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design; 5. Logicality, rigour, tight reasoning and deduction, pure thought; 6. Interconnectedness, links, unification; 7. Applicability, modelling power, empirical generality. He argues that individual mathematicians and communities of mathematicians will have preferred choices from this list. Some, like Hardy, will reject some (Hardy claimed that applied mathematics is ugly). However, Rentuya Sa and colleagues compared the views of
British mathematicians and undergraduates and Chinese mathematicians on the beauty of 20 well known equations and found a strong measure of agreement between their views.
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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 music or joy or sorrow. Moreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers.
1911:
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,
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Brain imaging experiments conducted by Semir Zeki and his colleagues 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
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designed for 2nd and 3rd graders, where students create their own snowflakes by folding a square piece of paper and cutting out designs of their choice along the edges of the folded paper. When the paper is unfolded, a symmetrical design reveals itself. In a day to day elementary school mathematics
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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
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proposes seven dimensions for any mathematical objects, including concepts, theorems, proofs and theories. These are 1. Economy, simplicity, brevity, succinctness, elegance; 2. Generality, abstraction, power; 3. Surprise, ingenuity, cleverness; 4. Pattern, structure, symmetry, regularity, visual
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381:'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. Brieskorn showed that these differential structures can be described in an extremely explicit and beautiful form.
588:, the study of counting, has artistic representations which some find mathematically beautiful. There are many visual examples which illustrate combinatorial concepts. Some of the topics and objects seen in combinatorics courses with visual representations include, among others
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by using algebra tiles. Cuisenaire rods can be used to teach fractions, and pattern blocks can be used to teach geometry. Using mathematical manipulatives helps students gain a conceptual understanding that might not be seen immediately in written mathematical formulas.
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In the search for an elegant proof, mathematicians may search for multiple 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
654:'s philosophy there were two worlds, the physical one in which we live and another abstract world which contained unchanging truth, including mathematics. He believed that the physical world was a mere reflection of the more perfect abstract world.
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In many cases, natural philosophers and other scientists who have made extensive use of mathematics have made leaps of inference between beauty and physical truth in ways that turned out to be erroneous. For example, at one stage in his life,
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There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture—that it came to him from outside, and that he did not consciously create it from within.
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created reliefs and paintings inspired by group theory. 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
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These mathematicians believe that the detailed and precise results of mathematics may be reasonably taken to be true without any dependence on the universe in which we live. For example, they would argue that the theory of the
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This disagreement illustrates both the subjective nature of mathematical beauty and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them.
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spoke of an imaginary 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 exclaim "This one's from The Book!"
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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.
538:. In a general Math Circle lesson, students use pattern finding, observation, and exploration to make their own mathematical discoveries. For example, mathematical beauty arises in a Math Circle activity on
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greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.
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is fundamentally valid, in a way that does not require any specific context. Some mathematicians have extrapolated this viewpoint that mathematical beauty is truth further, in some cases becoming
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Some mathematicians see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be unrelated. These results are often described as
337:. A trivial theorem may be a result that can be derived in an obvious and straightforward way from other known results, or which applies only to a specific set of particular objects such as the
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312:) in a surprising way. In particular, the area of a triangle on a curved surface is proportional to the excess of the triangle and the proportionality is curvature. Another example is the
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1460:"...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
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of another. As there are exactly five
Platonic solids, Kepler's hypothesis could only accommodate six planetary orbits and was disproved by the subsequent discovery of
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Conversely, results that are logically correct but involve laborious calculations, over-elaborate methods, highly conventional approaches or a large number of powerful
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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
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than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.
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A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago the proof of the existence of
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are after-school enrichment programs where students engage with mathematics through lectures and activities; there are also some teachers who encourage
169:. While it is difficult to find universal agreement on whether a result is deep, some examples are more commonly cited than others. One such example is
768:) relative to what the observer already knows. Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the
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analysed the beauty of mathematical proofs into these six dimensions: general, serious, deep, unexpected, inevitable, economical (simple).
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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.
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as information processing). Grundlagen und
Anwendungen der Informatik im Bereich ästhetischer Produktion und Kritik. Springer, 1974,
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class, symmetry can be presented as such in an artistic manner where students see aesthetically pleasing results in mathematics.
341:. In some occasions, a statement of a theorem can be original enough to be considered deep, though its proof is fairly obvious.
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1st edition, Princeton
University Press, Princeton, NJ. 2nd edition, 1949. Reprinted, Dover Publications, New York, NY, 1954.
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1790:"Rentuya Sa, Lara Alcock, Matthew Inglis and Fenner Stanley Tanswell: Do Mathematicians Agree about Mathematical Beauty?"
