Mathematical elegance
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Mathematical beauty is the
aesthetic Aesthetics, or esthetics, is a branch of philosophy that deals with the nature of beauty and taste, as well as the philosophy of art (its own area of philosophy that comes out of aesthetics). It examines aesthetic values, often expressed t ...
pleasure derived from the abstractness, purity, simplicity, depth or orderliness of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as
beautiful Beautiful, an adjective used to describe things as possessing beauty, may refer to: Film and theater * ''Beautiful'' (2000 film), an American film directed by Sally Field * ''Beautiful'' (2008 film), a South Korean film directed by Juhn Jai-h ...
or describe mathematics as an art form, e.g., a position taken by G. H. Hardy) or, at a minimum, as a creative activity. Comparisons are made with
music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...
and
poetry Poetry (derived from the Greek '' poiesis'', "making"), also called verse, is a form of literature that uses aesthetic and often rhythmic qualities of language − such as phonaesthetics, sound symbolism, and metre − to evoke meani ...
.


In method

Mathematicians describe an especially pleasing method of
proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a c ...
as ''
elegant Elegance is beauty that shows unusual effectiveness and simplicity. Elegance is frequently used as a standard of tastefulness, particularly in visual design, decorative arts, literature, science, and the aesthetics of mathematics. Elegant t ...
''. Depending on context, this may mean: * A proof that uses a minimum of additional assumptions or previous results. * A proof that is unusually succinct. * A proof that derives a result in a surprising way (e.g., from an apparently unrelated
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
or a collection of theorems). * A proof that is based on new and original insights. * A method of proof that can be easily generalized to solve a family of similar problems. 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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
, with hundreds of proofs being published up to date. Another theorem that has been proved in many different ways is the theorem of
quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...
. In fact,
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
alone had eight different proofs of this theorem, six of which he published. Conversely, results that are logically correct but involve laborious calculations, over-elaborate methods, highly conventional approaches or a large number of powerful
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s or previous results are usually not considered to be elegant, and may be even referred to as ''ugly'' or ''clumsy''.


In results

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 '' deep''. 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
Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the equality e^ + 1 = 0 where : is Euler's number, the base of natural logarithms, : is the imaginary unit, which by definition satisfies , and : is pi, the ratio of the circ ...
: \displaystyle e^ +1 = 0\, . This
elegant Elegance is beauty that shows unusual effectiveness and simplicity. Elegance is frequently used as a standard of tastefulness, particularly in visual design, decorative arts, literature, science, and the aesthetics of mathematics. Elegant t ...
expression ties together arguably the five most important
mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
s (''e'', ''i'', π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
, which the physicist
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
called "our jewel" and "the most remarkable formula in mathematics". Modern examples include the
modularity theorem The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. And ...
, which establishes an important connection between
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
s and
modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory o ...
s (work on which led to the awarding of the
Wolf Prize The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for ''"achievements in the interest of mankind and friendly relations among people ... irrespective of nati ...
to
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awa ...
and Robert Langlands), and "
monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
", which connects the
Monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order    24632059761121331719232931414759 ...
to
modular function In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
s via
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
(for which
Richard Borcherds Richard Ewen Borcherds (; born 29 November 1959) is a British mathematician currently working in quantum field theory. He is known for his work in lattices, group theory, and infinite-dimensional algebras, for which he was awarded the Fields Me ...
was awarded the
Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...
). Other examples of deep results include unexpected insights into mathematical structures. For example, Gauss's
Theorema Egregium Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
is a deep theorem which relates a local phenomenon (
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
) to a global phenomenon (
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
) 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
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
(and its vector versions including
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively orie ...
and
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
). The opposite of ''deep'' is
trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
. 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
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
. In some occasions, however, a statement of a theorem can be original enough to be considered deep—even though its proof is fairly obvious. In his 1940 essay '' A Mathematician's Apology'', G. H. Hardy suggested that a beautiful proof or result possesses "inevitability", "unexpectedness", and "economy". In 1997,
Gian-Carlo Rota Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, proba ...
, disagreed with unexpectedness as a necessary condition for beauty and proposed a counterexample: In contrast, Monastyrsky wrote in 2001: 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.


