%27s_identity "Euler's identity") is often given as an example of a beautiful result:[\[3\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-3): 1–3 [\[4\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-cain-form-4): 835–836 [\[5\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-5) e i π \+ 1 \= 0 . {\\displaystyle \\displaystyle \\mathrm {e} ^{\\mathrm {i} \\pi }+1=0\\,.} ![{\\displaystyle \\displaystyle \\mathrm {e} ^{\\mathrm {i} \\pi }+1=0\\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5213bb3f233806ff3212142214fee997434a15) This expression ties together arguably the five most important [mathematical constants](https://en.wikipedia.org/wiki/Mathematical_constant "Mathematical constant") (e, i, π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"), which the physicist [Richard Feynman](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman") called "our jewel" and "the most remarkable formula in mathematics".[\[6\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-6) Another example is [Fermat's theorem on sums of two squares](https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares "Fermat's theorem on sums of two squares"), which says that any [prime number](https://en.wikipedia.org/wiki/Prime_number "Prime number") such that p ≡ 1 ( mod 4 ) {\\displaystyle p\\equiv 1{\\pmod {4}}} ![{\\displaystyle p\\equiv 1{\\pmod {4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/125d70c2c8848e96954eddfc7f9283ec4f676ed7) can be written as a sum of two square numbers (for example, 5 \= 1 2 \+ 2 2 {\\displaystyle 5=1^{2}+2^{2}} ![{\\displaystyle 5=1^{2}+2^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b288581682d73e457bc6829e15be203280852a), 13 \= 2 2 \+ 3 2 {\\displaystyle 13=2^{2}+3^{2}} ![{\\displaystyle 13=2^{2}+3^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8be5ea38d26942831bf630c465218ef5e3189181), 37 \= 1 2 \+ 6 2 {\\displaystyle 37=1^{2}+6^{2}} ![{\\displaystyle 37=1^{2}+6^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f81cd69710c7c05bdc903449dc7e834540c797a)), which both [G.H. Hardy](https://en.wikipedia.org/wiki/G.H._Hardy "G.H. Hardy")[\[7\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-hardy-apology-7): §12 and [E.T. Bell](https://en.wikipedia.org/wiki/E.T._Bell "E.T. Bell")[\[8\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-8): ch.4 thought was a beautiful result. In a survey in which mathematicians were asked to evaluate 24 theorems for their beauty, the top-rated three theorems were:[\[9\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-9) Euler's equation; [Euler's polyhedron formula](https://en.wikipedia.org/wiki/Euler_characteristic "Euler characteristic"), which asserts that for a polyhedron with V vertices, E edges, and F faces, V − E \+ F \= 2 {\\displaystyle V-E+F=2} ![{\\displaystyle V-E+F=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/759601e482258ff7a359a7db381abf60372c5b06); and [Euclid's theorem](https://en.wikipedia.org/wiki/Euclid%27s_theorem "Euclid's theorem") that there are infinitely many prime numbers, which was also given by [Hardy](https://en.wikipedia.org/wiki/G.H._Hardy "G.H. Hardy") as an example of a beautiful theorem.[\[7\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-hardy-apology-7): §12 ### Proofs \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=3 "Edit section: Proofs")\] [![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Pythagorean_proof_%281%29.svg/250px-Pythagorean_proof_%281%29.svg.png)](https://en.wikipedia.org/wiki/File:Pythagorean_proof_\(1\).svg) An example of "beauty in method"—a simple and elegant visual descriptor of the [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem"). [Cantor's diagonal argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument "Cantor's diagonal argument"), which establishes that there are [infinite sets](https://en.wikipedia.org/wiki/Infinite_set "Infinite set") which cannot be put into [one-to-one correspondence](https://en.wikipedia.org/wiki/Bijection "Bijection") with the infinite set of [natural numbers](https://en.wikipedia.org/wiki/Natural_number "Natural number"), has been cited by both mathematicians[\[10\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-10) and philosophers[\[11\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-11) as an example of a beautiful proof. [![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Proofwithoutwords.svg/250px-Proofwithoutwords.svg.png)](https://en.wikipedia.org/wiki/File:Proofwithoutwords.svg) A proof without words for the sum of odd numbers theorem Visual proofs, such as the illustrated proof of the [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem"), and other [proofs without words](https://en.wikipedia.org/wiki/Proofs_without_words "Proofs without words") generally, such as the shown proof that the sum of all positive [odd numbers](https://en.wikipedia.org/wiki/Parity_\(mathematics\) "Parity (mathematics)") up to 2n − 1 is a [perfect square](https://en.wikipedia.org/wiki/Square_number "Square number"), have been thought beautiful.[\[12\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-12) The mathematician [Paul Erdős](https://en.wikipedia.org/wiki/Paul_Erd%C5%91s "Paul Erdős") spoke of The Book, an imaginary infinite book in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would proclaim it "straight from The Book!".[\[13\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-13): 35 His rhetorical device inspired the creation of [Proofs from THE BOOK](https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK "Proofs from THE BOOK"), a collection of such proofs, including many suggested by Erdős himself.[\[14\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-14): v ### Objects \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=4 "Edit section: Objects")\] In [Plato](https://en.wikipedia.org/wiki/Plato "Plato")'s [Timaeus](https://en.wikipedia.org/wiki/Timaeus_\(dialogue\) "Timaeus (dialogue)"), the five [regular convex polyhedra](https://en.wikipedia.org/wiki/Platonic_solids "Platonic solids"), called the Platonic solids for their role in this dialogue, are called the "most beautiful" ("κάλλιστα") bodies.[\[15\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-plato-timaeus-15): 53e In the Timaeus, they are described as having been used by the [demiurge](https://en.wikipedia.org/wiki/Demiurge "Demiurge"), or creator-craftsman who builds the cosmos, for the four [classical elements](https://en.wikipedia.org/wiki/Classical_elements "Classical elements") plus the heavens, because of their beauty.[\[15\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-plato-timaeus-15): 54e–55e [![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/19/Kepler-solar-system-1.png/250px-Kepler-solar-system-1.png)](https://en.wikipedia.org/wiki/File:Kepler-solar-system-1.png) Kepler's Platonic solid model of the solar system In his 1596 book [Mysterium Cosmographicum](https://en.wikipedia.org/wiki/Mysterium_Cosmographicum "Mysterium Cosmographicum"), [Johannes Kepler](https://en.wikipedia.org/wiki/Johannes_Kepler "Johannes Kepler") argued that the orbits of the then-known planets in the [Solar System](https://en.wikipedia.org/wiki/Solar_System "Solar System") have been arranged by [God](https://en.wikipedia.org/wiki/God "God") to correspond to a concentric arrangement of the five [Platonic solids](https://en.wikipedia.org/wiki/Platonic_solid "Platonic solid"), each orbit lying on the [circumsphere](https://en.wikipedia.org/wiki/Circumscribed_sphere "Circumscribed sphere") of one [polyhedron](https://en.wikipedia.org/wiki/Polyhedron "Polyhedron") and the [insphere](https://en.wikipedia.org/wiki/Inscribed_sphere "Inscribed sphere") of another. For Kepler, God had wanted to shape the universe according to the five regular solids because of their beauty, and this explained why there were six planets (according to the knowledge of the time).[\[16\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-16): ch.3 [\[4\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-cain-form-4): 280--285 [![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/250px-E8Petrie.svg.png)](https://en.wikipedia.org/wiki/File:E8Petrie.svg) Petrie projection of E 8 {\\displaystyle E\_{8}} ![{\\displaystyle E\_{8}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48479e96d90b4cfabc7784106cc3cfff907dda34) A more modern example is the exceptional [simple Lie group](https://en.wikipedia.org/wiki/Simple_Lie_group "Simple Lie group") E 8 {\\displaystyle E\_{8}} ![{\\displaystyle E\_{8}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48479e96d90b4cfabc7784106cc3cfff907dda34), which has been called "perhaps the most beautiful structure in all of mathematics".[\[17\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-17) ### Scientific theories \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=5 "Edit section: Scientific theories")\] The mathematical statements of scientific theories, especially in physics, are sometimes considered to be mathematically beautiful. For example, [Roger Penrose](https://en.wikipedia.org/wiki/Roger_Penrose "Roger Penrose") thought there was a "special beauty" in [Maxwell's equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations "Maxwell's equations") of electromagnetism:[\[18\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-18): 268 ∇ ⋅ E \= ρ ε 0 ∇ ⋅ B \= 0 ∇ × E \= − ∂ B ∂ t ∇ × B \= μ 0 ( J \+ ε 0 ∂ E ∂ t ) {\\displaystyle {\\begin{aligned}\\nabla \\cdot \\mathbf {E} \\,\\,\\,&={\\frac {\\rho }{\\varepsilon \_{0}}}\\\\\\nabla \\cdot \\mathbf {B} \\,\\,\\,&=0\\\\\\nabla \\times \\mathbf {E} &=-{\\frac {\\partial \\mathbf {B} }{\\partial t}}\\\\\\nabla \\times \\mathbf {B} &=\\mu \_{0}\\left(\\mathbf {J} +\\varepsilon \_{0}{\\frac {\\partial \\mathbf {E} }{\\partial t}}\\right)\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\nabla \\cdot \\mathbf {E} \\,\\,\\,&={\\frac {\\rho }{\\varepsilon \_{0}}}\\\\\\nabla \\cdot \\mathbf {B} \\,\\,\\,&=0\\\\\\nabla \\times \\mathbf {E} &=-{\\frac {\\partial \\mathbf {B} }{\\partial t}}\\\\\\nabla \\times \\mathbf {B} &=\\mu \_{0}\\left(\\mathbf {J} +\\varepsilon \_{0}{\\frac {\\partial \\mathbf {E} }{\\partial t}}\\right)\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1740d383a275f64105f457e209ff5c66eeeb21) Einstein's theory of [general relativity](https://en.wikipedia.org/wiki/General_relativity "General relativity") has been characterized as a work of art, and, among other aesthetic praise,[\[19\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-19): 148 was described by [Paul Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac") as having "great mathematical beauty"[\[20\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-20): 123 and by Penrose as having "supreme mathematical beauty".[\[21\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-21): 1038 (There can be more to the beauty of a scientific theory than just its mathematical statement. For example, whether a theory is visualizable or deterministic might have an influence on whether it is seen as beautiful.[\[22\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-22): 53 [\[4\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-cain-form-4): 873–877 ) ## Properties of beautiful mathematics \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=6 "Edit section: Properties of beautiful mathematics")\] Many mathematicians and philosophers who have written about mathematical beauty have tried to identify properties or criteria that are conducive to the perception of beauty in a piece of mathematics. It is debated whether beauty can be clarified or explained by such properties: [Paul Erdős](https://en.wikipedia.org/wiki/Paul_Erd%C5%91s "Paul Erdős") thought that it was no more possible to convince someone of the beauty of a piece of mathematics than to convince them of the beauty of [Beethoven's Ninth Symphony](https://en.wikipedia.org/wiki/Symphony_No._9_\(Beethoven\) "Symphony No. 9 (Beethoven)"), if they couldn't see it for themselves.[\[23\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-23) ### Results \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=7 "Edit section: Results")\] In his 1940 essay [A Mathematician's Apology](https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology "A Mathematician's Apology"), [G. H. Hardy](https://en.wikipedia.org/wiki/G._H._Hardy "G. H. Hardy") said that a beautiful result, including its proof, possesses three "purely aesthetic qualities", namely "inevitability", "unexpectedness", and "economy". He particularly excluded enumeration of cases as "one of the duller forms of mathematical argument".[\[7\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-hardy-apology-7): §18 In 1997, [Gian-Carlo Rota](https://en.wikipedia.org/wiki/Gian-Carlo_Rota "Gian-Carlo Rota") disagreed with unexpectedness as a sufficient condition for beauty and proposed a counterexample: > A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago \[from 1977\] the proof of the existence of [non-equivalent differentiable structures](https://en.wikipedia.org/wiki/Exotic_sphere "Exotic sphere") on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or now.[\[2\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-rota-phenomenology-2): 172 In contrast, Monastyrsky wrote in 2001: > It is very difficult to find an analogous invention in the past to [Milnor](https://en.wikipedia.org/wiki/Milnor "Milnor")'s beautiful construction of the different differential structures on the seven-dimensional sphere... The original proof of Milnor was not very constructive, but later E. Briscorn showed that these differential structures can be described in an extremely explicit and beautiful form.[\[24\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-24): 44 This disagreement illustrates both the subjective nature of mathematical beauty, [like other forms of beauty in general](https://en.wikipedia.org/wiki/Beauty "Beauty"), and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them. ### Proofs \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=8 "Edit section: Proofs")\] Besides Hardy's properties of "unexpectedness", "inevitability", "economy", which he applied to proofs as well as results, mathematicians have customarily thought beautiful proofs that are short and simple.[\[25\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-25): 22 In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly the [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem"), with hundreds of proofs having been published.[\[26\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-26) Another theorem that has been proved in many different ways is the theorem of [quadratic reciprocity](https://en.wikipedia.org/wiki/Quadratic_reciprocity "Quadratic reciprocity"). In fact, [Carl Friedrich Gauss](https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss "Carl Friedrich Gauss") alone had eight different proofs of this theorem, six of which he published.[\[27\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-27) In contrast, results that are logically correct but involve laborious calculations or consideration of many cases, are not usually considered beautiful, and may be even referred to as ugly or clumsy. For example, [Kenneth Appel](https://en.wikipedia.org/wiki/Kenneth_Appel "Kenneth Appel") and [Wolfgang Haken](https://en.wikipedia.org/wiki/Wolfgang_Haken "Wolfgang Haken")'s proof of the [four color theorem](https://en.wikipedia.org/wiki/Four_color_theorem "Four color theorem") made use of computer checking of over a thousand cases. [Philip J. Davis](https://en.wikipedia.org/wiki/Philip_J._Davis "Philip J. Davis") and [Reuben Hersh](https://en.wikipedia.org/wiki/Reuben_Hersh "Reuben Hersh") said that when they first heard that about the proof, they hoped it contained a new insight "whose beauty would transform my day", and were disheartened when informed the proof was by case enumeration and computer verification.[\[28\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-davis-experience-28): 384 [Paul Erdős](https://en.wikipedia.org/wiki/Paul_Erd%C5%91s "Paul Erdős") said it was "not beautiful" because it gave no insight into why the theorem was true.[\[29\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-29): 44 ## Philosophical analysis \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=9 "Edit section: Philosophical analysis")\] [Aristotle](https://en.wikipedia.org/wiki/Aristotle "Aristotle") thought that beauty was found especially in mathematics, writing in the [Metaphysics](https://en.wikipedia.org/wiki/Metaphysics_\(Aristotle\) "Metaphysics (Aristotle)") that > those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a very great deal about them; for if they do not expressly mention them, but prove attributes which are their results or their formulae, it is not true to say that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.[\[30\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-30): 1078a32–35 The logician and philosopher [Bertrand Russell](https://en.wikipedia.org/wiki/Bertrand_Russell "Bertrand Russell") made a now-famous statement characterizing mathematical beauty in terms of purity and austerity: > Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.[\[31\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-31) In the twentieth century, some philosophers questioned whether there was genuinely beauty in mathematics. The philosopher of science [Rom Harré](https://en.wikipedia.org/wiki/Rom_Harr%C3%A9 "Rom Harré") argued that there were no true aesthetic appraisals of mathematics, but only quasi-aesthetic appraisals. Any mathematical success described by an aesthetic term was a second-order success besides understanding and correctness. In contrast, aesthetic appraisal of a work of art was first-order. Harré considered this to be the difference between a quasi-aesthetic and a genuinely aesthetic appraisal.[\[32\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-32) [Nick Zangwill](https://en.wikipedia.org/wiki/Nick_Zangwill "Nick Zangwill") thought that there were no true aesthetic experiences of mathematics and that a proofs or theories could only be metaphorically beautiful. His argument had two bases. First, he thought that aesthetic properties depended on sensory properties, and so abstract entities could not have aesthetic properties. Second, he thought that proofs, theorems, theories, and so on had purposes such as demonstrating correctness or granting understanding, and that any praise of them reflected only how well they achieved their purpose.