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Some mathematicians are of the opinion that the doing of mathematics is closer to discovery than invention, for example:
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http://www.br-online.de/bayerisches-fernsehen/faszination-wissen/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml
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42:. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as
574:. Origami, the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the
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Other examples of deep results include unexpected insights into mathematical structures. For example, Gauss's
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1934:. International Joint Conference on Neural Networks. Vol. 2. Singapore: IEEE press. pp. 1458–1463.
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relative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an
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suggested that a beautiful proof or result possesses "inevitability", "unexpectedness", and "economy".
488:, among others) to its "great mathematical beauty". The beauty of mathematics is experienced when the
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is mathematics. Badiou also believes in deep connections between mathematics, poetry and philosophy.
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2258:; Brun, M.; Mitterndorfer, K. (2008). "The use of heuristics in intuitive mathematical judgment".
2204:, The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem.
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Hull, Thomas. "Project
Origami: Activities for Exploring Mathematics". Taylor & Francis, 2006.
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795:) leads to improved data compression such that the observation sequence can be described by fewer
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called "our jewel" and "the most remarkable formula in mathematics". Modern examples include the
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LNAI 4754, Springer, 2007. Joint invited lecture for DS 2007 and ALT 2007, Sendai, Japan, 2007.
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633:, from a lecture to the Royal Institution titled "Some of the conditions of mental development"
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John Ernest's use of mathematics and especially group theory in his art works is analysed in
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or previous results are usually not considered to be elegant, and may be even referred to as
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to present mathematics in an aesthetically pleasing way. Examples of a manipulative include
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Some believe that in order to appreciate mathematics, one must engage in doing mathematics.
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A method of proof that can be easily generalized to solve a family of similar problems.
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Some painters and sculptors create work distorted with the mathematical principles of
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A proof that derives a result in a surprising way (e.g., from an apparently unrelated
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1606:"Judgments of mathematical beauty are resistant to revision through external opinion"
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Zeki, Semir; Romaya, John Paul; Benincasa, Dionigi M. T.; Atiyah, Michael F. (2014).
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pleasure derived from the abstractness, purity, simplicity, depth or orderliness of
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Sowell, E (1989). "Effects of
Manipulative Materials in Mathematics Instruction".
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formulated a mathematical theory of observer-dependent subjective beauty based on
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A Philosophical
Enquiry into the Origin of Our Ideas of the Sublime and Beautiful
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of mathematics when he said, "Why are numbers beautiful? It's like asking why is
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The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip
1212:"Proof elegance - MAA Mathematical Communication MAA Mathematical Communication"
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of subjectively perceived beauty: the observer continually tries to improve the
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1866:, Journal of the International Society for the Arts, Sciences, and Technology (
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An example of "beauty in method"—a simple and elegant visual descriptor of the
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Examples of the use of mathematics in the visual arts include applications of
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760:: the most beautiful objects among subjectively comparable objects have short
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Cellucci, Carlo (2015), "Mathematical beauty, understanding, and discovery",
2155:(foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992.
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791:. Whenever the observer's learning process (possibly a predictive artificial
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believed that the proportions of the orbits of the then-known planets in the
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alone had eight different proofs of this theorem, six of which he published.
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2439:(2014), "The experience of mathematical beauty and its neural correlates",
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476:, who "did mathematics for the beauty of it". The aesthetic pleasure that
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A proof that uses a minimum of additional assumptions or previous results.
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1981:
https://www.exeter.ac.uk/research/groups/education/pmej/pome24/index.htm
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of the observations by discovering regularities such as repetitions and
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1671:(Spring 2018 ed.), Metaphysics Research Lab, Stanford University
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It is very difficult to find an analogous invention in the past to
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1952:
Schmidhuber's theory of beauty and curiosity in a German TV show:
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Mathematicians commonly describe an especially pleasing method of
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1547:"The experience of mathematical beauty and its neural correlates"
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Viewpoints: Mathematical Perspective and Fractal Geometry in Art
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published over 360 proofs in his book Pythagorean Proposition (
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2409:. New York: Basic Books, a member of the Perseus Books Group.
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numbers are beautiful. If they aren't beautiful, nothing is".