In experience

Interest in
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
that is separate from
empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
study has been part of the experience of various civilizations, including that of the
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...
, who "did mathematics for the beauty of it". The aesthetic pleasure that mathematical physicists tend to experience in Einstein's theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
has been attributed (by
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
, among others) to its "great mathematical beauty". The beauty of mathematics is experienced when the physical reality of objects are represented by
mathematical models A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physi ...
.
Group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, developed in the early 1800s for the sole purpose of solving
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
equations, became a fruitful way of categorizing
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, ...
s—the building blocks of matter. Similarly, the study of
knots A knot is a fastening in rope or interwoven lines. Knot may also refer to: Places * Knot, Nancowry, a village in India Archaeology * Knot of Isis (tyet), symbol of welfare/life. * Minoan snake goddess figurines#Sacral knot Arts, entertainme ...
provides important insights into
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
and
loop quantum gravity Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attem ...
. Some believe that in order to appreciate mathematics, one must engage in doing mathematics. For example,
Math Circle A math circle is a learning space where participants engage in the depths and intricacies of mathematical thinking, propagate the culture of doing mathematics, and create knowledge. To reach these goals, participants partake in problem-solving, mat ...
is an after-school enrichment program where students do mathematics through games and activities; there are also some teachers that encourage
student engagement Student engagement occurs when "students make a psychological investment in learning. They try hard to learn what school offers. They take pride not simply in earning the formal indicators of success (grades), but in understanding the material and ...
by teaching mathematics in
kinesthetic learning Kinesthetic learning (American English), kinaesthetic learning (British English), or tactile learning is learning that involves physical activity. As cited by Favre (2009), Dunn and Dunn define kinesthetic learners as students who prefer whole-bod ...
. 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
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
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 class, symmetry can be presented as such in an artistic manner where students see aesthetically pleasing results in mathematics. Some teachers prefer to use
mathematical manipulatives In mathematics education, a manipulative is an object which is designed so that a learner can perceive some mathematical concept by manipulating it, hence its name. The use of manipulatives provides a way for children to learn concepts throu ...
to present mathematics in an aesthetically pleasing way. Examples of a manipulative include
algebra tiles Algebra tiles are mathematical manipulatives that allow students to better understand ways of algebraic thinking and the concepts of algebra. These tiles have proven to provide concrete models for elementary school, middle school, high school, and ...
,
cuisenaire rods Cuisenaire rods are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisor ...
, and
pattern blocks Pattern Blocks are a set of mathematical manipulatives developed in the 1960s. The six shapes are both a play resource and a tool for learning in mathematics, which serve to develop spatial reasoning skills that are fundamental to the learning of ...
. For example, one can teach the method of
completing the square : In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form :ax^2 + bx + c to the form :a(x-h)^2 + k for some values of ''h'' and ''k''. In other words, completing the square places a perfe ...
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. Another example of beauty in experience involves the use of
origami ) is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a f ...
. Origami, the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the
mathematics of paper folding The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper f ...
by observing the
crease pattern A crease pattern is an origami diagram that consists of all or most of the creases in the final model, rendered into one image. This is useful for diagramming complex and super-complex models, where the model is often not simple enough to diagram e ...
on unfolded origami pieces.
Combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
, 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
Four color theorem In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions sha ...
,
Young tableau In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups a ...
, Permutohedron,
Graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
,
Partition of a set In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every part ...
. 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 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.