[\[33\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-33): 140–142 ## Scientific analysis \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=10 "Edit section: Scientific analysis")\] ### Information-theory model \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=11 "Edit section: Information-theory model")\] In the 1970s, [Abraham Moles](https://en.wikipedia.org/wiki/Abraham_Moles "Abraham Moles") and [Frieder Nake](https://en.wikipedia.org/wiki/Frieder_Nake "Frieder Nake") analyzed links between beauty, [information processing](https://en.wikipedia.org/wiki/Data_processing "Data processing"), and [information theory](https://en.wikipedia.org/wiki/Information_theory "Information theory").[\[34\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-34)[\[35\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-35) In the 1990s, [Jürgen Schmidhuber](https://en.wikipedia.org/wiki/J%C3%BCrgen_Schmidhuber "Jürgen Schmidhuber") formulated a mathematical theory of observer-dependent subjective beauty based on [algorithmic information theory](https://en.wikipedia.org/wiki/Algorithmic_information_theory "Algorithmic information theory"): the most beautiful objects among subjectively comparable objects have short [algorithmic](https://en.wikipedia.org/wiki/Algorithmic_information_theory "Algorithmic information theory") descriptions (i.e., [Kolmogorov complexity](https://en.wikipedia.org/wiki/Kolmogorov_complexity "Kolmogorov complexity")) relative to what the observer already knows.[\[36\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-36)[\[37\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-37)[\[38\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-38) Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the [first derivative](https://en.wikipedia.org/wiki/First_derivative "First derivative") of subjectively perceived beauty: the observer continually tries to improve the [predictability](https://en.wikipedia.org/wiki/Predictability "Predictability") and [compressibility](https://en.wikipedia.org/wiki/Data_compression "Data compression") of the observations by discovering regularities such as repetitions and [symmetries](https://en.wikipedia.org/wiki/Symmetries "Symmetries") and [fractal](https://en.wikipedia.org/wiki/Fractal "Fractal") [self-similarity](https://en.wikipedia.org/wiki/Self-similarity "Self-similarity"). Whenever the observer's learning process (possibly a predictive artificial [neural network](https://en.wikipedia.org/wiki/Neural_network "Neural network")) leads to improved data compression such that the observation sequence can be described by fewer [bits](https://en.wikipedia.org/wiki/Bit "Bit") than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.[\[39\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-39)[\[40\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-40) ### Neural correlates \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=12 "Edit section: Neural correlates")\] Brain imaging experiments conducted by [Semir Zeki](https://en.wikipedia.org/wiki/Semir_Zeki "Semir Zeki"), [Michael Atiyah](https://en.wikipedia.org/wiki/Michael_Atiyah "Michael Atiyah") and collaborators show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial [orbito-frontal cortex](https://en.wikipedia.org/wiki/Orbitofrontal_cortex "Orbitofrontal cortex") (mOFC) of the brain and that this activity is parametrically related to the declared intensity of beauty. The location of the activity is similar to the location of the activity that correlates with the experience of beauty from other sources, such as visual art or music.[\[41\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-41) Moreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers.[\[42\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-42) ## Mathematical beauty and the arts \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=13 "Edit section: Mathematical beauty and the arts")\] Main articles: [Mathematics and art](https://en.wikipedia.org/wiki/Mathematics_and_art "Mathematics and art"), [Mathematics and music](https://en.wikipedia.org/wiki/Mathematics_and_music "Mathematics and music"), and [Mathematics and architecture](https://en.wikipedia.org/wiki/Mathematics_and_architecture "Mathematics and architecture") ### Music \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=14 "Edit section: Music")\] Examples of the use of mathematics in music include the [stochastic music](https://en.wikipedia.org/wiki/Stochastic_music "Stochastic music") of [Iannis Xenakis](https://en.wikipedia.org/wiki/Iannis_Xenakis "Iannis Xenakis"),[\[43\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-43) the [Fibonacci sequence](https://en.wikipedia.org/wiki/Fibonacci_number "Fibonacci number") in [Tool](https://en.wikipedia.org/wiki/Tool_\(band\) "Tool (band)")'s [Lateralus](https://en.wikipedia.org/wiki/Lateralus_\(song\) "Lateralus (song)"),[\[44\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-44) and the [Metric modulation](https://en.wikipedia.org/wiki/Metric_modulation "Metric modulation") of [Elliott Carter](https://en.wikipedia.org/wiki/Elliott_Carter "Elliott Carter").[\[45\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-45) Other instances include the counterpoint of [Johann Sebastian Bach](https://en.wikipedia.org/wiki/Johann_Sebastian_Bach "Johann Sebastian Bach"), [polyrhythmic](https://en.wikipedia.org/wiki/Polyrhythm "Polyrhythm") structures (as in [Igor Stravinsky](https://en.wikipedia.org/wiki/Igor_Stravinsky "Igor Stravinsky")'s [The Rite of Spring](https://en.wikipedia.org/wiki/The_Rite_of_Spring "The Rite of Spring")), [permutation](https://en.wikipedia.org/wiki/Permutation "Permutation") theory in [serialism](https://en.wikipedia.org/wiki/Serialism "Serialism") beginning with [Arnold Schoenberg](https://en.wikipedia.org/wiki/Arnold_Schoenberg "Arnold Schoenberg"), the application of Shepard tones in [Karlheinz Stockhausen](https://en.wikipedia.org/wiki/Karlheinz_Stockhausen "Karlheinz Stockhausen")'s [Hymnen](https://en.wikipedia.org/wiki/Hymnen "Hymnen") and the application of [Group theory](https://en.wikipedia.org/wiki/Group_theory "Group theory") to transformations in music in the theoretical writings of [David Lewin](https://en.wikipedia.org/wiki/David_Lewin "David Lewin"). ### Visual arts \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=15 "Edit section: Visual arts")\] \| \| \| \|---\|---\| \| [![icon](https://upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/60px-Question_book-new.svg.png)](https://en.wikipedia.org/wiki/File:Question_book-new.svg) \| This section needs additional citations for [verification](https://en.wikipedia.org/wiki/Wikipedia:Verifiability "Wikipedia:Verifiability"). Please help [improve this article](https://en.wikipedia.org/wiki/Special:EditPage/Mathematical_beauty "Special:EditPage/Mathematical beauty") by [adding citations to reliable sources](https://en.wikipedia.org/wiki/Help:Referencing_for_beginners "Help:Referencing for beginners") in this section. Unsourced material may be challenged and removed. (September 2025) ([Learn how and when to remove this message](https://en.wikipedia.org/wiki/Help:Maintenance_template_removal "Help:Maintenance template removal")) \| [![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/38/Della_Pittura_Alberti_perspective_pillars_on_grid.jpg/250px-Della_Pittura_Alberti_perspective_pillars_on_grid.jpg)](https://en.wikipedia.org/wiki/File:Della_Pittura_Alberti_perspective_pillars_on_grid.jpg) Diagram from [Leon Battista Alberti](https://en.wikipedia.org/wiki/Leon_Battista_Alberti "Leon Battista Alberti")'s 1435 [Della Pittura](https://en.wikipedia.org/wiki/Della_Pittura "Della Pittura"), with pillars in [perspective](https://en.wikipedia.org/wiki/Perspective_\(graphical\) "Perspective (graphical)") on a grid Examples of the use of mathematics in the visual arts include applications of [chaos theory](https://en.wikipedia.org/wiki/Chaos_theory "Chaos theory") and [fractal](https://en.wikipedia.org/wiki/Fractal "Fractal") geometry to [computer-generated art](https://en.wikipedia.org/wiki/Digital_art "Digital art"), symmetry studies of [Leonardo da Vinci](https://en.wikipedia.org/wiki/Leonardo_da_Vinci "Leonardo da Vinci"), [projective geometries](https://en.wikipedia.org/wiki/Projective_geometry "Projective geometry") in development of the [perspective](https://en.wikipedia.org/wiki/Perspective_\(graphical\) "Perspective (graphical)") theory of [Renaissance](https://en.wikipedia.org/wiki/Renaissance "Renaissance") art, [grids](https://en.wikipedia.org/wiki/Grid_\(page_layout\) "Grid (page layout)") in [Op art](https://en.wikipedia.org/wiki/Op_art "Op art"), optical geometry in the [camera obscura](https://en.wikipedia.org/wiki/Camera_obscura "Camera obscura") of [Giambattista della Porta](https://en.wikipedia.org/wiki/Giambattista_della_Porta "Giambattista della Porta"), and multiple perspective in analytic [cubism](https://en.wikipedia.org/wiki/Cubism "Cubism") and [futurism](https://en.wikipedia.org/wiki/Futurism "Futurism"). [Sacred geometry](https://en.wikipedia.org/wiki/Sacred_geometry "Sacred geometry") is a field of its own, giving rise to countless art forms including some of the best known mystic symbols and religious motifs, and has a particularly rich history in [Islamic architecture](https://en.wikipedia.org/wiki/Islamic_architecture "Islamic architecture"). It also provides a means of meditation and comtemplation, for example the study of the [Kaballah](https://en.wikipedia.org/wiki/Kaballah "Kaballah") [Sefirot](https://en.wikipedia.org/wiki/Sefirot "Sefirot") (Tree Of Life) and [Metatron's Cube](https://en.wikipedia.org/wiki/Metatron%27s_Cube "Metatron's Cube"); and also the act of drawing itself. The Dutch graphic designer [M. C. Escher](https://en.wikipedia.org/wiki/M._C._Escher "M. C. Escher") created mathematically inspired [woodcuts](https://en.wikipedia.org/wiki/Woodcut "Woodcut"), [lithographs](https://en.wikipedia.org/wiki/Lithograph "Lithograph"), and [mezzotints](https://en.wikipedia.org/wiki/Mezzotint "Mezzotint"). These feature impossible constructions, explorations of [infinity](https://en.wikipedia.org/wiki/Infinity "Infinity"), architecture, visual [paradoxes](https://en.wikipedia.org/wiki/Paradox "Paradox") and [tessellations](https://en.wikipedia.org/wiki/Tessellation "Tessellation"). Some painters and sculptors create work distorted with the mathematical principles of [anamorphosis](https://en.wikipedia.org/wiki/Anamorphosis "Anamorphosis"), including South African sculptor [Jonty Hurwitz](https://en.wikipedia.org/wiki/Jonty_Hurwitz "Jonty Hurwitz"). [Origami](https://en.wikipedia.org/wiki/Origami "Origami"), the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the [mathematics of paper folding](https://en.wikipedia.org/wiki/Mathematics_of_paper_folding "Mathematics of paper folding") by observing the [crease pattern](https://en.wikipedia.org/wiki/Crease_pattern "Crease pattern") on unfolded origami pieces.[\[46\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-46) British constructionist artist [John Ernest](https://en.wikipedia.org/wiki/John_Ernest "John Ernest") created reliefs and paintings inspired by group theory.[\[47\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-47) A number of other British artists of the constructionist and systems schools of thought also draw on mathematics models and structures as a source of inspiration, including [Anthony Hill](https://en.wikipedia.org/wiki/Anthony_Hill_\(artist\) "Anthony Hill (artist)") and [Peter Lowe](https://en.wikipedia.org/wiki/Peter_Lowe_\(artist\) "Peter Lowe (artist)").[\[48\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-48) Computer-generated art is based on mathematical [algorithms](https://en.wikipedia.org/wiki/Algorithm "Algorithm"). ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=16 "Edit section: See also")\] - [Argument from beauty](https://en.wikipedia.org/wiki/Argument_from_beauty "Argument from beauty") - [Fluency heuristic](https://en.wikipedia.org/wiki/Fluency_heuristic "Fluency heuristic") - [Golden ratio](https://en.wikipedia.org/wiki/Golden_ratio "Golden ratio") - [Neuroesthetics](https://en.wikipedia.org/wiki/Neuroesthetics "Neuroesthetics") - [Philosophy of mathematics](https://en.wikipedia.org/wiki/Philosophy_of_mathematics "Philosophy of mathematics") - [Processing fluency theory of aesthetic pleasure](https://en.wikipedia.org/wiki/Processing_fluency_theory_of_aesthetic_pleasure "Processing fluency theory of aesthetic pleasure") - [Pythagoreanism](https://en.wikipedia.org/wiki/Pythagoreanism "Pythagoreanism") - [Sense of wonder](https://en.wikipedia.org/wiki/Sense_of_wonder "Sense of wonder") - [Theory of everything](https://en.wikipedia.org/wiki/Theory_of_everything "Theory of everything") ## Notes \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=17 "Edit section: Notes")\] 1. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-1) Phillips, George (2005). ["Preface"](https://books.google.com/books?id=psFwdN6V6icC&q=there+is+nothing+in+the+world+of+mathematics+that+corresponds+to+an+audience+in+a+concert+hall,+where+the+passive+listen+to+the+active.+Happily,+mathematicians+are+all+doers,+not+spectators.&pg=PR7). Mathematics Is Not a Spectator Sport. [Springer Science+Business Media](https://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Media "Springer Science+Business Media"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-387-25528-1](https://en.wikipedia.org/wiki/Special:BookSources/0-387-25528-1 "Special:BookSources/0-387-25528-1") . Retrieved 2008-08-22. ""...there is nothing in the world of mathematics that corresponds to an audience in a concert hall, where the passive listen to the active. Happily, mathematicians are all doers, not spectators." 2. ^ [*a](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-rota-phenomenology_2-0) [b](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-rota-phenomenology_2-1) Rota, Gian-Carlo (May 1997). "The Phenomenology of Mathematical Beauty". Synthese. 111* (2): 171–182\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1023/A:1004930722234](https://doi.org/10.1023%2FA%3A1004930722234). 3. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-3) Wilson, Robin (2018). Euler's Pioneering Equation: The most beautiful theorem in mathematics. Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-19-879492-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-879492-9 "Special:BookSources/978-0-19-879492-9") . 4. ^ [*a](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-cain-form_4-0) [b](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-cain-form_4-1) [c](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-cain-form_4-2) Cain, Alan J. (2024). [Form & Number: A History of Mathematical Beauty](https://archive.org/details/cain_formandnumber_ebook_large). Lisbon: EBook. 5. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-5)* Gallagher, James (13 February 2014). ["Mathematics: Why the brain sees maths as beauty"](https://www.bbc.co.uk/news/science-environment-26151062). BBC News online. Retrieved 13 February 2014. 6. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-6) Feynman, Richard P. (1977). [The Feynman Lectures on Physics](https://feynmanlectures.caltech.edu/I_22.html#Ch22-S6-p4). Vol. I. Addison-Wesley. p. 22-16. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-201-02010-6](https://en.wikipedia.org/wiki/Special:BookSources/0-201-02010-6 "Special:BookSources/0-201-02010-6") . 7. ^ [*a](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-hardy-apology_7-0) [b](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-hardy-apology_7-1) [c](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-hardy-apology_7-2) Hardy, G.H. (1967). [A Mathematician's Apology](https://archive.org/details/hardy_annotated). Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-107-60463-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-60463-6 "Special:BookSources/978-1-107-60463-6") . 8. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-8)* Bell, E.T. (1937). Men of Mathematics. Simon & Schuster. 9. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-9) Wells, David (June 1990). "Are these the most beautiful?". The Mathematical Intelligencer. 12 (3): 37–41\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF03024015](https://doi.org/10.1007%2FBF03024015). 10. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-10) Paulos, John Allen (1991). Beyond Numeracy. New York: Alfred A. Knopf. pp. 125–127\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-394-586409](https://en.wikipedia.org/wiki/Special:BookSources/0-394-586409 "Special:BookSources/0-394-586409") . 11. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-11) Dutilh Novaes, Catarina (2019). "The Beauty (?) of Mathematical Proofs". In Aberdein, Andrew; Inglis, Matthew (eds.). Advances in Experimental Philosophy of Logic and Mathematics. Bloomsbury Academic. pp. 69–71\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-350-03901-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-350-03901-8 "Special:BookSources/978-1-350-03901-8") . 12. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-12) Polster, Burkert (2004). Q.E.D.: Beauty in Mathematical Proof. Walker & Company. pp. 32–33\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-8027-1431-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8027-1431-2 "Special:BookSources/978-0-8027-1431-2") . 13. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-13) Schechter, Bruce (2000). My Brain is Open: The Mathematical Journeys of Paul Erdős. New York: [Simon & Schuster](https://en.wikipedia.org/wiki/Simon_%26_Schuster "Simon & Schuster"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-684-85980-7](https://en.wikipedia.org/wiki/Special:BookSources/0-684-85980-7 "Special:BookSources/0-684-85980-7") . 14. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-14) [Aigner, Martin](https://en.wikipedia.org/wiki/Martin_Aigner "Martin Aigner"); [Ziegler, Günter M.](https://en.wikipedia.org/wiki/G%C3%BCnter_M._Ziegler "Günter M. Ziegler") (2018). [Proofs from THE BOOK](https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK "Proofs from THE BOOK") (6th ed.). Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-662-57264-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-662-57264-1 "Special:BookSources/978-3-662-57264-1") . 15. ^ [*a](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-plato-timaeus_15-0) [b](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-plato-timaeus_15-1) Plato (1929). [Timaeus](https://www.loebclassics.com/view/LCL234/1929/volume.xml). Cambridge, MA: Harvard University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-674-99257-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-674-99257-3 "Special:BookSources/978-0-674-99257-3") . `{{cite book}}`: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date "Help:CS1 errors")) 16. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-16)* Field, J.V. (2013). Kepler's Geometrical Cosmology. Bloomsbury. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9781472507037](https://en.wikipedia.org/wiki/Special:BookSources/9781472507037 "Special:BookSources/9781472507037") . 17. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-17) Whitfield, John. ["Journey to the 248th dimension"](https://www.nature.com/articles/news070319-4). Nature Online. Retrieved 12 September 2025. 18. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-18) Penrose, Roger (1974). "The Rôle of Aesthetics in Pure and Applied Mathematical Research". Bulletin of the Institute of Mathematics and its Applications. 10: 226–271\. 19. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-19) [Chandrasekhar, Subrahmanyan](https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar "Subrahmanyan Chandrasekhar") (1987). Truth and Beauty: Aesthetics and Motivation in Science. Chicago, London: University of Chicago Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-226-10087-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-226-10087-6 "Special:BookSources/978-0-226-10087-6") . 20. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-20) Dirac, P.A.M. (1940). "The Relation between Mathematics and Physics". Proceedings of the Royal Society of Edinburgh. 59: 122–129\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1017/s0370164600012207](https://doi.org/10.1017%2Fs0370164600012207). 21. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-21) Penrose, Roger (2004). The Road to Reality. London: Jonathan Cape. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-224-04447-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-224-04447-9 "Special:BookSources/978-0-224-04447-9") . 22. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-22) McAllister, James W. (1996). [Beauty & Revolution in Science](https://hdl.handle.net/1887/10158). Ithaca: Cornell University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-8014-3240-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8014-3240-8 "Special:BookSources/978-0-8014-3240-8") . 23. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-23) [Devlin, Keith](https://en.wikipedia.org/wiki/Keith_Devlin "Keith Devlin") (2000). "Do Mathematicians Have Different Brains?". [The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip](https://archive.org/details/mathgene00keit). [Basic Books](https://en.wikipedia.org/wiki/Basic_Books "Basic Books"). p. [140](https://archive.org/details/mathgene00keit/page/140). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-465-01619-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-465-01619-8 "Special:BookSources/978-0-465-01619-8") . Retrieved 2008-08-22. 24. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-24) Monastyrsky, Michael (2001). ["Some Trends in Modern Mathematics and the Fields Medal"](http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf) (PDF). Can. Math. Soc. Notes. 33 (2 and 3). 25. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-25) McAllister, James W. (2005). "Mathematical Beauty and the Evolution of the Standards of Mathematical Proof". In Emmer, Michele (ed.). The Visual Mind II. MIT Press. pp. 15–34\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-262-05076-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-262-05076-0 "Special:BookSources/978-0-262-05076-0") . 26. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-26) [Loomis, Elisha Scott](https://en.wikipedia.org/wiki/Elisha_Scott_Loomis "Elisha Scott Loomis") (1968). The Pythagorean Proposition. Washington, DC: National Council of Teachers of Mathematics. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-873-53036-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-873-53036-1 "Special:BookSources/978-0-873-53036-1") . 27. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-27) Weisstein, Eric W. ["Quadratic Reciprocity Theorem"](http://mathworld.wolfram.com/QuadraticReciprocityTheorem.html). mathworld.wolfram.com. Retrieved 2019-10-31. 28. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-davis-experience_28-0) Davis, Philip J.; Hersh, Reuben (1981). The Mathematical Experience. Boston: Houghton Mifflin. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-395-32131-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-395-32131-7 "Special:BookSources/978-0-395-32131-7") . 29. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-29) Hoffman, Paul (1999). The Man Who Loved Only Numbers. London: Fourth Estate. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-85702-829-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-85702-829-4 "Special:BookSources/978-1-85702-829-4") . 30. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-30) [Aristotle](https://en.wikipedia.org/wiki/Aristotle "Aristotle") (1995). "Metaphysics". In Barnes, Jonathan (ed.). The Complete Works of Aristotle. Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-691-01650-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-01650-4 "Special:BookSources/978-0-691-01650-4") . 31. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-31) [Russell, Bertrand](https://en.wikipedia.org/wiki/Bertrand_Russell "Bertrand Russell") (1919). "The Study of Mathematics". [Mysticism and Logic: And Other Essays](https://archive.org/details/bub_gb_zwMQAAAAYAAJ). [Longman](https://en.wikipedia.org/wiki/Longman "Longman"). p. [60](https://archive.org/details/bub_gb_zwMQAAAAYAAJ/page/n68). Retrieved 2008-08-22. "Mathematics rightly viewed possesses not only truth but supreme beauty a beauty cold and austere like that of sculpture without appeal to any part of our weaker nature without the gorgeous trappings Russell." 32. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-32) [Harré, Rom](https://en.wikipedia.org/wiki/Rom_Harr%C3%A9 "Rom Harré") (1958). ["Quasi-Aesthetic Appraisals"](https://www.jstor.org/stable/3748562). Philosophy. 33: 132–137\. 33. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-33) [Zangwill, Nick](https://en.wikipedia.org/wiki/Nick_Zangwill "Nick Zangwill") (2001). The Metaphysics of Beauty. Ithaca, London: Cornell University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-8014-3820-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8014-3820-2 "Special:BookSources/978-0-8014-3820-2") . 34. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-34) A. Moles: Théorie de l'information et perception esthétique, Paris, Denoël, 1973 ([Information Theory](https://en.wikipedia.org/wiki/Information_Theory "Information Theory") and aesthetical perception) 35. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-35) F Nake (1974). Ästhetik als Informationsverarbeitung. ([Aesthetics](https://en.wikipedia.org/wiki/Aesthetics "Aesthetics") as information processing). Grundlagen und Anwendungen der Informatik im Bereich ästhetischer Produktion und Kritik. Springer, 1974, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [3-211-81216-4](https://en.wikipedia.org/wiki/Special:BookSources/3-211-81216-4 "Special:BookSources/3-211-81216-4") , [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-211-81216-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-211-81216-7 "Special:BookSources/978-3-211-81216-7") 36. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-36) J. Schmidhuber. [Low-complexity art](https://en.wikipedia.org/wiki/Low-complexity_art "Low-complexity art"). [Leonardo](https://en.wikipedia.org/wiki/Leonardo_\(journal\) "Leonardo (journal)"), Journal of the International Society for the Arts, Sciences, and Technology ([Leonardo/ISAST](https://en.wikipedia.org/wiki/Leonardo,_The_International_Society_of_the_Arts,_Sciences_and_Technology "Leonardo, The International Society of the Arts, Sciences and Technology")), 30(2):97–103, 1997. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2307/1576418](https://doi.org/10.2307%2F1576418). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [1576418](https://www.jstor.org/stable/1576418). 37. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-37) J. Schmidhuber. Papers on the theory of beauty and [low-complexity art](https://en.wikipedia.org/wiki/Low-complexity_art "Low-complexity art") since 1994: <http://www.idsia.ch/~juergen/beauty.html> 38. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-38) J. Schmidhuber. Simple Algorithmic Principles of Discovery, Subjective Beauty, Selective Attention, Curiosity & Creativity. Proc. 10th Intl. Conf. on Discovery Science (DS 2007) pp. 26–38, LNAI 4755, Springer, 2007. Also in Proc. 18th Intl. Conf. on Algorithmic Learning Theory (ALT 2007) p. 32, LNAI 4754, Springer, 2007. Joint invited lecture for DS 2007 and ALT 2007, Sendai, Japan, 2007. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[0709\.0674](https://arxiv.org/abs/0709.0674). 39. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-39) .J. Schmidhuber. Curious model-building control systems. International Joint Conference on Neural Networks, Singapore, vol 2, 1458–1463. IEEE press, 1991 40. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-40) Schmidhuber's theory of beauty and curiosity in a German TV show: <http://www.br-online.de/bayerisches-fernsehen/faszination-wissen/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml> [Archived](https://web.archive.org/web/20080603221058/http://www.br-online.de/bayerisches-fernsehen/faszination-wissen/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml) June 3, 2008, at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine") 41. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-41) Zeki, Semir; Romaya, John Paul; Benincasa, Dionigi M. T.; Atiyah, Michael F. (2014). ["The experience of mathematical beauty and its neural correlates"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3923150). Frontiers in Human Neuroscience. 8: 68. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3389/fnhum.2014.00068](https://doi.org/10.3389%2Ffnhum.2014.00068). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1662-5161](https://search.worldcat.org/issn/1662-5161). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [3923150](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3923150). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [24592230](https://pubmed.ncbi.nlm.nih.gov/24592230). 42. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-42) Zhang, Haoxuan; Zeki, Semir (May 2022). ["Judgments of mathematical beauty are resistant to revision through external opinion"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9790661). PsyCh Journal. 11 (5): 741–747\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/pchj.556](https://doi.org/10.1002%2Fpchj.556). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [2046-0252](https://search.worldcat.org/issn/2046-0252). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [9790661](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9790661). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [35491015](https://pubmed.ncbi.nlm.nih.gov/35491015). 43. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-43) [Luque, Sergio. 2009. "The Stochastic Synthesis of Iannis Xenakis." Leonardo Music Journal (19): 77–84](http://www.sergioluque.com/stochastics) 44. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-44) Norris, Chris (2001). ["Hammer Of The Gods"](https://web.archive.org/web/20111101195240/http://toolshed.down.net/articles/text/spinmag.jun.2001.html). Archived from [the original](http://toolshed.down.net/articles/text/spinmag.jun.2001.html) on November 1, 2011. Retrieved April 25, 2007. 45. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-45) Schell, Michael (December 11, 2018). ["Elliott Carter (1908–2012): Legacy of a Centenarian"](https://www.secondinversion.org/2018/12/11/elliott-carter-1908-2012-legacy-of-a-centenarian/). Second Inversion. [Archived](https://web.archive.org/web/20181224054514/https://www.secondinversion.org/2018/12/11/elliott-carter-1908-2012-legacy-of-a-centenarian/) from the original on December 24, 2018. Retrieved December 11, 2018. 46. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-46) Hull, Thomas. "Project Origami: Activities for Exploring Mathematics". Taylor & Francis, 2006. 47. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-47) John Ernest's use of mathematics and especially group theory in his art works is analysed in John Ernest, A Mathematical Artist by Paul Ernest in Philosophy of Mathematics Education Journal, No. 24 Dec. 2009 (Special Issue on Mathematics and Art): <http://people.exeter.ac.uk/PErnest/pome24/index.htm> 48. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-48) Franco, Francesca (2017-10-05). ["The Systems Group (Chapter 2)"](https://books.google.com/books?id=oJU4DwAAQBAJ&q=anthony+hill+and+peter+lowe&pg=PT61). Generative Systems Art: The Work of Ernest Edmonds. Routledge. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9781317137436](https://en.wikipedia.org/wiki/Special:BookSources/9781317137436 "Special:BookSources/9781317137436") . ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=18 "Edit section: References")\] - Cellucci, Carlo (2015), "Mathematical beauty, understanding, and discovery", [Foundations of Science](https://en.wikipedia.org/wiki/Foundations_of_Science "Foundations of Science"), 20 (4): 339–355, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/s10699-014-9378-7](https://doi.org/10.1007%2Fs10699-014-9378-7), [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [120068870](https://api.semanticscholar.org/CorpusID:120068870) - [Hoffman, Paul](https://en.wikipedia.org/wiki/Paul_Hoffman_\(science_writer\) "Paul Hoffman (science writer)") (1992), [The Man Who Loved Only Numbers](https://en.wikipedia.org/wiki/The_Man_Who_Loved_Only_Numbers "The Man Who Loved Only Numbers"), Hyperion. - [Loomis, Elisha Scott](https://en.wikipedia.org/wiki/Elisha_Scott_Loomis "Elisha Scott Loomis") (1968), The Pythagorean Proposition, The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem. - [Pandey, S.K.](https://en.wikipedia.org/w/index.php?title=S.K._Pandey&action=edit&redlink=1 "S.K. Pandey (page does not exist)") . [The Humming of Mathematics: Melody of Mathematics](https://books.google.com/books?id=BgYbzAEACAAJ). Independently Published, 2019. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [1710134437](https://en.wikipedia.org/wiki/Special:BookSources/1710134437 "Special:BookSources/1710134437") . - Peitgen, H.-O., and Richter, P.H. (1986), The Beauty of Fractals, Springer-Verlag. - [Reber, R.](https://en.wikipedia.org/wiki/Rolf_Reber "Rolf Reber"); Brun, M.; Mitterndorfer, K. (2008). "The use of heuristics in intuitive mathematical judgment". Psychonomic Bulletin & Review. 15 (6): 1174–1178\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3758/PBR.15.6.1174](https://doi.org/10.3758%2FPBR.15.6.1174). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[1956/2734](https://hdl.handle.net/1956%2F2734). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [19001586](https://pubmed.ncbi.nlm.nih.gov/19001586). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [5297500](https://api.semanticscholar.org/CorpusID:5297500). - Strohmeier, John, and Westbrook, Peter (1999), Divine Harmony, The Life and Teachings of Pythagoras, Berkeley Hills Books, Berkeley, CA. - [Zeki, S.](https://en.wikipedia.org/wiki/Semir_Zeki "Semir Zeki"); Romaya, J. P.; Benincasa, D. M. T.; [Atiyah, M. F.](https://en.wikipedia.org/wiki/Michael_Atiyah "Michael Atiyah") (2014), "The experience of mathematical beauty and its neural correlates", Frontiers in Human Neuroscience, 8: 68, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3389/fnhum.2014.00068](https://doi.org/10.3389%2Ffnhum.2014.00068), [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [3923150](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3923150), [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [24592230](https://pubmed.ncbi.nlm.nih.gov/24592230) ## Further reading \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=19 "Edit section: Further reading")\] - [Aigner, Martin](https://en.wikipedia.org/wiki/Martin_Aigner "Martin Aigner"); [Ziegler, Günter M.](https://en.wikipedia.org/wiki/G%C3%BCnter_M._Ziegler "Günter M. Ziegler") (2018). [Proofs from THE BOOK](https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK "Proofs from THE BOOK") (6th ed.). Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-662-57264-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-662-57264-1 "Special:BookSources/978-3-662-57264-1") . - Cain, Alan J. (2024). [Form & Number: A History of Mathematical Beauty](https://archive.org/details/cain_formandnumber_ebook_large). Lisbon: Ebook. - [Hadamard, Jacques](https://en.wikipedia.org/wiki/Jacques_Hadamard "Jacques Hadamard") (1949). [An Essay on the Psychology of Invention in the Mathematical Field](https://archive.org/details/essayonpsycholog00hada) (2nd enlarged ed.). Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-486-20107-4](https://en.wikipedia.org/wiki/Special:BookSources/0-486-20107-4 "Special:BookSources/0-486-20107-4") . `{{cite book}}`: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date "Help:CS1 errors")) - [Hardy, G.H.](https://en.wikipedia.org/wiki/G.H._Hardy "G.H. Hardy") (1967) \[1st published 1940\]. [A Mathematician's Apology](https://archive.org/details/hardy_annotated). Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-107-60463-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-60463-6 "Special:BookSources/978-1-107-60463-6") . - Huntley, H.E. (1970). [The Divine Proportion: A Study in Mathematical Beauty](https://archive.org/details/divineproportion0000hunt_o2w9). New York: Dover Publications. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-486-22254-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-22254-7 "Special:BookSources/978-0-486-22254-7") . - Stewart, Ian (2007). Why beauty is truth : a history of symmetry. New York: Basic Books, a member of the Perseus Books Group. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-465-08236-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-465-08236-0 "Special:BookSources/978-0-465-08236-0") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [76481488](https://search.worldcat.org/oclc/76481488). ## External links \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=20 "Edit section: External links")\] - [Mathematics, Poetry and Beauty](https://web.archive.org/web/20151209130947/http://raharoni.net.technion.ac.il/mathematics-poetry-and-beauty/) - [Is Mathematics Beautiful?](http://www.cut-the-knot.org/manifesto/beauty.shtml) cut-the-knot.org - [Justin Mullins.com](http://www.justinmullins.com/) - [Edna St. Vincent Millay (poet): Euclid alone has looked on beauty bare](http://www.the-athenaeum.org/poetry/detail.php?id=80) - [Terence Tao](https://en.wikipedia.org/wiki/Terence_Tao "Terence Tao"), [What is good mathematics?](https://arxiv.org/abs/math/0702396) - [Mathbeauty Blog](http://mathbeauty.wordpress.com/) - The [Aesthetic Appeal](https://archive.org/details/aestheticappeal) collection at the [Internet Archive](https://en.wikipedia.org/wiki/Internet_Archive "Internet Archive") - [A Mathematical Romance](http://www.nybooks.com/articles/archives/2013/dec/05/mathematical-romance/) [Jim Holt](https://en.wikipedia.org/wiki/Jim_Holt_\(philosopher\) "Jim Holt (philosopher)") December 5, 2013 issue of [The New York Review of Books](https://en.wikipedia.org/wiki/The_New_York_Review_of_Books "The New York Review of Books") review of Love and Math: The Heart of Hidden Reality by [Edward Frenkel](https://en.wikipedia.org/wiki/Edward_Frenkel "Edward Frenkel") \| [v](https://en.wikipedia.org/wiki/Template:Aesthetics "Template:Aesthetics") [t](https://en.wikipedia.org/wiki/Template_talk:Aesthetics "Template talk:Aesthetics") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Aesthetics "Special:EditPage/Template:Aesthetics")[Aesthetics](https://en.wikipedia.org/wiki/Aesthetics "Aesthetics") \| \| \|---\|---\| \| Areas \| [African](https://en.wikipedia.org/wiki/African_aesthetic "African aesthetic") [Ancient](https://en.wikipedia.org/wiki/Ancient_aesthetics "Ancient aesthetics") [Indian](https://en.wikipedia.org/wiki/Indian_aesthetics "Indian aesthetics") [Japanese](https://en.wikipedia.org/wiki/Japanese_aesthetics "Japanese aesthetics") [Mathematics](https://en.wikipedia.org/wiki/Mathematics_and_art "Mathematics and art") [Medieval](https://en.wikipedia.org/wiki/Medieval_aesthetics "Medieval aesthetics") [Music](https://en.wikipedia.org/wiki/Aesthetics_of_music "Aesthetics of music") [Nature](https://en.wikipedia.org/wiki/Aesthetics_of_nature "Aesthetics of nature") [Science](https://en.wikipedia.org/wiki/Aesthetics_of_science "Aesthetics of science") [Theology](https://en.wikipedia.org/wiki/Theological_aesthetics "Theological aesthetics") \| \| Schools \| [Aestheticism](https://en.wikipedia.org/wiki/Aestheticism "Aestheticism") [Classicism](https://en.wikipedia.org/wiki/Classicism "Classicism") [Fascism](https://en.wikipedia.org/wiki/Fascism#Aesthetics "Fascism") [Feminism](https://en.wikipedia.org/wiki/Feminist_aesthetics "Feminist aesthetics") [Formalism](https://en.wikipedia.org/wiki/Formalism_\(art\) "Formalism (art)") [Historicism](https://en.wikipedia.org/wiki/Historicism_\(art\) "Historicism (art)") [Marxism](https://en.wikipedia.org/wiki/Marxist_aesthetics "Marxist aesthetics") [Modernism](https://en.wikipedia.org/wiki/Modernism "Modernism") [Postmodernism](https://en.wikipedia.org/wiki/Postmodernism "Postmodernism") [Psychoanalysis](https://en.wikipedia.org/wiki/Psychoanalytic_theory "Psychoanalytic theory") [Realism](https://en.wikipedia.org/wiki/Aesthetic_Realism "Aesthetic Realism") [Romanticism](https://en.wikipedia.org/wiki/Romanticism "Romanticism") [Symbolism](https://en.wikipedia.org/wiki/Symbolism_\(movement\) "Symbolism (movement)") [Theosophy](https://en.wikipedia.org/wiki/Theosophy_and_visual_arts "Theosophy and visual arts") [more...](https://en.wikipedia.org/wiki/List_of_art_movements "List of art movements") \| \| Philosophers \| [Abhinavagupta](https://en.wikipedia.org/wiki/Abhinavagupta "Abhinavagupta") [Adorno](https://en.wikipedia.org/wiki/Theodor_W._Adorno "Theodor W. Adorno") [Alberti](https://en.wikipedia.org/wiki/Leon_Battista_Alberti "Leon Battista Alberti") [Akhundov](https://en.wikipedia.org/wiki/Mirza_Fatali_Akhundov "Mirza Fatali Akhundov") [Ibn Arabi](https://en.wikipedia.org/wiki/Ibn_Arabi "Ibn Arabi") [Aristotle](https://en.wikipedia.org/wiki/Aristotle "Aristotle") [Aquinas](https://en.wikipedia.org/wiki/Thomas_Aquinas "Thomas Aquinas") [Balázs](https://en.wikipedia.org/wiki/B%C3%A9la_Bal%C3%A1zs "Béla Balázs") [Balthasar](https://en.wikipedia.org/wiki/Hans_Urs_von_Balthasar "Hans Urs von Balthasar") [Bataille](https://en.wikipedia.org/wiki/Georges_Bataille "Georges Bataille") [Baudelaire](https://en.wikipedia.org/wiki/Charles_Baudelaire "Charles Baudelaire") [Baudrillard](https://en.wikipedia.org/wiki/Jean_Baudrillard "Jean Baudrillard") [Baumgarten](https://en.wikipedia.org/wiki/Alexander_Gottlieb_Baumgarten "Alexander Gottlieb Baumgarten") [Bell](https://en.wikipedia.org/wiki/Clive_Bell "Clive Bell") [Benjamin](https://en.wikipedia.org/wiki/Walter_Benjamin "Walter Benjamin") [Burke](https://en.wikipedia.org/wiki/Edmund_Burke "Edmund Burke") [Coleridge](https://en.wikipedia.org/wiki/Samuel_Taylor_Coleridge "Samuel Taylor Coleridge") [Collingwood](https://en.wikipedia.org/wiki/R._G._Collingwood "R. G. Collingwood") [Coomaraswamy](https://en.wikipedia.org/wiki/Ananda_Coomaraswamy "Ananda Coomaraswamy") [Danto](https://en.wikipedia.org/wiki/Arthur_Danto "Arthur Danto") [Deleuze](https://en.wikipedia.org/wiki/Gilles_Deleuze "Gilles Deleuze") [Dewey](https://en.wikipedia.org/wiki/John_Dewey "John Dewey") [Flaubert](https://en.wikipedia.org/wiki/Gustave_Flaubert "Gustave Flaubert") [Foucault](https://en.wikipedia.org/wiki/Michel_Foucault "Michel Foucault") [Fry](https://en.wikipedia.org/wiki/Roger_Fry "Roger Fry") [Goethe](https://en.wikipedia.org/wiki/Johann_Wolfgang_von_Goethe "Johann Wolfgang von Goethe") [Goodman](https://en.wikipedia.org/wiki/Nelson_Goodman "Nelson Goodman") [Gramsci](https://en.wikipedia.org/wiki/Antonio_Gramsci "Antonio Gramsci") [Greenberg](https://en.wikipedia.org/wiki/Clement_Greenberg "Clement Greenberg") [Hanslick](https://en.wikipedia.org/wiki/Eduard_Hanslick "Eduard Hanslick") [Hegel](https://en.wikipedia.org/wiki/Georg_Wilhelm_Friedrich_Hegel "Georg Wilhelm Friedrich Hegel") [Heidegger](https://en.wikipedia.org/wiki/Martin_Heidegger "Martin Heidegger") [Hume](https://en.wikipedia.org/wiki/David_Hume "David Hume") [Hutcheson](https://en.wikipedia.org/wiki/Francis_Hutcheson_\(philosopher\) "Francis Hutcheson (philosopher)") [Kant](https://en.wikipedia.org/wiki/Immanuel_Kant "Immanuel Kant") [Kierkegaard](https://en.wikipedia.org/wiki/S%C3%B8ren_Kierkegaard "Søren Kierkegaard") [Klee](https://en.wikipedia.org/wiki/Paul_Klee "Paul Klee") [Langer](https://en.wikipedia.org/wiki/Susanne_Langer "Susanne Langer") [Lipps](https://en.wikipedia.org/wiki/Theodor_Lipps "Theodor Lipps") [Liu](https://en.wikipedia.org/wiki/Liu_Xie "Liu Xie") [Lukács](https://en.wikipedia.org/wiki/Gy%C3%B6rgy_Luk%C3%A1cs "György Lukács") [Lyotard](https://en.wikipedia.org/wiki/Jean-Fran%C3%A7ois_Lyotard "Jean-François Lyotard") [de Man](https://en.wikipedia.org/wiki/Paul_de_Man "Paul de Man") [Marcuse](https://en.wikipedia.org/wiki/Herbert_Marcuse "Herbert Marcuse") [Maritain](https://en.wikipedia.org/wiki/Jacques_Maritain "Jacques Maritain") [Merleau-Ponty](https://en.wikipedia.org/wiki/Maurice_Merleau-Ponty "Maurice Merleau-Ponty") [Nietzsche](https://en.wikipedia.org/wiki/Friedrich_Nietzsche "Friedrich Nietzsche") [Ortega y Gasset](https://en.wikipedia.org/wiki/Jos%C3%A9_Ortega_y_Gasset "José Ortega y Gasset") [Orwell](https://en.wikipedia.org/wiki/George_Orwell "George Orwell") [Pater](https://en.wikipedia.org/wiki/Walter_Pater "Walter Pater") [Petrarch](https://en.wikipedia.org/wiki/Petrarch "Petrarch") [Plato](https://en.wikipedia.org/wiki/Plato "Plato") [Pythagoras](https://en.wikipedia.org/wiki/Pythagoras "Pythagoras") [Quintilian](https://en.wikipedia.org/wiki/Quintilian "Quintilian") [Rancière](https://en.wikipedia.org/wiki/Jacques_Ranci%C3%A8re "Jacques Rancière") [Rand](https://en.wikipedia.org/wiki/Ayn_Rand "Ayn Rand") [Richards](https://en.wikipedia.org/wiki/I._A._Richards "I. A. Richards") [Ruskin](https://en.wikipedia.org/wiki/John_Ruskin "John Ruskin") [Said](https://en.wikipedia.org/wiki/Edward_Said "Edward Said") [Santayana](https://en.wikipedia.org/wiki/George_Santayana "George Santayana") [Schiller](https://en.wikipedia.org/wiki/Friedrich_Schiller "Friedrich Schiller") [Schopenhauer](https://en.wikipedia.org/wiki/Arthur_Schopenhauer "Arthur Schopenhauer") [Scruton](https://en.wikipedia.org/wiki/Roger_Scruton "Roger Scruton") [Sontag](https://en.wikipedia.org/wiki/Susan_Sontag "Susan Sontag") [Tagore](https://en.wikipedia.org/wiki/Rabindranath_Tagore "Rabindranath Tagore") [Tanizaki](https://en.wikipedia.org/wiki/Jun%27ichir%C5%8D_Tanizaki "Jun'ichirō Tanizaki") [Tolkien](https://en.wikipedia.org/wiki/J._R._R._Tolkien "J. R. R. Tolkien") [Vasari](https://en.wikipedia.org/wiki/Giorgio_Vasari "Giorgio Vasari") [Wilde](https://en.wikipedia.org/wiki/Oscar_Wilde "Oscar Wilde") [Winckelmann](https://en.wikipedia.org/wiki/Johann_Joachim_Winckelmann "Johann Joachim Winckelmann") [Woolf](https://en.wikipedia.org/wiki/Virginia_Woolf "Virginia Woolf") [Zola](https://en.wikipedia.org/wiki/%C3%89mile_Zola "Émile Zola") [more...](https://en.wikipedia.org/wiki/List_of_philosophers_of_art "List of philosophers of art") \| \| Concepts \| [Apollonian and Dionysian](https://en.wikipedia.org/wiki/Apollonian_and_Dionysian "Apollonian and Dionysian") [Appropriation](https://en.wikipedia.org/wiki/Appropriation_\(art\) "Appropriation (art)") [Art for art's sake](https://en.wikipedia.org/wiki/Art_for_art%27s_sake "Art for art's sake") [Art manifesto](https://en.wikipedia.org/wiki/Art_manifesto "Art manifesto") [Artistic merit](https://en.wikipedia.org/wiki/Artistic_merit "Artistic merit") [Authenticity](https://en.wikipedia.org/wiki/Authenticity_in_art "Authenticity in art") [Avant-garde](https://en.wikipedia.org/wiki/Avant-garde "Avant-garde") [Beauty](https://en.wikipedia.org/wiki/Beauty "Beauty") [Feminine](https://en.wikipedia.org/wiki/Feminine_beauty_ideal "Feminine beauty ideal") [Masculine](https://en.wikipedia.org/wiki/Masculine_beauty_ideal "Masculine beauty ideal") [Cool](https://en.wikipedia.org/wiki/Cool_\(aesthetic\) "Cool (aesthetic)") [Camp](https://en.wikipedia.org/wiki/Camp_\(style\) "Camp (style)") [Comedy](https://en.wikipedia.org/wiki/Comedy "Comedy") [Creativity](https://en.wikipedia.org/wiki/Creativity "Creativity") [Cuteness](https://en.wikipedia.org/wiki/Cuteness "Cuteness") [Depiction](https://en.wikipedia.org/wiki/Depiction "Depiction") [Disgust](https://en.wikipedia.org/wiki/Disgust "Disgust") [Ecstasy](https://en.wikipedia.org/wiki/Ecstasy_\(philosophy\) "Ecstasy (philosophy)") [Elegance](https://en.wikipedia.org/wiki/Elegance "Elegance") [Emotions](https://en.wikipedia.org/wiki/Aesthetic_emotions "Aesthetic emotions") [Entertainment](https://en.wikipedia.org/wiki/Entertainment "Entertainment") [Eroticism](https://en.wikipedia.org/wiki/Eroticism "Eroticism") [Exoticism](https://en.wikipedia.org/wiki/Exoticism "Exoticism") [Fashion](https://en.wikipedia.org/wiki/Fashion "Fashion") [Gaze](https://en.wikipedia.org/wiki/Gaze "Gaze") [Harmony](https://en.wikipedia.org/wiki/Harmony "Harmony") [Hegemony](https://en.wikipedia.org/wiki/Cultural_hegemony "Cultural hegemony") [Imperialism](https://en.wikipedia.org/wiki/Cultural_imperialism "Cultural imperialism") [Humour](https://en.wikipedia.org/wiki/Humour "Humour") [Iconography](https://en.wikipedia.org/wiki/Iconography "Iconography") [Aniconism](https://en.wikipedia.org/wiki/Aniconism "Aniconism") [Integrity](https://en.wikipedia.org/wiki/Artistic_integrity "Artistic integrity") [Interpretation](https://en.wikipedia.org/wiki/Aesthetic_interpretation "Aesthetic interpretation") [Judgment](https://en.wikipedia.org/wiki/Judgment "Judgment") [Kama](https://en.wikipedia.org/wiki/Kama "Kama") [Kitsch](https://en.wikipedia.org/wiki/Kitsch "Kitsch") [Life imitating art](https://en.wikipedia.org/wiki/Life_imitating_art "Life imitating art") [Magnificence](https://en.wikipedia.org/wiki/Magnificence_\(history_of_ideas\) "Magnificence (history of ideas)") [Mimesis](https://en.wikipedia.org/wiki/Mimesis "Mimesis") [Morality](https://en.wikipedia.org/wiki/Art_and_morality "Art and morality") [Perception](https://en.wikipedia.org/wiki/Perception "Perception") [Picturesque](https://en.wikipedia.org/wiki/Picturesque "Picturesque") [Quality](https://en.wikipedia.org/wiki/Quality_\(philosophy\) "Quality (philosophy)") [Rasa](https://en.wikipedia.org/wiki/Rasa_\(aesthetics\) "Rasa (aesthetics)") [Recreation](https://en.wikipedia.org/wiki/Recreation "Recreation") [Reverence](https://en.wikipedia.org/wiki/Reverence_\(emotion\) "Reverence (emotion)") [Satire](https://en.wikipedia.org/wiki/Satire "Satire") [Style](https://en.wikipedia.org/wiki/Style_\(visual_arts\) "Style (visual arts)") [Sublime](https://en.wikipedia.org/wiki/Sublime_\(philosophy\) "Sublime (philosophy)") [Taste](https://en.wikipedia.org/wiki/Taste_\(sociology\) "Taste (sociology)") [Tragedy](https://en.wikipedia.org/wiki/Tragedy "Tragedy") [Work of art](https://en.wikipedia.org/wiki/Work_of_art "Work of art") \| \| Works \| [Hippias Major](https://en.wikipedia.org/wiki/Hippias_Major "Hippias Major") (c. 390 BCE) [Poetics](https://en.wikipedia.org/wiki/Poetics_\(Aristotle\) "Poetics (Aristotle)") (c. 335 BCE) [Ars Poetica](https://en.wikipedia.org/wiki/Ars_Poetica_\(Horace\) "Ars Poetica (Horace)") (c. 19 BCE) [The Literary Mind and the Carving of Dragons](https://en.wikipedia.org/wiki/The_Literary_Mind_and_the_Carving_of_Dragons "The Literary Mind and the Carving of Dragons") (c. 100) [On the Sublime](https://en.wikipedia.org/wiki/On_the_Sublime "On the Sublime") (c. 500) [Asrar al-Balagha](https://en.wikipedia.org/wiki/Asrar_al-Balagha "Asrar al-Balagha") (11th century) [Mumyōzōshi](https://en.wikipedia.org/wiki/Mumy%C5%8Dz%C5%8Dshi "Mumyōzōshi") (13th century) [On the Sublime and Beautiful](https://en.wikipedia.org/wiki/On_the_Sublime_and_Beautiful "On the Sublime and Beautiful") (1757) [Lectures on Aesthetics](https://en.wikipedia.org/wiki/Lectures_on_Aesthetics "Lectures on Aesthetics") (1835) "[The Critic as Artist](https://en.wikipedia.org/wiki/The_Critic_as_Artist "The Critic as Artist")" (1891) "[A Room of One's Own](https://en.wikipedia.org/wiki/A_Room_of_One%27s_Own "A Room of One's Own")" (1929) [In Praise of Shadows](https://en.wikipedia.org/wiki/In_Praise_of_Shadows "In Praise of Shadows") (1933) [Art as Experience](https://en.wikipedia.org/wiki/Art_as_Experience "Art as Experience") (1934) "[The Work of Art in the Age of Mechanical Reproduction](https://en.wikipedia.org/wiki/The_Work_of_Art_in_the_Age_of_Mechanical_Reproduction "The Work of Art in the Age of Mechanical Reproduction")" (1935) "[Avant-Garde and Kitsch](https://en.wikipedia.org/wiki/Avant-Garde_and_Kitsch "Avant-Garde and Kitsch")" (1939) [Prison Notebooks](https://en.wikipedia.org/wiki/Prison_Notebooks "Prison Notebooks") (Gramsci) 1947 [Critical Essays](https://en.wikipedia.org/wiki/Critical_Essays_\(Orwell\) "Critical Essays (Orwell)") (1946) [Against Interpretation](https://en.wikipedia.org/wiki/Against_Interpretation "Against Interpretation") (1966) "[Notes on 'Camp'](https://en.wikipedia.org/wiki/Notes_on_%22Camp%22 "Notes on \"Camp\"")" (1964) [The Aesthetic Dimension](https://en.wikipedia.org/wiki/The_Aesthetic_Dimension "The Aesthetic Dimension") (1977) [Orientalism](https://en.wikipedia.org/wiki/Orientalism_\(book\) "Orientalism (book)") (1978) [Why Beauty Matters](https://en.wikipedia.org/wiki/Why_Beauty_Matters "Why Beauty Matters") (2009) \| \| Related \| [Applied aesthetics](https://en.wikipedia.org/wiki/Applied_aesthetics "Applied aesthetics") [Arts criticism](https://en.wikipedia.org/wiki/Arts_criticism "Arts criticism") [Art criticism](https://en.wikipedia.org/wiki/Art_criticism "Art criticism") [The arts and politics](https://en.wikipedia.org/wiki/The_arts_and_politics "The arts and politics") [Aestheticization of politics](https://en.wikipedia.org/wiki/Aestheticization_of_politics "Aestheticization of politics") [Artistic freedom](https://en.wikipedia.org/wiki/Artistic_freedom "Artistic freedom") [Axiology](https://en.wikipedia.org/wiki/Axiology "Axiology") [Economics of the arts and literature](https://en.wikipedia.org/wiki/Economics_of_the_arts_and_literature "Economics of the arts and literature") [Artistic patronage](https://en.wikipedia.org/wiki/Patronage#Arts "Patronage") [Cultural economics](https://en.wikipedia.org/wiki/Cultural_economics "Cultural economics") [Evolutionary aesthetics](https://en.wikipedia.org/wiki/Evolutionary_aesthetics "Evolutionary aesthetics") [Mathematical beauty]() [Neuroesthetics](https://en.wikipedia.org/wiki/Neuroesthetics "Neuroesthetics") [Patterns in nature](https://en.wikipedia.org/wiki/Patterns_in_nature "Patterns in nature") [Philosophy of design](https://en.wikipedia.org/wiki/Philosophy_of_design "Philosophy of design") [Philosophy of film](https://en.wikipedia.org/wiki/Philosophy_of_film "Philosophy of film") [Philosophy of