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or describe mathematics as an art form, (a position taken by
1979:, No. 24 Dec. 2009 (Special Issue on Mathematics and Art):
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Another example of beauty in experience involves the use of
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beautiful. If you don't see why, someone can't tell you. I
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expressed his sense of mathematical beauty in these words:
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to transformations in music in the theoretical writings of
1024:. These feature impossible constructions, explorations of
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expression ties together arguably the five most important
1690:
My brain is open: The mathematical journeys of Paul Erdős
2210:"Some Trends in Modern Mathematics and the Fields Medal"
2129:
Truth and Beauty: Aesthetics and Motivations in Science,
826:
Examples of the use of mathematics in music include the
508:—the building blocks of matter. Similarly, the study of
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The Beauty of Doing Mathematics: Three Public Dialogues
1830:
F Nake (1974). Ästhetik als Informationsverarbeitung. (
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A "cold and austere beauty" has been attributed to the
2139:
The Psychology of Invention in the Mathematical Field,
1400:
Some Trends in Modern Mathematics and the Fields Medal
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to correspond to a concentric arrangement of the five
3370:
The Work of Art in the Age of Mechanical Reproduction
2172:
The Divine Proportion: A Study in Mathematical Beauty
182:
181:
2346:
Divine Harmony, The Life and Teachings of Pythagoras
2308:(1997). "The phenomenology of mathematical beauty".
2061:(2000). "Do Mathematicians Have Different Brains?".
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is a deep theorem which relates a local phenomenon (
253:, which establishes an important connection between
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1895:J. Schmidhuber. Papers on the theory of beauty and
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A proof that is based on new and original insights.
4553:The Drawing of Geometric Patterns in Saracenic Art
2062:
1998:Generative Systems Art: The Work of Ernest Edmonds
1063:. Computer-generated art is based on mathematical
215:
4688:Goudreau Museum of Mathematics in Art and Science
2233:The Humming of Mathematics: Melody of Mathematics
1815:Théorie de l'information et perception esthétique
1291:"Mathematics: Why the brain sees maths as beauty"
504:equations, became a fruitful way of categorizing
2406:Why beauty is truth : a history of symmetry
2344:Strohmeier, John, and Westbrook, Peter (1999),
1158:Processing fluency theory of aesthetic pleasure
622:
375:
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216:{\displaystyle \displaystyle e^{i\pi }+1=0\,.}
4683:European Society for Mathematics and the Arts
3857:Mathematica: A World of Numbers... and Beyond
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2575:
1478:Journal for Research in Mathematics Education
8:
3837:List of works designed with the golden ratio
3300:The Literary Mind and the Carving of Dragons
1665:"Platonism in the Philosophy of Mathematics"
1977:Philosophy of Mathematics Education Journal
562:. For example, one can teach the method of
480:tend to experience in Einstein's theory of
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3558:
3544:
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2582:
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2551:Love and Math: The Heart of Hidden Reality
2247:Peitgen, H.-O., and Richter, P.H. (1986),
1718:"Alain Badiou: Ontology and Structuralism"
261:(work on which led to the awarding of the
3897:Cathedral of Saint Mary of the Assumption
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2131:University of Chicago Press, Chicago, IL.
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1621:
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448:Learn how and when to remove this message
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1901:http://www.idsia.ch/~juergen/beauty.html
1604:Zhang, Haoxuan; Zeki, Semir (May 2022).
894:. They also include the application of
393:
373:In contrast, Monastyrsky wrote in 2001:
367:non-equivalent differentiable structures
4443:Vier Bücher von Menschlicher Proportion
1669:The Stanford Encyclopedia of Philosophy
1179:
976:, and multiple perspective in analytic
83:. Depending on context, this may mean:
2504:Euclid alone has looked on beauty bare
2435:; Romaya, J. P.; Benincasa, D. M. T.;
1932:Curious model-building control systems
1005:; and also the act of drawing itself.
884:, and application of Shepard tones in
468:study has been part of the experience
2348:, Berkeley Hills Books, Berkeley, CA.
2027:Mysticism and Logic: And Other Essays
1762:"Paul Ernest: Mathematics and Beauty"
1289:Gallagher, James (13 February 2014).
668:Twentieth-century French philosopher
7:
2024:(1919). "The Study of Mathematics".
1436:Mathematics Is Not a Spectator Sport
1385:
1276:
4628:Journal of Mathematics and the Arts
2392:"Is Beauty Truth and Truth Beauty?"
2174:, Dover Publications, New York, NY.
1794:Review of Philosophy and Psychology
1043:, including South African sculptor
316:(and its vector versions including
90:A proof that is unusually succinct.
4642:Making Mathematics with Needlework
2151:, 1st published, 1940. Reprinted,
1973:John Ernest, A Mathematical Artist
1344:"Fundamental Theorems of Calculus"
420:tone or style may not reflect the
14:
4468:I quattro libri dell'architettura
2260:Psychonomic Bulletin & Review
2236:. Independently Published, 2019.
716:Analysis of Beauty in Mathematics
512:provides important insights into
4747:
4746:
3992:Self-portrait in a Convex Mirror
3676:
3518:
2502:Edna St. Vincent Millay (poet):
2390:Martin Gardner (April 1, 2007).
1992:Franco, Francesca (2017-10-05).