In philosophy

Some mathematicians are of the opinion that the doing of mathematics is closer to discovery than invention, for example: 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
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
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
mysticism Mysticism is popularly known as becoming one with God or the Absolute, but may refer to any kind of ecstasy or altered state of consciousness which is given a religious or spiritual meaning. It may also refer to the attainment of insight in ...
. In
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
'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. Hungarian mathematician
Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
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!" Twentieth-century French philosopher
Alain Badiou Alain Badiou (; ; born 17 January 1937) is a French philosopher, formerly chair of Philosophy at the École normale supérieure (ENS) and founder of the faculty of Philosophy of the Université de Paris VIII with Gilles Deleuze, Michel Fouc ...
claimed that
ontology In metaphysics, ontology is the philosophy, philosophical study of being, as well as related concepts such as existence, Becoming (philosophy), becoming, and reality. Ontology addresses questions like how entities are grouped into Category ...
is mathematics. Badiou also believes in deep connections between mathematics, poetry and philosophy. In many cases, however, 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,
Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
believed that the proportions of the orbits of the then-known planets in the
Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
have been arranged by
God In monotheistic thought, God is usually viewed as the supreme being, creator, and principal object of faith. Swinburne, R.G. "God" in Honderich, Ted. (ed)''The Oxford Companion to Philosophy'', Oxford University Press, 1995. God is typically ...
to correspond to a concentric arrangement of the five
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s, each orbit lying on the
circumsphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcirc ...
of one
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...
and the
insphere In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and i ...
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
Uranus Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus ( Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars), grandfather of Zeus (Jupiter) and father of ...
.


In information theory

In the 1970s,
Abraham Moles Abraham Moles (19 August 1920 – 22 May 1992) was a pioneer in information science and communication studies in France, He was a professor at Ulm school of design and University of Strasbourg. He is known for his work on kitsch. Biography Mo ...
and
Frieder Nake Frieder Nake (born December 16, 1938 in Stuttgart, Germany) is a mathematician, computer scientist, and pioneer of computer art. He is best known internationally for his contributions to the earliest manifestations of computer art, a field of co ...
analyzed links between beauty,
information processing Information processing is the change (processing) of information in any manner detectable by an observer. As such, it is a process that ''describes'' everything that happens (changes) in the universe, from the falling of a rock (a change in posi ...
, and
information theory Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...
. In the 1990s,
Jürgen Schmidhuber Jürgen Schmidhuber (born 17 January 1963) is a German computer scientist most noted for his work in the field of artificial intelligence, deep learning and artificial neural networks. He is a co-director of the Dalle Molle Institute for Artific ...
formulated a mathematical theory of observer-dependent subjective beauty based on
algorithmic information theory Algorithmic information theory (AIT) is a branch of theoretical computer science that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as str ...
: the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e.,
Kolmogorov complexity In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produ ...
) relative to what the observer already knows. 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 Predictability is the degree to which a correct prediction or forecast of a system's state can be made, either qualitatively or quantitatively. Predictability and causality Causal determinism has a strong relationship with predictability. Per ...
and
compressibility In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a f ...
of the observations by discovering regularities such as repetitions and symmetries and
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...
self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
. Whenever the observer's learning process (possibly a predictive artificial
neural network A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
) leads to improved data compression such that the observation sequence can be described by fewer
bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.