language](https://en.wikipedia.org/wiki/Philosophy_of_language "Philosophy of language") [Semantics](https://en.wikipedia.org/wiki/Semantics "Semantics") [Philosophy of music](https://en.wikipedia.org/wiki/Philosophy_of_music "Philosophy of music") [Psychology of art](https://en.wikipedia.org/wiki/Psychology_of_art "Psychology of art") [Religious art](https://en.wikipedia.org/wiki/Religious_art "Religious art") [Theory of art](https://en.wikipedia.org/wiki/Theory_of_art "Theory of art") \| \| [Outline](https://en.wikipedia.org/wiki/Outline_of_aesthetics "Outline of aesthetics") [Category](https://en.wikipedia.org/wiki/Category:Aesthetics "Category:Aesthetics") ![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/20px-Socrates.png) [Philosophy portal](https://en.wikipedia.org/wiki/Portal:Philosophy "Portal:Philosophy") \| \| \| [v](https://en.wikipedia.org/wiki/Template:Mathematics_and_art "Template:Mathematics and art") [t](https://en.wikipedia.org/wiki/Template_talk:Mathematics_and_art "Template talk:Mathematics and art") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Mathematics_and_art "Special:EditPage/Template:Mathematics and art")[Mathematics and art](https://en.wikipedia.org/wiki/Mathematics_and_art "Mathematics and art") \| \| \| \|---\|---\|---\| \| Concepts \| [Algorithm](https://en.wikipedia.org/wiki/Algorithm "Algorithm") [Catenary](https://en.wikipedia.org/wiki/Catenary "Catenary") [Fractal](https://en.wikipedia.org/wiki/Fractal "Fractal") [Golden ratio](https://en.wikipedia.org/wiki/Golden_ratio "Golden ratio") [Hyperboloid structure](https://en.wikipedia.org/wiki/Hyperboloid_structure "Hyperboloid structure") [Minimal surface](https://en.wikipedia.org/wiki/Minimal_surface "Minimal surface") [Paraboloid](https://en.wikipedia.org/wiki/Paraboloid "Paraboloid") [Perspective](https://en.wikipedia.org/wiki/Perspective_\(graphical\) "Perspective (graphical)") [Camera lucida](https://en.wikipedia.org/wiki/Camera_lucida "Camera lucida") [Camera obscura](https://en.wikipedia.org/wiki/Camera_obscura "Camera obscura") [Plastic ratio](https://en.wikipedia.org/wiki/Plastic_ratio "Plastic ratio") [Projective geometry](https://en.wikipedia.org/wiki/Projective_geometry "Projective geometry") Proportion [Architecture](https://en.wikipedia.org/wiki/Proportion_\(architecture\) "Proportion (architecture)") [Human](https://en.wikipedia.org/wiki/Body_proportions "Body proportions") [Symmetry](https://en.wikipedia.org/wiki/Symmetry "Symmetry") [Tessellation](https://en.wikipedia.org/wiki/Tessellation "Tessellation") [Wallpaper group](https://en.wikipedia.org/wiki/Wallpaper_group "Wallpaper group") \| [![Fibonacci word: detail of artwork by Samuel Monnier, 2009](https://upload.wikimedia.org/wikipedia/commons/thumb/9/99/FWF_Samuel_Monnier_%28vertical_detail%29.jpg/120px-FWF_Samuel_Monnier_%28vertical_detail%29.jpg)](https://en.wikipedia.org/wiki/File:FWF_Samuel_Monnier_\(vertical_detail\).jpg "Fibonacci word: detail of artwork by Samuel Monnier, 2009") \| \| Forms \| [Algorithmic art](https://en.wikipedia.org/wiki/Algorithmic_art "Algorithmic art") [Anamorphic art](https://en.wikipedia.org/wiki/Anamorphosis "Anamorphosis") [Architecture](https://en.wikipedia.org/wiki/Mathematics_and_architecture "Mathematics and architecture") [Geodesic dome](https://en.wikipedia.org/wiki/Geodesic_dome "Geodesic dome") [Pyramid](https://en.wikipedia.org/wiki/Pyramid "Pyramid") [Vastu shastra](https://en.wikipedia.org/wiki/Vastu_shastra "Vastu shastra") [Computer art](https://en.wikipedia.org/wiki/Computer_art "Computer art") [Fiber arts](https://en.wikipedia.org/wiki/Mathematics_and_fiber_arts "Mathematics and fiber arts") [4D art](https://en.wikipedia.org/wiki/Fourth_dimension_in_art "Fourth dimension in art") [Fractal art](https://en.wikipedia.org/wiki/Fractal_art "Fractal art") [Islamic geometric patterns](https://en.wikipedia.org/wiki/Islamic_geometric_patterns "Islamic geometric patterns") [Girih](https://en.wikipedia.org/wiki/Girih "Girih") [Jali](https://en.wikipedia.org/wiki/Jali "Jali") [Muqarnas](https://en.wikipedia.org/wiki/Muqarnas "Muqarnas") [Zellij](https://en.wikipedia.org/wiki/Zellij "Zellij") [Knotting](https://en.wikipedia.org/wiki/Knot "Knot") [Celtic knot](https://en.wikipedia.org/wiki/Celtic_knot "Celtic knot") [Croatian interlace](https://en.wikipedia.org/wiki/Croatian_interlace "Croatian interlace") [Interlace](https://en.wikipedia.org/wiki/Interlace_\(art\) "Interlace (art)") [Music](https://en.wikipedia.org/wiki/Music_and_mathematics "Music and mathematics") [Origami](https://en.wikipedia.org/wiki/Origami "Origami") [Mathematics](https://en.wikipedia.org/wiki/Mathematics_of_paper_folding "Mathematics of paper folding") [Sculpture](https://en.wikipedia.org/wiki/Mathematical_sculpture "Mathematical sculpture") [String art](https://en.wikipedia.org/wiki/String_art "String art") [String figure](https://en.wikipedia.org/wiki/String_figure "String figure") [Tiling](https://en.wikipedia.org/wiki/Tessellation "Tessellation") \| \| \| Artworks \| [List of works designed with the golden ratio](https://en.wikipedia.org/wiki/List_of_works_designed_with_the_golden_ratio "List of works designed with the golden ratio") [Continuum](https://en.wikipedia.org/wiki/Continuum_\(sculpture\) "Continuum (sculpture)") [Mathemalchemy](https://en.wikipedia.org/wiki/Mathemalchemy "Mathemalchemy") [Mathematica: A World of Numbers... and Beyond](https://en.wikipedia.org/wiki/Mathematica:_A_World_of_Numbers..._and_Beyond "Mathematica: A World of Numbers... and Beyond") [Octacube](https://en.wikipedia.org/wiki/Octacube_\(sculpture\) "Octacube (sculpture)") [Pi](https://en.wikipedia.org/wiki/Pi_\(art_project\) "Pi (art project)") [Pi in the Sky](https://en.wikipedia.org/wiki/Pi_in_the_Sky "Pi in the Sky") \| \| \| [Buildings](https://en.wikipedia.org/wiki/Mathematics_and_architecture "Mathematics and architecture") \| [Cathedral of Saint Mary of the Assumption](https://en.wikipedia.org/wiki/Cathedral_of_Saint_Mary_of_the_Assumption_\(San_Francisco\) "Cathedral of Saint Mary of the Assumption (San Francisco)") [Hagia Sophia](https://en.wikipedia.org/wiki/Hagia_Sophia "Hagia Sophia") [Kresge Auditorium](https://en.wikipedia.org/wiki/Kresge_Auditorium "Kresge Auditorium") [Pantheon](https://en.wikipedia.org/wiki/Pantheon,_Rome "Pantheon, Rome") [Parthenon](https://en.wikipedia.org/wiki/Parthenon "Parthenon") [Pyramid of Khufu](https://en.wikipedia.org/wiki/Great_Pyramid_of_Giza "Great Pyramid of Giza") [Sagrada Família](https://en.wikipedia.org/wiki/Sagrada_Fam%C3%ADlia "Sagrada Família") [Sydney Opera House](https://en.wikipedia.org/wiki/Sydney_Opera_House "Sydney Opera House") [Taj Mahal](https://en.wikipedia.org/wiki/Taj_Mahal "Taj Mahal") \| \| \| [Artists](https://en.wikipedia.org/wiki/List_of_mathematical_artists "List of mathematical artists") \| \| \| \| \| \| \| \| Renaissance \| [Paolo Uccello](https://en.wikipedia.org/wiki/Paolo_Uccello "Paolo Uccello") [Piero della Francesca](https://en.wikipedia.org/wiki/Piero_della_Francesca "Piero della Francesca") [Leonardo da Vinci](https://en.wikipedia.org/wiki/Leonardo_da_Vinci "Leonardo da Vinci") [Vitruvian Man](https://en.wikipedia.org/wiki/Vitruvian_Man "Vitruvian Man") [Albrecht Dürer](https://en.wikipedia.org/wiki/Albrecht_D%C3%BCrer "Albrecht Dürer") [Parmigianino](https://en.wikipedia.org/wiki/Parmigianino "Parmigianino") [Self-portrait in a Convex Mirror](https://en.wikipedia.org/wiki/Self-portrait_in_a_Convex_Mirror "Self-portrait in a Convex Mirror") \| \| \| 19th–20th Century \| [William Blake](https://en.wikipedia.org/wiki/William_Blake "William Blake") [The Ancient of Days](https://en.wikipedia.org/wiki/The_Ancient_of_Days "The Ancient of Days") [Newton](https://en.wikipedia.org/wiki/Newton_\(Blake\) "Newton (Blake)") [Jean Metzinger](https://en.wikipedia.org/wiki/Jean_Metzinger "Jean Metzinger") [Danseuse au café](https://en.wikipedia.org/wiki/Dancer_in_a_Caf%C3%A9 "Dancer in a Café") [L'Oiseau bleu](https://en.wikipedia.org/wiki/L%27Oiseau_bleu_\(Metzinger\) "L'Oiseau bleu (Metzinger)") [Giorgio de Chirico](https://en.wikipedia.org/wiki/Giorgio_de_Chirico "Giorgio de Chirico") [Man Ray](https://en.wikipedia.org/wiki/Man_Ray "Man Ray") [M. C. Escher](https://en.wikipedia.org/wiki/M._C._Escher "M. C. Escher") [Circle Limit III](https://en.wikipedia.org/wiki/Circle_Limit_III "Circle Limit III") [Print Gallery](https://en.wikipedia.org/wiki/Print_Gallery_\(M._C._Escher\) "Print Gallery (M. C. Escher)") [Relativity](https://en.wikipedia.org/wiki/Relativity_\(M._C._Escher\) "Relativity (M. C. Escher)") [Reptiles](https://en.wikipedia.org/wiki/Reptiles_\(M._C._Escher\) "Reptiles (M. C. Escher)") [Waterfall](https://en.wikipedia.org/wiki/Waterfall_\(M._C._Escher\) "Waterfall (M. C. Escher)") [René Magritte](https://en.wikipedia.org/wiki/Ren%C3%A9_Magritte "René Magritte") [La condition humaine](https://en.wikipedia.org/wiki/The_Human_Condition_\(Magritte\) "The Human Condition (Magritte)") [Salvador Dalí](https://en.wikipedia.org/wiki/Salvador_Dal%C3%AD "Salvador Dalí") [Crucifixion](https://en.wikipedia.org/wiki/Crucifixion_\(Corpus_Hypercubus\) "Crucifixion (Corpus Hypercubus)") [The Swallow's Tail](https://en.wikipedia.org/wiki/The_Swallow%27s_Tail "The Swallow's Tail") [Crockett Johnson](https://en.wikipedia.org/wiki/Crockett_Johnson "Crockett Johnson") \| \| \| Contemporary \| [Max Bill](https://en.wikipedia.org/wiki/Max_Bill "Max Bill") [Martin](https://en.wikipedia.org/wiki/Martin_Demaine "Martin Demaine") and [Erik Demaine](https://en.wikipedia.org/wiki/Erik_Demaine "Erik Demaine") [Scott Draves](https://en.wikipedia.org/wiki/Scott_Draves "Scott Draves") [Jan Dibbets](https://en.wikipedia.org/wiki/Jan_Dibbets "Jan Dibbets") [John Ernest](https://en.wikipedia.org/wiki/John_Ernest "John Ernest") [Helaman Ferguson](https://en.wikipedia.org/wiki/Helaman_Ferguson "Helaman Ferguson") [Peter Forakis](https://en.wikipedia.org/wiki/Peter_Forakis "Peter Forakis") [Susan Goldstine](https://en.wikipedia.org/wiki/Susan_Goldstine "Susan Goldstine") [Bathsheba Grossman](https://en.wikipedia.org/wiki/Bathsheba_Grossman "Bathsheba Grossman") [George W. Hart](https://en.wikipedia.org/wiki/George_W._Hart "George W. Hart") [Desmond Paul Henry](https://en.wikipedia.org/wiki/Desmond_Paul_Henry "Desmond Paul Henry") [Anthony Hill](https://en.wikipedia.org/wiki/Anthony_Hill_\(artist\) "Anthony Hill (artist)") [Charles Jencks](https://en.wikipedia.org/wiki/Charles_Jencks "Charles Jencks") [Garden of Cosmic Speculation](https://en.wikipedia.org/wiki/Garden_of_Cosmic_Speculation "Garden of Cosmic Speculation") [Andy Lomas](https://en.wikipedia.org/wiki/Andy_Lomas "Andy Lomas") [Robert Longhurst](https://en.wikipedia.org/wiki/Robert_Longhurst "Robert Longhurst") [Jeanette McLeod](https://en.wikipedia.org/wiki/Jeanette_McLeod "Jeanette McLeod") [Hamid Naderi Yeganeh](https://en.wikipedia.org/wiki/Hamid_Naderi_Yeganeh "Hamid Naderi Yeganeh") [István Orosz](https://en.wikipedia.org/wiki/Istv%C3%A1n_Orosz "István Orosz") [Hinke Osinga](https://en.wikipedia.org/wiki/Hinke_Osinga "Hinke Osinga") [Antoine Pevsner](https://en.wikipedia.org/wiki/Antoine_Pevsner "Antoine Pevsner") [Tony Robbin](https://en.wikipedia.org/wiki/Tony_Robbin "Tony Robbin") [Alba Rojo Cama](https://en.wikipedia.org/wiki/Alba_Rojo_Cama "Alba Rojo Cama") [Reza Sarhangi](https://en.wikipedia.org/wiki/Reza_Sarhangi "Reza Sarhangi") [Oliver Sin](https://en.wikipedia.org/wiki/Oliver_Sin "Oliver Sin") [Hiroshi Sugimoto](https://en.wikipedia.org/wiki/Hiroshi_Sugimoto "Hiroshi Sugimoto") [Daina Taimiņa](https://en.wikipedia.org/wiki/Daina_Taimi%C5%86a "Daina Taimiņa") [Roman Verostko](https://en.wikipedia.org/wiki/Roman_Verostko "Roman Verostko") [Margaret Wertheim](https://en.wikipedia.org/wiki/Margaret_Wertheim "Margaret Wertheim") \| \| \| Theorists \| \| \| \| \| \| \| \| Ancient \| [Polykleitos](https://en.wikipedia.org/wiki/Polykleitos "Polykleitos") Canon [Vitruvius](https://en.wikipedia.org/wiki/Vitruvius "Vitruvius") [De architectura](https://en.wikipedia.org/wiki/De_architectura "De architectura") \| \| \| Renaissance \| [Filippo Brunelleschi](https://en.wikipedia.org/wiki/Filippo_Brunelleschi "Filippo Brunelleschi") [Leon Battista Alberti](https://en.wikipedia.org/wiki/Leon_Battista_Alberti "Leon Battista Alberti") [De pictura](https://en.wikipedia.org/wiki/De_pictura "De pictura") [De re aedificatoria](https://en.wikipedia.org/wiki/De_re_aedificatoria "De re aedificatoria") [Piero della Francesca](https://en.wikipedia.org/wiki/Piero_della_Francesca "Piero della Francesca") [De prospectiva pingendi](https://en.wikipedia.org/wiki/De_prospectiva_pingendi "De prospectiva pingendi") [Luca Pacioli](https://en.wikipedia.org/wiki/Luca_Pacioli "Luca Pacioli") [De divina proportione](https://en.wikipedia.org/wiki/Divina_proportione "Divina proportione") [Leonardo da Vinci](https://en.wikipedia.org/wiki/Leonardo_da_Vinci "Leonardo da Vinci") [A Treatise on Painting](https://en.wikipedia.org/wiki/A_Treatise_on_Painting "A Treatise on Painting") [Albrecht Dürer](https://en.wikipedia.org/wiki/Albrecht_D%C3%BCrer "Albrecht Dürer") Vier Bücher von Menschlicher Proportion [Sebastiano Serlio](https://en.wikipedia.org/wiki/Sebastiano_Serlio "Sebastiano Serlio") Regole generali d'architettura [Andrea Palladio](https://en.wikipedia.org/wiki/Andrea_Palladio "Andrea Palladio") [I quattro libri dell'architettura](https://en.wikipedia.org/wiki/I_quattro_libri_dell%27architettura "I quattro libri dell'architettura") \| \| \| Romantic \| [Samuel Colman](https://en.wikipedia.org/wiki/Samuel_Colman "Samuel Colman") Nature's Harmonic Unity [Frederik Macody Lund](https://en.wikipedia.org/wiki/Frederik_Macody_Lund "Frederik Macody Lund") Ad Quadratum [Jay Hambidge](https://en.wikipedia.org/wiki/Jay_Hambidge "Jay Hambidge") The Greek Vase \| \| \| Modern \| [Owen Jones](https://en.wikipedia.org/wiki/Owen_Jones_\(architect\) "Owen Jones (architect)") [The Grammar of Ornament](https://en.wikipedia.org/wiki/Owen_Jones_\(architect\)#The_Grammar_of_Ornament "Owen Jones (architect)") [Ernest Hanbury Hankin](https://en.wikipedia.org/wiki/Ernest_Hanbury_Hankin "Ernest Hanbury Hankin") The Drawing of Geometric Patterns in Saracenic Art [G. H. Hardy](https://en.wikipedia.org/wiki/G._H._Hardy "G. H. Hardy") [A Mathematician's Apology](https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology "A Mathematician's Apology") [George David Birkhoff](https://en.wikipedia.org/wiki/George_David_Birkhoff "George David Birkhoff") Aesthetic Measure [Douglas Hofstadter](https://en.wikipedia.org/wiki/Douglas_Hofstadter "Douglas Hofstadter") [Gödel, Escher, Bach](https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach "Gödel, Escher, Bach") [Nikos Salingaros](https://en.wikipedia.org/wiki/Nikos_Salingaros "Nikos Salingaros") The 'Life' of a Carpet \| \| \| Publications \| [Journal of Mathematics and the Arts](https://en.wikipedia.org/wiki/Journal_of_Mathematics_and_the_Arts "Journal of Mathematics and the Arts") [Lumen Naturae](https://en.wikipedia.org/wiki/Lumen_Naturae "Lumen Naturae") [Making Mathematics with Needlework](https://en.wikipedia.org/wiki/Making_Mathematics_with_Needlework "Making Mathematics with Needlework") [Rhythm of Structure](https://en.wikipedia.org/wiki/Rhythm_of_Structure "Rhythm of Structure") [Viewpoints: Mathematical Perspective and Fractal Geometry in Art](https://en.wikipedia.org/wiki/Viewpoints:_Mathematical_Perspective_and_Fractal_Geometry_in_Art "Viewpoints: Mathematical Perspective and Fractal Geometry in Art") \| \| \| Organizations \| [Ars Mathematica](https://en.wikipedia.org/wiki/Ars_Mathematica_\(organization\) "Ars Mathematica (organization)") [The Bridges Organization](https://en.wikipedia.org/wiki/The_Bridges_Organization "The Bridges Organization") [European Society for Mathematics and the Arts](https://en.wikipedia.org/wiki/European_Society_for_Mathematics_and_the_Arts "European Society for Mathematics and the Arts") [Goudreau Museum of Mathematics in Art and Science](https://en.wikipedia.org/wiki/Goudreau_Museum_of_Mathematics_in_Art_and_Science "Goudreau Museum of Mathematics in Art and Science") [Institute For Figuring](https://en.wikipedia.org/wiki/Institute_For_Figuring "Institute For Figuring") [Mathemalchemy](https://en.wikipedia.org/wiki/Mathemalchemy "Mathemalchemy") [National Museum of Mathematics](https://en.wikipedia.org/wiki/National_Museum_of_Mathematics "National Museum of Mathematics") \| \| \| Related \| [Droste effect](https://en.wikipedia.org/wiki/Droste_effect "Droste effect") [Mathematical beauty]() [Patterns in nature](https://en.wikipedia.org/wiki/Patterns_in_nature "Patterns in nature") [Sacred geometry](https://en.wikipedia.org/wiki/Sacred_geometry "Sacred geometry") \| \| \| ![](https://upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/20px-Symbol_category_class.svg.png) [Category](https://en.wikipedia.org/wiki/Category:Mathematics_and_art "Category:Mathematics and art") \| \| \| ![](https://en.wikipedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1) Retrieved from "<https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&oldid=1343696221>" [Categories](https://en.wikipedia.org/wiki/Help:Category "Help:Category"): - 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Readable Markdown	Mathematical beauty is a type of [aesthetic](https://en.wikipedia.org/wiki/Aesthetics "Aesthetics") value that is experienced in doing or contemplating [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"). The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such as [G.H. Hardy](https://en.wikipedia.org/wiki/G.H._Hardy "G.H. Hardy"), have characterized mathematics as an [art](https://en.wikipedia.org/wiki/Art "Art") form that seeks beauty. Beauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understanding [beauty](https://en.wikipedia.org/wiki/Beauty "Beauty") in general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstract ideas which can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than (for example) listening to music.[\[1\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-1) Furthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors identify mathematical elegance with mathematical beauty; others distinguish elegance as a separate aesthetic value, or as being, for instance, limited to the form mathematical exposition.