1192:www-history.mcs.st-andrews.ac.uk
1012:created mathematically inspired
430:guide to writing better articles
409:
151:= 1, travelling at the velocity
2441:Frontiers in Human Neuroscience
1994:"The Systems Group (Chapter 2)"
1551:Frontiers in Human Neuroscience
1440:Springer Science+Business Media
1321:. Vol. I. Addison-Wesley.
1318:The Feynman Lectures on Physics
1254:"Quadratic Reciprocity Theorem"
1050:British constructionist artist
744:analyzed links between beauty,
314:fundamental theorem of calculus
4703:National Museum of Mathematics
4455:Regole generali d'architettura
2486:Mathematics, Poetry and Beauty
2164:The Man Who Loved Only Numbers
758:algorithmic information theory
492:of objects are represented by
16:Aesthetic value of mathematics
1:
2208:Monastyrsky, Michael (2001).
2186:. New York: Springer-Verlag.
2121:3rd edition, Springer-Verlag.
1667:, in Zalta, Edward N. (ed.),
1375:– via Internet Archive.
97:or a collection of theorems).
4228:Garden of Cosmic Speculation
3430:Aestheticization of politics
2547:The New York Review of Books
1315:Feynman, Richard P. (1977).
1138:Mathematics and architecture
817:Mathematics and architecture
582:on unfolded origami pieces.
576:mathematics of paper folding
546:Some teachers prefer to use
2202:The Pythagorean Proposition
2125:Chandrasekhar, Subrahmanyan
1821:and aesthetical perception)
1090:expressed his views on the
1008:The Dutch graphic designer
534:by teaching mathematics in
57:Comparisons are made with
4811:
3745:Islamic geometric patterns
2545:December 5, 2013 issue of
1748:Cambridge University Press
1522:Introductory Combinatorics
1096:Beethoven's Ninth Symphony
968:, optical geometry in the
806:
696:, each orbit lying on the
548:mathematical manipulatives
308:) to a global phenomenon (
4785:Philosophy of mathematics
4742:
4566:A Mathematician's Apology
3674:
3498:
2515:What is good mathematics?
2491:Is Mathematics Beautiful?
2376:10.1007/s10699-014-9378-7
2149:A Mathematician's Apology
1940:10.1109/IJCNN.1991.170605
1744:A Mathematician's Apology
1688:Schechter, Bruce (2000).
1663:Linnebo, Øystein (2018),
1520:Brualdi, Richard (2009).
1430:Phillips, George (2005).
1373:A Mathematician's Apology
1153:Philosophy of mathematics
347:A Mathematician's Apology
50:) or, at a minimum, as a
4790:Mathematical terminology
4678:The Bridges Organization
2454:10.3389/fnhum.2014.00068
1930:Schmidhuber, J. (1991).
1564:10.3389/fnhum.2014.00068
1071:Quotes by mathematicians
974:Giambattista della Porta
631:William Kingdon Clifford
484:has been attributed (by
472:, including that of the
470:of various civilizations
4540:The Grammar of Ornament
4492:Nature's Harmonic Unity
4402:De prospectiva pingendi
3450:Evolutionary aesthetics
3400:The Aesthetic Dimension
2322:10.1023/A:1004930722234
1870:), 30(2):97–103, 1997.
1817:, Paris, Denoël, 1973 (
1028:, architecture, visual
478:mathematical physicists
424:used on Knowledge (XXG)
4780:Elementary mathematics
4693:Institute For Figuring
4605:The 'Life' of a Carpet
4430:A Treatise on Painting
3380:Avant-Garde and Kitsch
3330:Lectures on Aesthetics
2539:A Mathematical Romance
2363:Foundations of Science
2249:The Beauty of Fractals
2170:Huntley, H.E. (1970),
1085:
952:in development of the
944:, symmetry studies of
942:computer-generated art
929:
746:information processing
688:have been arranged by
636:
464:that is separate from
428:See Knowledge (XXG)'s
402:
400:compound of five cubes
383:
371:
277:", which connects the
245:, which the physicist
231:mathematical constants
217:
160:
28:
4574:George David Birkhoff
4548:Ernest Hanbury Hankin
4416:De divina proportione
4396:Piero della Francesca
4375:Leon Battista Alberti
3962:Piero della Francesca
3601:Hyperboloid structure
3525:Philosophy portal
2403:Stewart, Ian (2007).
2272:10.3758/PBR.15.6.1174
2217:Can. Math. Soc. Notes
1959:June 3, 2008, at the
1742:Hardy, G. H. (1940).
1420:Chandrasekhar, p. 148
1348:mathworld.wolfram.com
1258:mathworld.wolfram.com
1188:"Quotations by Hardy"
1080:
950:projective geometries
916:Leon Battista Alberti
913:
886:Karlheinz Stockhausen
848:Johann Sebastian Bach
813:Mathematics and music
766:Kolmogorov complexity
732:In information theory
564:completing the square
397:
218:
146:
114:quadratic reciprocity
22:
4499:Frederik Macody Lund
4370:Filippo Brunelleschi
4251:Hamid Naderi Yeganeh
4113:La condition humaine
3470:Philosophy of design
3350:In Praise of Shadows
3340:The Critic as Artist
2198:Loomis, Elisha Scott
2117:Proofs from THE BOOK
1769:Mathematics Teaching
1694:Simon & Schuster
1398:Monastyrsky (2001),
1168:Theory of everything
1113:Argument from beauty
992:Islamic architecture
764:descriptions (i.e.,
536:kinesthetic learning
518:loop quantum gravity
506:elementary particles
179:
118:Carl Friedrich Gauss
4795:Mathematics and art
4724:Mathematical beauty
4649:Rhythm of Structure
4592:Gödel, Escher, Bach
4388:De re aedificatoria
4019:The Ancient of Days
3638:Projective geometry
3567:Mathematics and art
3480:Philosophy of music
3455:Mathematical beauty
2396:Scientific American
1342:Weisstein, Eric W.
1252:Weisstein, Eric W.
1232:Elisha Scott Loomis
1123:Descriptive science
1001:(Tree Of Life) and
809:Mathematics and art
494:mathematical models
344:In his 1940 essay
275:monstrous moonshine
110:Pythagorean theorem
32:Mathematical beauty
25:Pythagorean theorem
4729:Patterns in nature
4586:Douglas Hofstadter
4212:Desmond Paul Henry
4202:Bathsheba Grossman
4134:The Swallow's Tail
4055:Giorgio de Chirico
3927:Sydney Opera House
3782:Croatian interlace
3475:Philosophy of film
3465:Patterns in nature
3435:Applied aesthetics
3410:Why Beauty Matters
3196:Life imitating art
3057:Art for art's sake
2531:collection at the
2497:Justin Mullins.com
2251:, Springer-Verlag.
2112:Ziegler, Gunter M.
1975:by Paul Ernest in
1897:low-complexity art
1860:Low-complexity art
1819:Information Theory
1722:Ceasefire Magazine
1696:. pp. 70–71.
1118:Cellular automaton
930:
924:, with pillars in
861:The Rite of Spring
854:structures (as in
846:, counterpoint of
836:Fibonacci sequence
754:Jürgen Schmidhuber
750:information theory
606:Partition of a set
590:Four color theorem
532:student engagement
482:general relativity
403:
251:modularity theorem
213:
212:
161:
29:
4762:
4761:
4615:
4614:
4579:Aesthetic Measure
4450:Sebastiano Serlio
4424:Leonardo da Vinci
4314:
4313:
4306:Margaret Wertheim
3967:Leonardo da Vinci
3533:
3532:
3485:Psychology of art
3360:Art as Experience
2416:978-0-465-08236-0
2135:Hadamard, Jacques
2084:978-0-465-01619-8
2022:Russell, Bertrand
1848:978-3-211-81216-7
1464:, not spectators.
1148:Normative science
1128:Fluency heuristic
946:Leonardo da Vinci
882:Arnold Schoenberg
866:Metric modulation
578:by observing the
458:
457:
450:
422:encyclopedic tone
302:Theorema Egregium
291:Richard Borcherds
283:modular functions
52:creative activity
4802:
4775:Aesthetic beauty
4750:
4749:
4600:Nikos Salingaros
4323:
4291:Hiroshi Sugimoto
4241:Robert Longhurst
4187:Helaman Ferguson
4142:Crockett Johnson
4071:Circle Limit III
4040:Danseuse au café
3947:
3917:Pyramid of Khufu
3680:
3560:
3553:
3546:
3537:
3523:
3522:
3521:
3415:
3405:
3395:
3385:
3375:
3365:
3355:
3345:
3335:
3325:
3315:
3305:
3295:
3285:
2584:
2577:
2570:
2561:
2533:Internet Archive
2528:Aesthetic Appeal
2493:cut-the-knot.org
2475:
2466:
2456:
2428:
2399:
2386:
2341:
2306:Rota, Gian-Carlo
2301:
2283:
2266:(6): 1174–1178.
2224:
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2018:
2012:
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1989:
1983:
1969:
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1944:
1943:
1927:
1921:
1909:
1903:
1893:
1887:
1858:J. Schmidhuber.