[\[2\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-rota-phenomenology-2): 177–178 Beauty itself is often linked to, or thought to be dependent on, the abstractness, purity, simplicity, depth or order of mathematics. ## Examples of beautiful mathematics \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=1 "Edit section: Examples of beautiful mathematics")\] [![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/3b/EulerIdentity2.svg/250px-EulerIdentity2.svg.png)](https://en.wikipedia.org/wiki/File:EulerIdentity2.svg) Starting at e0 = 1, travelling at the velocity i relative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an [Argand diagram](https://en.wikipedia.org/wiki/Argand_diagram "Argand diagram").) [Euler's identity](https://en.wikipedia.org/wiki/Euler%27s_identity "Euler's identity") is often given as an example of a beautiful result:[\[3\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-3): 1–3 [\[4\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-cain-form-4): 835–836 [\[5\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-5) ![{\\displaystyle \\displaystyle \\mathrm {e} ^{\\mathrm {i} \\pi }+1=0\\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5213bb3f233806ff3212142214fee997434a15) This expression ties together arguably the five most important [mathematical constants](https://en.wikipedia.org/wiki/Mathematical_constant "Mathematical constant") (e, i, π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"), which the physicist [Richard Feynman](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman") called "our jewel" and "the most remarkable formula in mathematics".[\[6\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-6) Another example is [Fermat's theorem on sums of two squares](https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares "Fermat's theorem on sums of two squares"), which says that any [prime number](https://en.wikipedia.org/wiki/Prime_number "Prime number") such that ![{\\displaystyle p\\equiv 1{\\pmod {4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/125d70c2c8848e96954eddfc7f9283ec4f676ed7) can be written as a sum of two square numbers (for example, ![{\\displaystyle 5=1^{2}+2^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b288581682d73e457bc6829e15be203280852a), ![{\\displaystyle 13=2^{2}+3^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8be5ea38d26942831bf630c465218ef5e3189181), ![{\\displaystyle 37=1^{2}+6^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f81cd69710c7c05bdc903449dc7e834540c797a)), which both [G.H. Hardy](https://en.wikipedia.org/wiki/G.H._Hardy "G.H. Hardy")[\[7\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-hardy-apology-7): §12 and [E.T. Bell](https://en.wikipedia.org/wiki/E.T._Bell "E.T. Bell")[\[8\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-8): ch.4 thought was a beautiful result. In a survey in which mathematicians were asked to evaluate 24 theorems for their beauty, the top-rated three theorems were:[\[9\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-9) Euler's equation; [Euler's polyhedron formula](https://en.wikipedia.org/wiki/Euler_characteristic "Euler characteristic"), which asserts that for a polyhedron with V vertices, E edges, and F faces, ![{\\displaystyle V-E+F=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/759601e482258ff7a359a7db381abf60372c5b06); and [Euclid's theorem](https://en.wikipedia.org/wiki/Euclid%27s_theorem "Euclid's theorem") that there are infinitely many prime numbers, which was also given by [Hardy](https://en.wikipedia.org/wiki/G.H._Hardy "G.H. Hardy") as an example of a beautiful theorem.[\[7\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-hardy-apology-7): §12 [![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Pythagorean_proof_%281%29.svg/250px-Pythagorean_proof_%281%29.svg.png)](https://en.wikipedia.org/wiki/File:Pythagorean_proof_\(1\).svg) An example of "beauty in method"—a simple and elegant visual descriptor of the [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem"). [Cantor's diagonal argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument "Cantor's diagonal argument"), which establishes that there are [infinite sets](https://en.wikipedia.org/wiki/Infinite_set "Infinite set") which cannot be put into [one-to-one correspondence](https://en.wikipedia.org/wiki/Bijection "Bijection") with the infinite set of [natural numbers](https://en.wikipedia.org/wiki/Natural_number "Natural number"), has been cited by both mathematicians[\[10\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-10) and philosophers[\[11\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-11) as an example of a beautiful proof. [![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Proofwithoutwords.svg/250px-Proofwithoutwords.svg.png)](https://en.wikipedia.org/wiki/File:Proofwithoutwords.svg) A proof without words for the sum of odd numbers theorem Visual proofs, such as the illustrated proof of the [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem"), and other [proofs without words](https://en.wikipedia.org/wiki/Proofs_without_words "Proofs without words") generally, such as the shown proof that the sum of all positive [odd numbers](https://en.wikipedia.org/wiki/Parity_\(mathematics\) "Parity (mathematics)") up to 2n − 1 is a [perfect square](https://en.wikipedia.org/wiki/Square_number "Square number"), have been thought beautiful.[\[12\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-12) The mathematician [Paul Erdős](https://en.wikipedia.org/wiki/Paul_Erd%C5%91s "Paul Erdős") spoke of The Book, an imaginary infinite book in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would proclaim it "straight from The Book!".[\[13\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-13): 35 His rhetorical device inspired the creation of [Proofs from THE BOOK](https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK "Proofs from THE BOOK"), a collection of such proofs, including many suggested by Erdős himself.[\[14\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-14): v In [Plato](https://en.wikipedia.org/wiki/Plato "Plato")'s [Timaeus](https://en.wikipedia.org/wiki/Timaeus_\(dialogue\) "Timaeus (dialogue)"), the five [regular convex polyhedra](https://en.wikipedia.org/wiki/Platonic_solids "Platonic solids"), called the Platonic solids for their role in this dialogue, are called the "most beautiful" ("κάλλιστα") bodies.[\[15\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-plato-timaeus-15): 53e In the Timaeus, they are described as having been used by the [demiurge](https://en.wikipedia.org/wiki/Demiurge "Demiurge"), or creator-craftsman who builds the cosmos, for the four [classical elements](https://en.wikipedia.org/wiki/Classical_elements "Classical elements") plus the heavens, because of their beauty.[\[15\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-plato-timaeus-15): 54e–55e [![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/19/Kepler-solar-system-1.png/250px-Kepler-solar-system-1.png)](https://en.wikipedia.org/wiki/File:Kepler-solar-system-1.png) Kepler's Platonic solid model of the solar system In his 1596 book [Mysterium Cosmographicum](https://en.wikipedia.org/wiki/Mysterium_Cosmographicum "Mysterium Cosmographicum"), [Johannes Kepler](https://en.wikipedia.org/wiki/Johannes_Kepler "Johannes Kepler") argued that the orbits of the then-known planets in the [Solar System](https://en.wikipedia.org/wiki/Solar_System "Solar System") have been arranged by [God](https://en.wikipedia.org/wiki/God "God") to correspond to a concentric arrangement of the five [Platonic solids](https://en.wikipedia.org/wiki/Platonic_solid "Platonic solid"), each orbit lying on the [circumsphere](https://en.wikipedia.org/wiki/Circumscribed_sphere "Circumscribed sphere") of one [polyhedron](https://en.wikipedia.org/wiki/Polyhedron "Polyhedron") and the [insphere](https://en.wikipedia.org/wiki/Inscribed_sphere "Inscribed sphere") of another. For Kepler, God had wanted to shape the universe according to the five regular solids because of their beauty, and this explained why there were six planets (according to the knowledge of the time).[\[16\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-16): ch.3 [\[4\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-cain-form-4): 280--285 [![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/250px-E8Petrie.svg.png)](https://en.wikipedia.org/wiki/File:E8Petrie.svg) Petrie projection of ![{\\displaystyle E\_{8}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48479e96d90b4cfabc7784106cc3cfff907dda34) A more modern example is the exceptional [simple Lie group](https://en.wikipedia.org/wiki/Simple_Lie_group "Simple Lie group") ![{\\displaystyle E\_{8}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48479e96d90b4cfabc7784106cc3cfff907dda34), which has been called "perhaps the most beautiful structure in all of mathematics".[\[17\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-17) ### Scientific theories \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=5 "Edit section: Scientific theories")\] The mathematical statements of scientific theories, especially in physics, are sometimes considered to be mathematically beautiful. For example, [Roger Penrose](https://en.wikipedia.org/wiki/Roger_Penrose "Roger Penrose") thought there was a "special beauty" in [Maxwell's equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations "Maxwell's equations") of electromagnetism:[\[18\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-18): 268 ![{\\displaystyle {\\begin{aligned}\\nabla \\cdot \\mathbf {E} \\,\\,\\,&={\\frac {\\rho }{\\varepsilon \_{0}}}\\\\\\nabla \\cdot \\mathbf {B} \\,\\,\\,&=0\\\\\\nabla \\times \\mathbf {E} &=-{\\frac {\\partial \\mathbf {B} }{\\partial t}}\\\\\\nabla \\times \\mathbf {B} &=\\mu \_{0}\\left(\\mathbf {J} +\\varepsilon \_{0}{\\frac {\\partial \\mathbf {E} }{\\partial t}}\\right)\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1740d383a275f64105f457e209ff5c66eeeb21) Einstein's theory of [general relativity](https://en.wikipedia.org/wiki/General_relativity "General relativity") has been characterized as a work of art, and, among other aesthetic praise,[\[19\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-19): 148 was described by [Paul Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac") as having "great mathematical beauty"[\[20\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-20): 123 and by Penrose as having "supreme mathematical beauty".[\[21\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-21): 1038 (There can be more to the beauty of a scientific theory than just its mathematical statement. For example, whether a theory is visualizable or deterministic might have an influence on whether it is seen as beautiful.[\[22\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-22): 53 [\[4\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-cain-form-4): 873–877 ) ## Properties of beautiful mathematics \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=6 "Edit section: Properties of beautiful mathematics")\] Many mathematicians and philosophers who have written about mathematical beauty have tried to identify properties or criteria that are conducive to the perception of beauty in a piece of mathematics. It is debated whether beauty can be clarified or explained by such properties: [Paul Erdős](https://en.wikipedia.org/wiki/Paul_Erd%C5%91s "Paul Erdős") thought that it was no more possible to convince someone of the beauty of a piece of mathematics than to convince them of the beauty of [Beethoven's Ninth Symphony](https://en.wikipedia.org/wiki/Symphony_No._9_\(Beethoven\) "Symphony No. 9 (Beethoven)"), if they couldn't see it for themselves.[\[23\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-23) In his 1940 essay [A Mathematician's Apology](https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology "A Mathematician's Apology"), [G. H. Hardy](https://en.wikipedia.org/wiki/G._H._Hardy "G. H. Hardy") said that a beautiful result, including its proof, possesses three "purely aesthetic qualities", namely "inevitability", "unexpectedness", and "economy". He particularly excluded enumeration of cases as "one of the duller forms of mathematical argument".[\[7\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-hardy-apology-7): §18 In 1997, [Gian-Carlo Rota](https://en.wikipedia.org/wiki/Gian-Carlo_Rota "Gian-Carlo Rota") disagreed with unexpectedness as a sufficient condition for beauty and proposed a counterexample: > A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago \[from 1977\] the proof of the existence of [non-equivalent differentiable structures](https://en.wikipedia.org/wiki/Exotic_sphere "Exotic sphere") on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or now.[\[2\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-rota-phenomenology-2): 172 In contrast, Monastyrsky wrote in 2001: > It is very difficult to find an analogous invention in the past to [Milnor](https://en.wikipedia.org/wiki/Milnor "Milnor")'s beautiful construction of the different differential structures on the seven-dimensional sphere... The original proof of Milnor was not very constructive, but later E. Briscorn showed that these differential structures can be described in an extremely explicit and beautiful form.[\[24\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-24): 44 This disagreement illustrates both the subjective nature of mathematical beauty, [like other forms of beauty in general](https://en.wikipedia.org/wiki/Beauty "Beauty"), and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them. Besides Hardy's properties of "unexpectedness", "inevitability", "economy", which he applied to proofs as well as results, mathematicians have customarily thought beautiful proofs that are short and simple.[\[25\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-25): 22 In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly the [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem"), with hundreds of proofs having been published.[\[26\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-26) Another theorem that has been proved in many different ways is the theorem of [quadratic reciprocity](https://en.wikipedia.org/wiki/Quadratic_reciprocity "Quadratic reciprocity"). In fact, [Carl Friedrich Gauss](https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss "Carl Friedrich Gauss") alone had eight different proofs of this theorem, six of which he published.[\[27\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-27) In contrast, results that are logically correct but involve laborious calculations or consideration of many cases, are not usually considered beautiful, and may be even referred to as ugly or clumsy. For example, [Kenneth Appel](https://en.wikipedia.org/wiki/Kenneth_Appel "Kenneth Appel") and [Wolfgang Haken](https://en.wikipedia.org/wiki/Wolfgang_Haken "Wolfgang Haken")'s proof of the [four color theorem](https://en.wikipedia.org/wiki/Four_color_theorem "Four color theorem") made use of computer checking of over a thousand cases. [Philip J. Davis](https://en.wikipedia.org/wiki/Philip_J._Davis "Philip J. Davis") and [Reuben Hersh](https://en.wikipedia.org/wiki/Reuben_Hersh "Reuben Hersh") said that when they first heard that about the proof, they hoped it contained a new insight "whose beauty would transform my day", and were disheartened when informed the proof was by case enumeration and computer verification.[\[28\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-davis-experience-28): 384 [Paul Erdős](https://en.wikipedia.org/wiki/Paul_Erd%C5%91s "Paul Erdős") said it was "not beautiful" because it gave no insight into why the theorem was true.[\[29\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-29): 44 ## Philosophical analysis \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=9 "Edit section: Philosophical analysis")\] [Aristotle](https://en.wikipedia.org/wiki/Aristotle "Aristotle") thought that beauty was found especially in mathematics, writing in the [Metaphysics](https://en.wikipedia.org/wiki/Metaphysics_\(Aristotle\) "Metaphysics (Aristotle)") that > those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a very great deal about them; for if they do not expressly mention them, but prove attributes which are their results or their formulae, it is not true to say that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.[\[30\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-30): 1078a32–35 The logician and philosopher [Bertrand Russell](https://en.wikipedia.org/wiki/Bertrand_Russell "Bertrand Russell") made a now-famous statement characterizing mathematical beauty in terms of purity and austerity: > Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.[\[31\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-31) In the twentieth century, some philosophers questioned whether there was genuinely beauty in mathematics. The philosopher of science [Rom Harré](https://en.wikipedia.org/wiki/Rom_Harr%C3%A9 "Rom Harré") argued that there were no true aesthetic appraisals of mathematics, but only quasi-aesthetic appraisals. Any mathematical success described by an aesthetic term was a second-order success besides understanding and correctness. In contrast, aesthetic appraisal of a work of art was first-order. Harré considered this to be the difference between a quasi-aesthetic and a genuinely aesthetic appraisal.[\[32\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-32) [Nick Zangwill](https://en.wikipedia.org/wiki/Nick_Zangwill "Nick Zangwill") thought that there were no true aesthetic experiences of mathematics and that a proofs or theories could only be metaphorically beautiful. His argument had two bases. First, he thought that aesthetic properties depended on sensory properties, and so abstract entities could not have aesthetic properties. Second, he thought that proofs, theorems, theories, and so on had purposes such as demonstrating correctness or granting understanding, and that any praise of them reflected only how well they achieved their purpose.