1856:
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1623:10.1002/pchj.556
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1219:
1208:
1202:
1201:
1199:
1198:
1184:
1076:Bertrand Russell
828:stochastic music
770:first derivative
752:. In the 1990s,
634:
490:physical reality
462:pure mathematics
453:
446:
442:
439:
433:
432:for suggestions.
413:
412:
405:
327:The opposite of
293:was awarded the
271:Robert Langlands
222:
220:
219:
214:
195:
194:
171:Euler's identity
4810:
4809:
4805:
4804:
4803:
4801:
4800:
4799:
4765:
4764:
4763:
4758:
4738:
4734:Sacred geometry
4707:
4673:Ars Mathematica
4661:
4611:
4522:
4475:
4462:Andrea Palladio
4358:
4351:De architectura
4310:
4266:Antoine Pevsner
4246:Jeanette McLeod
4197:Susan Goldstine
4146:
4005:
3999:
3936:
3922:Sagrada Família
3883:
3825:
3693:Algorithmic art
3681:
3672:
3668:Wallpaper group
3606:Minimal surface
3569:
3564:
3534:
3529:
3519:
3517:
3494:
3418:
3413:
3403:
3393:
3390:Critical Essays
3383:
3373:
3363:
3353:
3343:
3333:
3323:
3313:
3303:
3293:
3283:
3267:
3040:
2954:Ortega y Gasset
2747:
2659:
2593:
2588:
2521:Mathbeauty Blog
2482:
2431:
2417:
2402:
2389:
2359:
2356:
2354:Further reading
2351:
2304:
2254:
2212:
2207:
2103:
2098:
2089:
2087:
2085:
2057:
2056:
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2020:
2019:
2015:
2008:
1991:
1990:
1986:
1970:
1966:
1961:Wayback Machine
1951:
1947:
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1928:
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1876:10.2307/1576418
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1299:
1297:
1295:BBC News online
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1196:
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1186:
1185:
1181:
1177:
1172:
1108:
1073:
1003:Metatron's Cube
988:Sacred geometry
908:
880:beginning with
856:Igor Stravinsky
824:
819:
807:Main articles:
805:
789:self-similarity
778:compressibility
734:
718:
694:Platonic solids
682:Johannes Kepler
641:natural numbers
635:
629:
618:
556:cuisenaire rods
454:
443:
437:
434:
427:
418:This section's
414:
410:
392:
359:Gian-Carlo Rota
322:Stokes' theorem
318:Green's theorem
255:elliptic curves
247:Richard Feynman
243:Euler's formula
183:
177:
176:
141:
71:
17:
12:
11:
5:
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4806:
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4520:
4519:
4518:
4516:The Greek Vase
4508:
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4447:
4446:
4445:
4438:Albrecht Dürer
4435:
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4420:
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4407:
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4312:
4311:
4309:
4308:
4303:
4301:Roman Verostko
4298:
4293:
4288:
4283:
4278:
4276:Alba Rojo Cama
4273:
4268:
4263:
4258:
4253:
4248:
4243:
4238:
4233:
4232:
4231:
4222:Charles Jencks
4219:
4214:
4209:
4207:George W. Hart
4204:
4199:
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4189:
4184:
4179:
4174:
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4102:
4095:
4088:
4081:
4074:
4062:
4057:
4052:
4051:
4050:
4043:
4034:Jean Metzinger
4031:
4030:
4029:
4022:
4009:
4007:
4001:
4000:
3998:
3997:
3996:
3995:
3983:
3981:Albrecht Dürer
3978:
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3698:Anamorphic art
3695:
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3665:
3660:
3655:
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3628:
3626:Camera obscura
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3505:
3499:
3496:
3495:
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3492:
3487:
3482:
3477:
3472:
3467:
3462:
3460:Neuroesthetics
3457:
3452:
3447:
3442:
3440:Arts criticism
3437:
3432:
3426:
3424:
3420:
3419:
3417:
3416:
3406:
3396:
3386:
3376:
3366:
3356:
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3336:
3326:
3316:
3310:On the Sublime
3306:
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3198:
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3188:
3181:
3176:
3174:Interpretation
3171:
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3141:
3136:
3131:
3126:
3121:
3116:
3111:
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3096:
3091:
3090:
3089:
3084:
3074:
3069:
3067:Artistic merit
3064:
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2716:Psychoanalysis
2713:
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2555:Edward Frenkel
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2481:
2480:External links
2478:
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2370:(4): 339–355,
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2108:Aigner, Martin
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2013:
2006:
1984:
1964:
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1868:Leonardo/ISAST
1851:
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1655:
1616:(5): 741–747.
1596:
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1531:978-0136020400
1530:
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1490:10.2307/749423
1484:(5): 498–505.
1468:
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1388:, p. 172.
1378:
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1279:, p. 173.
1269:
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1163:Pythagoreanism
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1143:Neuroesthetics
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970:camera obscura
907:
904:
870:Elliott Carter
832:Iannis Xenakis
823:
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804:
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793:neural network
774:predictability
736:In the 1970s,
733:
730:
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660:mathematician
627:
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580:crease pattern
560:pattern blocks
474:ancient Greeks
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4296:Daina Taimiņa
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4121:Salvador Dalí
4119:
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4107:René Magritte
4105:
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4078:Print Gallery
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3957:Paolo Uccello
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3878:Pi in the Sky
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3062:Art manifesto
3060:
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3052:Appropriation
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2759:Abhinavagupta
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2711:Postmodernism
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2437:Atiyah, M. F.
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2017:
2014:
2009:
2007:9781317137436
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2000:. Routledge.
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1610:PsyCh Journal
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1045:Jonty Hurwitz
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1034:tessellations
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921:Della Pittura
917:
914:Diagram from
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738:Abraham Moles
731:
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675:
672:claimed that
671:
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648:
646:
642:
632:
626:
621:
616:In philosophy
615:
613:
609:
607:
603:
599:
598:Permutohedron
595:
594:Young tableau
591:
587:
586:Combinatorics
583:
581:
577:
573:
568:
565:
561:
557:
553:
552:algebra tiles
549:
544:
541:
537:
533:
529:
526:For example,
524:
521:
519:
515:
514:string theory
511:
507:
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390:In experience
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287:string theory
284:
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279:Monster group
276:
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259:modular forms
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4620:Publications
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4511:Jay Hambidge
4504:Ad Quadratum
4503:
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4428:
4414:
4410:Luca Pacioli
4400:
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4261:Hinke Osinga
4256:István Orosz
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4217:Anthony Hill
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4167:Erik Demaine
4151:Contemporary
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4069:
4065:M. C. Escher
4045:
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3986:Parmigianino
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3902:Hagia Sophia
3876:
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3725:Computer art
3703:Architecture
3663:Tessellation
3646:Architecture
3596:Golden ratio
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2834:Coomaraswamy
2752:Philosophers
2740:
2671:Aestheticism
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2228:Pandey, S.K.
2220:
2216:
2201:
2182:
2171:
2162:
2148:
2138:
2128:
2115:
2088:. Retrieved
2064:
2053:
2045:
2039:. Retrieved
2026:
2016:
1997:
1987:
1976:
1972:
1967:
1948:
1931:
1925:
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1899:since 1994:
1891:
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1809:
1798:. Retrieved
1796:. 2024-02-21
1793:
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1773:. Retrieved
1771:. 2015-09-01
1768:
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1743:
1737:
1726:. Retrieved
1724:. 2014-04-02
1721:
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1692:. New York:
1689:
1683:
1673:, retrieved
1668:
1658:
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1453:. Retrieved
1435:
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1399:
1393:
1381:
1372:
1367:Hardy, G.H.
1362:
1351:. Retrieved
1347:
1337:
1317:
1310:
1298:. Retrieved
1294:
1284:
1272:
1261:. Retrieved
1257:
1247:
1227:
1216:. Retrieved
1214:. 2011-04-01
1206:
1195:. Retrieved
1191:
1182:
1133:Golden ratio
1099:
1092:ineffability
1086:
1081:
1074:
1057:Anthony Hill
1049:
1041:anamorphosis
1038:
1010:M. C. Escher
1007:
986:
940:geometry to
934:chaos theory
931:
919:
896:Group theory
889:
859:
852:polyrhythmic
825:
742:Frieder Nake
735:
719:
698:circumsphere
686:Solar System
678:
670:Alain Badiou
667:
656:
649:
637:
623:
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610:
602:Graph theory
584:
569:
545:
528:Math Circles
525:
522:
498:Group theory
460:Interest in
459:
444:
435:
419:
384:
376:
372:
364:
356:
345:
343:
332:
328:
326:
299:
295:Fields Medal
267:Andrew Wiles
238:
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162:
152:
148:
147:Starting at
132:
128:
122:
106:
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56:
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4560:G. H. Hardy
4363:Renaissance
4333:Polykleitos
4271:Tony Robbin
4182:John Ernest
4177:Jan Dibbets
4127:Crucifixion
3950:Renaissance
3804:Mathematics
3777:Celtic knot
3740:Fractal art
3642:Proportion
3616:Perspective
3294:(c. 335 BC)
3284:(c. 390 BC)
3263:Work of art
3216:Picturesque
3072:Avant-garde
3029:Winckelmann
2904:Kierkegaard
2829:Collingwood
2799:Baudrillard
2726:Romanticism
2696:Historicism
2630:Mathematics
2510:Terence Tao
2178:Lang, Serge
2167:, Hyperion.