[\[33\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-33): 140–142 ## Scientific analysis \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=10 "Edit section: Scientific analysis")\] ### Information-theory model \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=11 "Edit section: Information-theory model")\] In the 1970s, [Abraham Moles](https://en.wikipedia.org/wiki/Abraham_Moles "Abraham Moles") and [Frieder Nake](https://en.wikipedia.org/wiki/Frieder_Nake "Frieder Nake") analyzed links between beauty, [information processing](https://en.wikipedia.org/wiki/Data_processing "Data processing"), and [information theory](https://en.wikipedia.org/wiki/Information_theory "Information theory").[\[34\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-34)[\[35\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-35) In the 1990s, [Jürgen Schmidhuber](https://en.wikipedia.org/wiki/J%C3%BCrgen_Schmidhuber "Jürgen Schmidhuber") formulated a mathematical theory of observer-dependent subjective beauty based on [algorithmic information theory](https://en.wikipedia.org/wiki/Algorithmic_information_theory "Algorithmic information theory"): the most beautiful objects among subjectively comparable objects have short [algorithmic](https://en.wikipedia.org/wiki/Algorithmic_information_theory "Algorithmic information theory") descriptions (i.e., [Kolmogorov complexity](https://en.wikipedia.org/wiki/Kolmogorov_complexity "Kolmogorov complexity")) relative to what the observer already knows.[\[36\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-36)[\[37\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-37)[\[38\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-38) Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the [first derivative](https://en.wikipedia.org/wiki/First_derivative "First derivative") of subjectively perceived beauty: the observer continually tries to improve the [predictability](https://en.wikipedia.org/wiki/Predictability "Predictability") and [compressibility](https://en.wikipedia.org/wiki/Data_compression "Data compression") of the observations by discovering regularities such as repetitions and [symmetries](https://en.wikipedia.org/wiki/Symmetries "Symmetries") and [fractal](https://en.wikipedia.org/wiki/Fractal "Fractal") [self-similarity](https://en.wikipedia.org/wiki/Self-similarity "Self-similarity"). Whenever the observer's learning process (possibly a predictive artificial [neural network](https://en.wikipedia.org/wiki/Neural_network "Neural network")) leads to improved data compression such that the observation sequence can be described by fewer [bits](https://en.wikipedia.org/wiki/Bit "Bit") than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.[\[39\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-39)[\[40\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-40) Brain imaging experiments conducted by [Semir Zeki](https://en.wikipedia.org/wiki/Semir_Zeki "Semir Zeki"), [Michael Atiyah](https://en.wikipedia.org/wiki/Michael_Atiyah "Michael Atiyah") and collaborators show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial [orbito-frontal cortex](https://en.wikipedia.org/wiki/Orbitofrontal_cortex "Orbitofrontal cortex") (mOFC) of the brain and that this activity is parametrically related to the declared intensity of beauty. The location of the activity is similar to the location of the activity that correlates with the experience of beauty from other sources, such as visual art or music.[\[41\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-41) Moreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers.[\[42\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-42) ## Mathematical beauty and the arts \[[edit](https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&action=edit&section=13 "Edit section: Mathematical beauty and the arts")\] Examples of the use of mathematics in music include the [stochastic music](https://en.wikipedia.org/wiki/Stochastic_music "Stochastic music") of [Iannis Xenakis](https://en.wikipedia.org/wiki/Iannis_Xenakis "Iannis Xenakis"),[\[43\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-43) the [Fibonacci sequence](https://en.wikipedia.org/wiki/Fibonacci_number "Fibonacci number") in [Tool](https://en.wikipedia.org/wiki/Tool_\(band\) "Tool (band)")'s [Lateralus](https://en.wikipedia.org/wiki/Lateralus_\(song\) "Lateralus (song)"),[\[44\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-44) and the [Metric modulation](https://en.wikipedia.org/wiki/Metric_modulation "Metric modulation") of [Elliott Carter](https://en.wikipedia.org/wiki/Elliott_Carter "Elliott Carter").[\[45\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-45) Other instances include the counterpoint of [Johann Sebastian Bach](https://en.wikipedia.org/wiki/Johann_Sebastian_Bach "Johann Sebastian Bach"), [polyrhythmic](https://en.wikipedia.org/wiki/Polyrhythm "Polyrhythm") structures (as in [Igor Stravinsky](https://en.wikipedia.org/wiki/Igor_Stravinsky "Igor Stravinsky")'s [The Rite of Spring](https://en.wikipedia.org/wiki/The_Rite_of_Spring "The Rite of Spring")), [permutation](https://en.wikipedia.org/wiki/Permutation "Permutation") theory in [serialism](https://en.wikipedia.org/wiki/Serialism "Serialism") beginning with [Arnold Schoenberg](https://en.wikipedia.org/wiki/Arnold_Schoenberg "Arnold Schoenberg"), the application of Shepard tones in [Karlheinz Stockhausen](https://en.wikipedia.org/wiki/Karlheinz_Stockhausen "Karlheinz Stockhausen")'s [Hymnen](https://en.wikipedia.org/wiki/Hymnen "Hymnen") and the application of [Group theory](https://en.wikipedia.org/wiki/Group_theory "Group theory") to transformations in music in the theoretical writings of [David Lewin](https://en.wikipedia.org/wiki/David_Lewin "David Lewin"). [![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/38/Della_Pittura_Alberti_perspective_pillars_on_grid.jpg/250px-Della_Pittura_Alberti_perspective_pillars_on_grid.jpg)](https://en.wikipedia.org/wiki/File:Della_Pittura_Alberti_perspective_pillars_on_grid.jpg) Diagram from [Leon Battista Alberti](https://en.wikipedia.org/wiki/Leon_Battista_Alberti "Leon Battista Alberti")'s 1435 [Della Pittura](https://en.wikipedia.org/wiki/Della_Pittura "Della Pittura"), with pillars in [perspective](https://en.wikipedia.org/wiki/Perspective_\(graphical\) "Perspective (graphical)") on a grid Examples of the use of mathematics in the visual arts include applications of [chaos theory](https://en.wikipedia.org/wiki/Chaos_theory "Chaos theory") and [fractal](https://en.wikipedia.org/wiki/Fractal "Fractal") geometry to [computer-generated art](https://en.wikipedia.org/wiki/Digital_art "Digital art"), symmetry studies of [Leonardo da Vinci](https://en.wikipedia.org/wiki/Leonardo_da_Vinci "Leonardo da Vinci"), [projective geometries](https://en.wikipedia.org/wiki/Projective_geometry "Projective geometry") in development of the [perspective](https://en.wikipedia.org/wiki/Perspective_\(graphical\) "Perspective (graphical)") theory of [Renaissance](https://en.wikipedia.org/wiki/Renaissance "Renaissance") art, [grids](https://en.wikipedia.org/wiki/Grid_\(page_layout\) "Grid (page layout)") in [Op art](https://en.wikipedia.org/wiki/Op_art "Op art"), optical geometry in the [camera obscura](https://en.wikipedia.org/wiki/Camera_obscura "Camera obscura") of [Giambattista della Porta](https://en.wikipedia.org/wiki/Giambattista_della_Porta "Giambattista della Porta"), and multiple perspective in analytic [cubism](https://en.wikipedia.org/wiki/Cubism "Cubism") and [futurism](https://en.wikipedia.org/wiki/Futurism "Futurism"). [Sacred geometry](https://en.wikipedia.org/wiki/Sacred_geometry "Sacred geometry") is a field of its own, giving rise to countless art forms including some of the best known mystic symbols and religious motifs, and has a particularly rich history in [Islamic architecture](https://en.wikipedia.org/wiki/Islamic_architecture "Islamic architecture"). It also provides a means of meditation and comtemplation, for example the study of the [Kaballah](https://en.wikipedia.org/wiki/Kaballah "Kaballah") [Sefirot](https://en.wikipedia.org/wiki/Sefirot "Sefirot") (Tree Of Life) and [Metatron's Cube](https://en.wikipedia.org/wiki/Metatron%27s_Cube "Metatron's Cube"); and also the act of drawing itself. The Dutch graphic designer [M. C. Escher](https://en.wikipedia.org/wiki/M._C._Escher "M. C. Escher") created mathematically inspired [woodcuts](https://en.wikipedia.org/wiki/Woodcut "Woodcut"), [lithographs](https://en.wikipedia.org/wiki/Lithograph "Lithograph"), and [mezzotints](https://en.wikipedia.org/wiki/Mezzotint "Mezzotint"). These feature impossible constructions, explorations of [infinity](https://en.wikipedia.org/wiki/Infinity "Infinity"), architecture, visual [paradoxes](https://en.wikipedia.org/wiki/Paradox "Paradox") and [tessellations](https://en.wikipedia.org/wiki/Tessellation "Tessellation"). Some painters and sculptors create work distorted with the mathematical principles of [anamorphosis](https://en.wikipedia.org/wiki/Anamorphosis "Anamorphosis"), including South African sculptor [Jonty Hurwitz](https://en.wikipedia.org/wiki/Jonty_Hurwitz "Jonty Hurwitz"). [Origami](https://en.wikipedia.org/wiki/Origami "Origami"), the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the [mathematics of paper folding](https://en.wikipedia.org/wiki/Mathematics_of_paper_folding "Mathematics of paper folding") by observing the [crease pattern](https://en.wikipedia.org/wiki/Crease_pattern "Crease pattern") on unfolded origami pieces.[\[46\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-46) British constructionist artist [John Ernest](https://en.wikipedia.org/wiki/John_Ernest "John Ernest") created reliefs and paintings inspired by group theory.[\[47\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-47) A number of other British artists of the constructionist and systems schools of thought also draw on mathematics models and structures as a source of inspiration, including [Anthony Hill](https://en.wikipedia.org/wiki/Anthony_Hill_\(artist\) "Anthony Hill (artist)") and [Peter Lowe](https://en.wikipedia.org/wiki/Peter_Lowe_\(artist\) "Peter Lowe (artist)").[\[48\]](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_note-48) Computer-generated art is based on mathematical [algorithms](https://en.wikipedia.org/wiki/Algorithm "Algorithm"). - [Argument from beauty](https://en.wikipedia.org/wiki/Argument_from_beauty "Argument from beauty") - [Fluency heuristic](https://en.wikipedia.org/wiki/Fluency_heuristic "Fluency heuristic") - [Golden ratio](https://en.wikipedia.org/wiki/Golden_ratio "Golden ratio") - [Neuroesthetics](https://en.wikipedia.org/wiki/Neuroesthetics "Neuroesthetics") - [Philosophy of mathematics](https://en.wikipedia.org/wiki/Philosophy_of_mathematics "Philosophy of mathematics") - [Processing fluency theory of aesthetic pleasure](https://en.wikipedia.org/wiki/Processing_fluency_theory_of_aesthetic_pleasure "Processing fluency theory of aesthetic pleasure") - [Pythagoreanism](https://en.wikipedia.org/wiki/Pythagoreanism "Pythagoreanism") - [Sense of wonder](https://en.wikipedia.org/wiki/Sense_of_wonder "Sense of wonder") - [Theory of everything](https://en.wikipedia.org/wiki/Theory_of_everything "Theory of everything") 1. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-1) Phillips, George (2005). ["Preface"](https://books.google.com/books?id=psFwdN6V6icC&q=there+is+nothing+in+the+world+of+mathematics+that+corresponds+to+an+audience+in+a+concert+hall,+where+the+passive+listen+to+the+active.+Happily,+mathematicians+are+all+doers,+not+spectators.&pg=PR7). Mathematics Is Not a Spectator Sport. [Springer Science+Business Media](https://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Media "Springer Science+Business Media"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-387-25528-1](https://en.wikipedia.org/wiki/Special:BookSources/0-387-25528-1 "Special:BookSources/0-387-25528-1") . Retrieved 2008-08-22. ""...there is nothing in the world of mathematics that corresponds to an audience in a concert hall, where the passive listen to the active. Happily, mathematicians are all doers, not spectators." 2. ^ [*a](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-rota-phenomenology_2-0) [b](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-rota-phenomenology_2-1) Rota, Gian-Carlo (May 1997). "The Phenomenology of Mathematical Beauty". Synthese. 111* (2): 171–182\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1023/A:1004930722234](https://doi.org/10.1023%2FA%3A1004930722234). 3. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-3) Wilson, Robin (2018). Euler's Pioneering Equation: The most beautiful theorem in mathematics. Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-19-879492-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-879492-9 "Special:BookSources/978-0-19-879492-9") . 4. ^ [*a](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-cain-form_4-0) [b](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-cain-form_4-1) [c](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-cain-form_4-2) Cain, Alan J. (2024). [Form & Number: A History of Mathematical Beauty](https://archive.org/details/cain_formandnumber_ebook_large). Lisbon: EBook. 5. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-5)* Gallagher, James (13 February 2014). ["Mathematics: Why the brain sees maths as beauty"](https://www.bbc.co.uk/news/science-environment-26151062). BBC News online. Retrieved 13 February 2014. 6. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-6) Feynman, Richard P. (1977). [The Feynman Lectures on Physics](https://feynmanlectures.caltech.edu/I_22.html#Ch22-S6-p4). Vol. I. Addison-Wesley. p. 22-16. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-201-02010-6](https://en.wikipedia.org/wiki/Special:BookSources/0-201-02010-6 "Special:BookSources/0-201-02010-6") . 7. ^ [*a](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-hardy-apology_7-0) [b](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-hardy-apology_7-1) [c](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-hardy-apology_7-2) Hardy, G.H. (1967). [A Mathematician's Apology](https://archive.org/details/hardy_annotated). Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-107-60463-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-60463-6 "Special:BookSources/978-1-107-60463-6") . 8. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-8)* Bell, E.T. (1937). Men of Mathematics. Simon & Schuster. 9. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-9) Wells, David (June 1990). "Are these the most beautiful?". The Mathematical Intelligencer. 12 (3): 37–41\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF03024015](https://doi.org/10.1007%2FBF03024015). 10. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-10) Paulos, John Allen (1991). Beyond Numeracy. New York: Alfred A. Knopf. pp. 125–127\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-394-586409](https://en.wikipedia.org/wiki/Special:BookSources/0-394-586409 "Special:BookSources/0-394-586409") . 11. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-11) Dutilh Novaes, Catarina (2019). "The Beauty (?) of Mathematical Proofs". In Aberdein, Andrew; Inglis, Matthew (eds.). Advances in Experimental Philosophy of Logic and Mathematics. Bloomsbury Academic. pp. 69–71\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-350-03901-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-350-03901-8 "Special:BookSources/978-1-350-03901-8") . 12. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-12) Polster, Burkert (2004). Q.E.D.: Beauty in Mathematical Proof. Walker & Company. pp. 32–33\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-8027-1431-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8027-1431-2 "Special:BookSources/978-0-8027-1431-2") . 13. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-13) Schechter, Bruce (2000). My Brain is Open: The Mathematical Journeys of Paul Erdős. New York: [Simon & Schuster](https://en.wikipedia.org/wiki/Simon_%26_Schuster "Simon & Schuster"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-684-85980-7](https://en.wikipedia.org/wiki/Special:BookSources/0-684-85980-7 "Special:BookSources/0-684-85980-7") . 14. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-14) [Aigner, Martin](https://en.wikipedia.org/wiki/Martin_Aigner "Martin Aigner"); [Ziegler, Günter M.](https://en.wikipedia.org/wiki/G%C3%BCnter_M._Ziegler "Günter M. Ziegler") (2018). [Proofs from THE BOOK](https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK "Proofs from THE BOOK") (6th ed.). Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-662-57264-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-662-57264-1 "Special:BookSources/978-3-662-57264-1") . 15. ^ [*a](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-plato-timaeus_15-0) [b](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-plato-timaeus_15-1) Plato (1929). [Timaeus](https://www.loebclassics.com/view/LCL234/1929/volume.xml). Cambridge, MA: Harvard University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-674-99257-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-674-99257-3 "Special:BookSources/978-0-674-99257-3") . 16. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-16)* Field, J.V. (2013). Kepler's Geometrical Cosmology. Bloomsbury. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9781472507037](https://en.wikipedia.org/wiki/Special:BookSources/9781472507037 "Special:BookSources/9781472507037") . 17. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-17) Whitfield, John. ["Journey to the 248th dimension"](https://www.nature.com/articles/news070319-4). Nature Online. Retrieved 12 September 2025. 18. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-18) Penrose, Roger (1974). "The Rôle of Aesthetics in Pure and Applied Mathematical Research". Bulletin of the Institute of Mathematics and its Applications. 10: 226–271\. 19. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-19) [Chandrasekhar, Subrahmanyan](https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar "Subrahmanyan Chandrasekhar") (1987). Truth and Beauty: Aesthetics and Motivation in Science. Chicago, London: University of Chicago Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-226-10087-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-226-10087-6 "Special:BookSources/978-0-226-10087-6") . 20. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-20) Dirac, P.A.M. (1940). "The Relation between Mathematics and Physics". Proceedings of the Royal Society of Edinburgh. 59: 122–129\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1017/s0370164600012207](https://doi.org/10.1017%2Fs0370164600012207). 21. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-21) Penrose, Roger (2004). The Road to Reality. London: Jonathan Cape. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-224-04447-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-224-04447-9 "Special:BookSources/978-0-224-04447-9") . 22. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-22) McAllister, James W. (1996). [Beauty & Revolution in Science](https://hdl.handle.net/1887/10158). Ithaca: Cornell University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-8014-3240-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8014-3240-8 "Special:BookSources/978-0-8014-3240-8") . 23. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-23) [Devlin, Keith](https://en.wikipedia.org/wiki/Keith_Devlin "Keith Devlin") (2000). "Do Mathematicians Have Different Brains?". [The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip](https://archive.org/details/mathgene00keit). [Basic Books](https://en.wikipedia.org/wiki/Basic_Books "Basic Books"). p. [140](https://archive.org/details/mathgene00keit/page/140). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-465-01619-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-465-01619-8 "Special:BookSources/978-0-465-01619-8") . Retrieved 2008-08-22. 24. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-24) Monastyrsky, Michael (2001). ["Some Trends in Modern Mathematics and the Fields Medal"](http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf) (PDF). Can. Math. Soc. Notes. 33 (2 and 3). 25. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-25) McAllister, James W. (2005). "Mathematical Beauty and the Evolution of the Standards of Mathematical Proof". In Emmer, Michele (ed.). The Visual Mind II. MIT Press. pp. 15–34\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-262-05076-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-262-05076-0 "Special:BookSources/978-0-262-05076-0") . 26. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-26) [Loomis, Elisha Scott](https://en.wikipedia.org/wiki/Elisha_Scott_Loomis "Elisha Scott Loomis") (1968). The Pythagorean Proposition. Washington, DC: National Council of Teachers of Mathematics. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-873-53036-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-873-53036-1 "Special:BookSources/978-0-873-53036-1") . 27. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-27) Weisstein, Eric W. ["Quadratic Reciprocity Theorem"](http://mathworld.wolfram.com/QuadraticReciprocityTheorem.html). mathworld.wolfram.com. Retrieved 2019-10-31. 28. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-davis-experience_28-0) Davis, Philip J.; Hersh, Reuben (1981). The Mathematical Experience. Boston: Houghton Mifflin. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-395-32131-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-395-32131-7 "Special:BookSources/978-0-395-32131-7") . 29. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-29) Hoffman, Paul (1999). The Man Who Loved Only Numbers. London: Fourth Estate. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-85702-829-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-85702-829-4 "Special:BookSources/978-1-85702-829-4") . 30. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-30) [Aristotle](https://en.wikipedia.org/wiki/Aristotle "Aristotle") (1995). "Metaphysics". In Barnes, Jonathan (ed.). The Complete Works of Aristotle. Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-691-01650-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-01650-4 "Special:BookSources/978-0-691-01650-4") . 31. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-31) [Russell, Bertrand](https://en.wikipedia.org/wiki/Bertrand_Russell "Bertrand Russell") (1919). "The Study of Mathematics". [Mysticism and Logic: And Other Essays](https://archive.org/details/bub_gb_zwMQAAAAYAAJ). [Longman](https://en.wikipedia.org/wiki/Longman "Longman"). p. [60](https://archive.org/details/bub_gb_zwMQAAAAYAAJ/page/n68). Retrieved 2008-08-22. "Mathematics rightly viewed possesses not only truth but supreme beauty a beauty cold and austere like that of sculpture without appeal to any part of our weaker nature without the gorgeous trappings Russell." 32. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-32) [Harré, Rom](https://en.wikipedia.org/wiki/Rom_Harr%C3%A9 "Rom Harré") (1958). ["Quasi-Aesthetic Appraisals"](https://www.jstor.org/stable/3748562). Philosophy. 33: 132–137\. 33. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-33) [Zangwill, Nick](https://en.wikipedia.org/wiki/Nick_Zangwill "Nick Zangwill") (2001). The Metaphysics of Beauty. Ithaca, London: Cornell University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-8014-3820-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8014-3820-2 "Special:BookSources/978-0-8014-3820-2") . 34. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-34) A. Moles: Théorie de l'information et perception esthétique, Paris, Denoël, 1973 ([Information Theory](https://en.wikipedia.org/wiki/Information_Theory "Information Theory") and aesthetical perception) 35. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-35) F Nake (1974). Ästhetik als Informationsverarbeitung. ([Aesthetics](https://en.wikipedia.org/wiki/Aesthetics "Aesthetics") as information processing). Grundlagen und Anwendungen der Informatik im Bereich ästhetischer Produktion und Kritik. Springer, 1974, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [3-211-81216-4](https://en.wikipedia.org/wiki/Special:BookSources/3-211-81216-4 "Special:BookSources/3-211-81216-4") , [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-211-81216-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-211-81216-7 "Special:BookSources/978-3-211-81216-7") 36. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-36) J. Schmidhuber. [Low-complexity art](https://en.wikipedia.org/wiki/Low-complexity_art "Low-complexity art"). [Leonardo](https://en.wikipedia.org/wiki/Leonardo_\(journal\) "Leonardo (journal)"), Journal of the International Society for the Arts, Sciences, and Technology ([Leonardo/ISAST](https://en.wikipedia.org/wiki/Leonardo,_The_International_Society_of_the_Arts,_Sciences_and_Technology "Leonardo, The International Society of the Arts, Sciences and Technology")), 30(2):97–103, 1997. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2307/1576418](https://doi.org/10.2307%2F1576418). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [1576418](https://www.jstor.org/stable/1576418). 37. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-37) J. Schmidhuber. Papers on the theory of beauty and [low-complexity art](https://en.wikipedia.org/wiki/Low-complexity_art "Low-complexity art") since 1994: <http://www.idsia.ch/~juergen/beauty.html> 38. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-38) J. Schmidhuber. Simple Algorithmic Principles of Discovery, Subjective Beauty, Selective Attention, Curiosity & Creativity. Proc. 10th Intl. Conf. on Discovery Science (DS 2007) pp. 26–38, LNAI 4755, Springer, 2007. Also in Proc. 18th Intl. Conf. on Algorithmic Learning Theory (ALT 2007) p. 32, LNAI 4754, Springer, 2007. Joint invited lecture for DS 2007 and ALT 2007, Sendai, Japan, 2007. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[0709\.0674](https://arxiv.org/abs/0709.0674). 39. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-39) .J. Schmidhuber. Curious model-building control systems. International Joint Conference on Neural Networks, Singapore, vol 2, 1458–1463. IEEE press, 1991 40. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-40) Schmidhuber's theory of beauty and curiosity in a German TV show: <http://www.br-online.de/bayerisches-fernsehen/faszination-wissen/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml> [Archived](https://web.archive.org/web/20080603221058/http://www.br-online.de/bayerisches-fernsehen/faszination-wissen/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml) June 3, 2008, at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine") 41. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-41) Zeki, Semir; Romaya, John Paul; Benincasa, Dionigi M. T.; Atiyah, Michael F. (2014). ["The experience of mathematical beauty and its neural correlates"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3923150). Frontiers in Human Neuroscience. 8: 68. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3389/fnhum.2014.00068](https://doi.org/10.3389%2Ffnhum.2014.00068). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1662-5161](https://search.worldcat.org/issn/1662-5161). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [3923150](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3923150). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [24592230](https://pubmed.ncbi.nlm.nih.gov/24592230). 42. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-42) Zhang, Haoxuan; Zeki, Semir (May 2022). ["Judgments of mathematical beauty are resistant to revision through external opinion"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9790661). PsyCh Journal. 11 (5): 741–747\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/pchj.556](https://doi.org/10.1002%2Fpchj.556). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [2046-0252](https://search.worldcat.org/issn/2046-0252). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [9790661](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9790661). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [35491015](https://pubmed.ncbi.nlm.nih.gov/35491015). 43. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-43) [Luque, Sergio. 2009. "The Stochastic Synthesis of Iannis Xenakis." Leonardo Music Journal (19): 77–84](http://www.sergioluque.com/stochastics) 44. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-44) Norris, Chris (2001). ["Hammer Of The Gods"](https://web.archive.org/web/20111101195240/http://toolshed.down.net/articles/text/spinmag.jun.2001.html). Archived from [the original](http://toolshed.down.net/articles/text/spinmag.jun.2001.html) on November 1, 2011. Retrieved April 25, 2007. 45. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-45) Schell, Michael (December 11, 2018). ["Elliott Carter (1908–2012): Legacy of a Centenarian"](https://www.secondinversion.org/2018/12/11/elliott-carter-1908-2012-legacy-of-a-centenarian/). Second Inversion. [Archived](https://web.archive.org/web/20181224054514/https://www.secondinversion.org/2018/12/11/elliott-carter-1908-2012-legacy-of-a-centenarian/) from the original on December 24, 2018. Retrieved December 11, 2018. 46. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-46) Hull, Thomas. "Project Origami: Activities for Exploring Mathematics". Taylor & Francis, 2006. 47. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-47) John Ernest's use of mathematics and especially group theory in his art works is analysed in John Ernest, A Mathematical Artist by Paul Ernest in Philosophy of Mathematics Education Journal, No. 24 Dec. 2009 (Special Issue on Mathematics and Art): <http://people.exeter.ac.uk/PErnest/pome24/index.htm> 48. [^](https://en.wikipedia.org/wiki/Mathematical_beauty#cite_ref-48) Franco, Francesca (2017-10-05). ["The Systems Group (Chapter 2)"](https://books.google.com/books?id=oJU4DwAAQBAJ&q=anthony+hill+and+peter+lowe&pg=PT61). Generative Systems Art: The Work of Ernest Edmonds. Routledge. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9781317137436](https://en.wikipedia.org/wiki/Special:BookSources/9781317137436 "Special:BookSources/9781317137436") . - Cellucci, Carlo (2015), "Mathematical beauty, understanding, and discovery", [Foundations of Science](https://en.wikipedia.org/wiki/Foundations_of_Science "Foundations of Science"), 20 (4): 339–355, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/s10699-014-9378-7](https://doi.org/10.1007%2Fs10699-014-9378-7), [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [120068870](https://api.semanticscholar.org/CorpusID:120068870) - [Hoffman, Paul](https://en.wikipedia.org/wiki/Paul_Hoffman_\(science_writer\) "Paul Hoffman (science writer)") (1992), [The Man Who Loved Only Numbers](https://en.wikipedia.org/wiki/The_Man_Who_Loved_Only_Numbers "The Man Who Loved Only Numbers"), Hyperion. - [Loomis, Elisha Scott](https://en.wikipedia.org/wiki/Elisha_Scott_Loomis "Elisha Scott Loomis") (1968), The Pythagorean Proposition, The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem. - [Pandey, S.K.](https://en.wikipedia.org/w/index.php?title=S.K._Pandey&action=edit&redlink=1 "S.K. Pandey (page does not exist)") . [The Humming of Mathematics: Melody of Mathematics](https://books.google.com/books?id=BgYbzAEACAAJ). Independently Published, 2019. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [1710134437](https://en.wikipedia.org/wiki/Special:BookSources/1710134437 "Special:BookSources/1710134437") . - Peitgen, H.-O., and Richter, P.H. (1986), The Beauty of Fractals, Springer-Verlag. - [Reber, R.](https://en.wikipedia.org/wiki/Rolf_Reber "Rolf Reber"); Brun, M.; Mitterndorfer, K. (2008). "The use of heuristics in intuitive mathematical judgment". Psychonomic Bulletin & Review. 15 (6): 1174–1178\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3758/PBR.15.6.1174](https://doi.org/10.3758%2FPBR.15.6.1174). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[1956/2734](https://hdl.handle.net/1956%2F2734). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [19001586](https://pubmed.ncbi.nlm.nih.gov/19001586). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [5297500](https://api.semanticscholar.org/CorpusID:5297500). - Strohmeier, John, and Westbrook, Peter (1999), Divine Harmony, The Life and Teachings of Pythagoras, Berkeley Hills Books, Berkeley, CA. - [Zeki, S.](https://en.wikipedia.org/wiki/Semir_Zeki "Semir Zeki"); Romaya, J. P.; Benincasa, D. M. T.; [Atiyah, M. F.](https://en.wikipedia.org/wiki/Michael_Atiyah "Michael Atiyah") (2014), "The experience of mathematical beauty and its neural correlates", Frontiers in Human Neuroscience, 8: 68, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3389/fnhum.2014.00068](https://doi.org/10.3389%2Ffnhum.2014.00068), [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [3923150](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3923150), [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [24592230](https://pubmed.ncbi.nlm.nih.gov/24592230) - [Aigner, Martin](https://en.wikipedia.org/wiki/Martin_Aigner "Martin Aigner"); [Ziegler, Günter M.](https://en.wikipedia.org/wiki/G%C3%BCnter_M._Ziegler "Günter M. Ziegler") (2018). [Proofs from THE BOOK](https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK "Proofs from THE BOOK") (6th ed.). Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-662-57264-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-662-57264-1 "Special:BookSources/978-3-662-57264-1") . - Cain, Alan J. (2024). [Form & Number: A History of Mathematical Beauty](https://archive.org/details/cain_formandnumber_ebook_large). Lisbon: Ebook. - [Hadamard, Jacques](https://en.wikipedia.org/wiki/Jacques_Hadamard "Jacques Hadamard") (1949). [An Essay on the Psychology of Invention in the Mathematical Field](https://archive.org/details/essayonpsycholog00hada) (2nd enlarged ed.). Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-486-20107-4](https://en.wikipedia.org/wiki/Special:BookSources/0-486-20107-4 "Special:BookSources/0-486-20107-4") . - [Hardy, G.H.](https://en.wikipedia.org/wiki/G.H._Hardy "G.H. Hardy") (1967) \[1st published 1940\]. [A Mathematician's Apology](https://archive.org/details/hardy_annotated). Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-107-60463-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-60463-6 "Special:BookSources/978-1-107-60463-6") . - Huntley, H.E. (1970). [The Divine Proportion: A Study in Mathematical Beauty](https://archive.org/details/divineproportion0000hunt_o2w9). New York: Dover Publications. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-486-22254-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-22254-7 "Special:BookSources/978-0-486-22254-7") . - Stewart, Ian (2007). Why beauty is truth : a history of symmetry. New York: Basic Books, a member of the Perseus Books Group. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-465-08236-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-465-08236-0 "Special:BookSources/978-0-465-08236-0") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [76481488](https://search.worldcat.org/oclc/76481488). - [Mathematics, Poetry and Beauty](https://web.archive.org/web/20151209130947/http://raharoni.net.technion.ac.il/mathematics-poetry-and-beauty/) - [Is Mathematics Beautiful?](http://www.cut-the-knot.org/manifesto/beauty.shtml) cut-the-knot.org - [Justin Mullins.com](http://www.justinmullins.com/) - [Edna St. Vincent Millay (poet): Euclid alone has looked on beauty bare](http://www.the-athenaeum.org/poetry/detail.php?id=80) - [Terence Tao](https://en.wikipedia.org/wiki/Terence_Tao "Terence Tao"), [What is good mathematics?](https://arxiv.org/abs/math/0702396) - [Mathbeauty Blog](http://mathbeauty.wordpress.com/) - The [Aesthetic Appeal](https://archive.org/details/aestheticappeal) collection at the [Internet Archive](https://en.wikipedia.org/wiki/Internet_Archive "Internet Archive") - [A Mathematical Romance](http://www.nybooks.com/articles/archives/2013/dec/05/mathematical-romance/) [Jim Holt](https://en.wikipedia.org/wiki/Jim_Holt_\(philosopher\) "Jim Holt (philosopher)") December 5, 2013 issue of [The New York Review of Books](https://en.wikipedia.org/wiki/The_New_York_Review_of_Books "The New York Review of Books") review of Love and Math: The Heart of Hidden Reality by [Edward Frenkel](https://en.wikipedia.org/wiki/Edward_Frenkel "Edward Frenkel")
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