2145:Hardy, G.H.
2071:Basic Books
1524:. Pearson.
1386:Rota (1997)
1300:13 February
1277:Rota (1997)
1052:John Ernest
1018:lithographs
958:Renaissance
954:perspective
926:perspective
906:Visual arts
900:David Lewin
874:permutation
803:In the arts
762:algorithmic
725:Paul Ernest
721:G. H. Hardy
352:G. H. Hardy
289:(for which
116:. In fact,
48:G. H. Hardy
40:mathematics
4769:Categories
4534:Owen Jones
4381:De pictura
4286:Oliver Sin
4236:Andy Lomas
4085:Relativity
3816:String art
3730:Fiber arts
3611:Paraboloid
3233:Recreation
3211:Perception
3104:Creativity
2804:Baumgarten
2794:Baudelaire
2676:Classicism
2591:Aesthetics
2549:review of
2242:1710134437
2223:(2 and 3).
2153:C. P. Snow
2101:References
2090:2008-08-22
2073:. p.
2041:2008-08-22
2034:. p.
1832:Aesthetics
1813:A. Moles:
1800:2024-09-15
1775:2024-09-15
1728:2019-10-31
1675:2019-10-31
1455:2008-08-22
1411:Lang, p. 3
1353:2019-10-31
1263:2019-10-31
1218:2024-04-28
1197:2019-10-31
1088:Paul Erdős
1065:algorithms
1061:Peter Lowe
1022:mezzotints
956:theory of
876:theory in
782:symmetries
702:polyhedron
662:Paul Erdős
502:polynomial
486:Paul Dirac
438:March 2013
263:Wolf Prize
139:In results
4345:Vitruvius
4319:Theorists
4099:Waterfall
4004:19th–20th
3932:Taj Mahal
3912:Parthenon
3889:Buildings
3843:Continuum
3811:Sculpture
3787:Interlace
3581:Algorithm
3238:Reverence
3144:Eroticism
3114:Depiction
3087:Masculine
2989:Santayana
2949:Nietzsche
2894:Hutcheson
2884:Heidegger
2869:Greenberg
2824:Coleridge
2789:Balthasar
2774:Aristotle
2736:Theosophy
2731:Symbolism
2706:Modernism
2691:Formalism
2384:120068870
2281:1956/2734
2256:Reber, R.
1918:0709.0674
1632:2046-0252
1573:1662-5161
1432:"Preface"
1030:paradoxes
928:on a grid
878:serialism
844:Lateralus
658:Hungarian
645:mysticism
466:empirical
357:In 1997,
339:empty set
306:curvature
192:π
69:In method
44:beautiful
36:aesthetic
4753:Category
4480:Romantic
4158:Max Bill
4092:Reptiles
3907:Pantheon
3864:Octacube
3830:Artworks
3772:Knotting
3760:Muqarnas
3658:Symmetry
3586:Catenary
3574:Concepts
3513:Category
3445:Axiology
3314:(c. 500)
3304:(c. 100)
3179:Judgment
3134:Emotions
3129:Elegance
3109:Cuteness
3082:Feminine
3045:Concepts
3014:Tanizaki
2994:Schiller
2979:Richards
2969:Rancière
2939:Maritain
2874:Hanslick
2814:Benjamin
2686:Feminism
2655:Theology
2635:Medieval
2625:Japanese
2620:Internet
2543:Jim Holt
2473:24592230
2433:Zeki, S.
2425:76481488
2338:44064821
2330:20117626
2310:Synthese
2290:19001586
2200:(1968),
2180:(1985).
2161:(1992),
2147:(1940),
2137:(1949),
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1957:Archived
1864:Leonardo
1650:35491015
1591:24592230
1106:See also
1026:infinity
1014:woodcuts
996:Kaballah
982:futurism
918:'s 1435
706:insphere
704:and the
674:ontology
628:—
540:symmetry
273:), and "
4712:Related
4326:Ancient
4060:Man Ray
4006:Century
3942:Artists
3799:Origami
3713:Pyramid
3591:Fractal
3508:Outline
3423:Related
3290:Poetics
3258:Tragedy
3248:Sublime
3221:Quality
3206:Mimesis
3164:Harmony
3149:Fashion
3124:Ecstasy
3119:Disgust
3035:more...
3004:Scruton
2929:Lyotard
2864:Goodman
2844:Deleuze
2779:Aquinas
2769:Alberti
2742:more...
2721:Realism
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2681:Fascism
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2650:Science
2605:Ancient
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2032:Longman
1884:1576418
1641:9790661
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864:), the
786:fractal
700:of one
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80:elegant
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3009:Tagore
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2326:JSTOR
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