In the arts


Music

Examples of the use of mathematics in music include the
stochastic music Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...
of
Iannis Xenakis Giannis Klearchou Xenakis (also spelled for professional purposes as Yannis or Iannis Xenakis; el, Γιάννης "Ιωάννης" Κλέαρχου Ξενάκης, ; 29 May 1922 – 4 February 2001) was a Romanian-born Greek-French avant-garde c ...
, the
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
in
Tool A tool is an object that can extend an individual's ability to modify features of the surrounding environment or help them accomplish a particular task. Although many animals use simple tools, only human beings, whose use of stone tools dates b ...
's
Lateralus ''Lateralus'' () is the third studio album by American rock band Tool. It was released on May 15, 2001, through Volcano Entertainment. The album was recorded at Cello Studios in Hollywood and The Hook, Big Empty Space, and The Lodge, in Nor ...
, counterpoint of
Johann Sebastian Bach Johann Sebastian Bach (28 July 1750) was a German composer and musician of the late Baroque period. He is known for his orchestral music such as the '' Brandenburg Concertos''; instrumental compositions such as the Cello Suites; keyboard wo ...
,
polyrhythm Polyrhythm is the simultaneous use of two or more rhythms that are not readily perceived as deriving from one another, or as simple manifestations of the same meter. The rhythmic layers may be the basis of an entire piece of music ( cross-rhyt ...
ic structures (as in
Igor Stravinsky Igor Fyodorovich Stravinsky (6 April 1971) was a Russian composer, pianist and conductor, later of French (from 1934) and American (from 1945) citizenship. He is widely considered one of the most important and influential 20th-century clas ...
's ''
The Rite of Spring , image = Roerich Rite of Spring.jpg , image_size = 350px , caption = Concept design for act 1, part of Nicholas Roerich's designs for Diaghilev's 1913 production of ' , composer = Igor Stravinsky , based_on ...
''), the Metric modulation of
Elliott Carter Elliott Cook Carter Jr. (December 11, 1908 – November 5, 2012) was an American modernist composer. One of the most respected composers of the second half of the 20th century, he combined elements of European modernism and American "ultra- ...
,
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
theory in
serialism In music, serialism is a method of composition using series of pitches, rhythms, dynamics, timbres or other musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though some of his contemporaries were al ...
beginning with
Arnold Schoenberg Arnold Schoenberg or Schönberg (, ; ; 13 September 187413 July 1951) was an Austrian-American composer, music theorist, teacher, writer, and painter. He is widely considered one of the most influential composers of the 20th century. He was as ...
, and application of Shepard tones in
Karlheinz Stockhausen Karlheinz Stockhausen (; 22 August 1928 – 5 December 2007) was a German composer, widely acknowledged by critics as one of the most important but also controversial composers of the 20th and early 21st centuries. He is known for his groundb ...
's ''
Hymnen ''Hymnen'' (German for "Anthems") is an electronic and concrete work, with optional live performers, by Karlheinz Stockhausen, composed in 1966–67, and elaborated in 1969. In the composer's catalog of works, it is No. 22. The extended work is ...
''. They also include the application of
Group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
to transformations in music in the theoretical writings of David Lewin.


Visual arts

Examples of the use of mathematics in the visual arts include applications of
chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to hav ...
and
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...
geometry to
computer-generated art Algorithmic art or algorithm art is art, mostly visual art, in which the design is generated by an algorithm. Algorithmic artists are sometimes called ''algorists''. Overview Algorithmic art, also known as computer-generated art, is a subset o ...
, symmetry studies of
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested on ...
, projective geometries in development of the perspective theory of
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ide ...
art,
grids AIDS is caused by a human immunodeficiency virus (HIV), which originated in non-human primates in Central and West Africa. While various sub-groups of the virus acquired human infectivity at different times, the present pandemic had its origins i ...
in
Op art Op art, short for optical art, is a style of visual art that uses optical illusions. Op artworks are abstract, with many better-known pieces created in black and white. Typically, they give the viewer the impression of movement, hidden image ...
, optical geometry in the
camera obscura A camera obscura (; ) is a darkened room with a small hole or lens at one side through which an image is projected onto a wall or table opposite the hole. ''Camera obscura'' can also refer to analogous constructions such as a box or tent in w ...
of
Giambattista della Porta Giambattista della Porta (; 1535 – 4 February 1615), also known as Giovanni Battista Della Porta, was an Italian scholar, polymath and playwright who lived in Naples at the time of the Renaissance, Scientific Revolution and Reformation. Giamb ...
, and multiple perspective in analytic
cubism Cubism is an early-20th-century avant-garde art movement that revolutionized European painting and sculpture, and inspired related movements in music, literature and architecture. In Cubist artwork, objects are analyzed, broken up and reassemble ...
and
futurism Futurism ( it, Futurismo, link=no) was an artistic and social movement that originated in Italy, and to a lesser extent in other countries, in the early 20th century. It emphasized dynamism, speed, technology, youth, violence, and objects suc ...
. The Dutch graphic designer
M. C. Escher Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints. Despite wide popular interest, Escher was for most of his life neglected in t ...
created mathematically inspired
woodcut Woodcut is a relief printing technique in printmaking. An artist carves an image into the surface of a block of wood—typically with gouges—leaving the printing parts level with the surface while removing the non-printing parts. Areas tha ...
s,
lithograph Lithography () is a planographic method of printing originally based on the immiscibility of oil and water. The printing is from a stone (lithographic limestone) or a metal plate with a smooth surface. It was invented in 1796 by the German a ...
s, and
mezzotint Mezzotint is a monochrome printmaking process of the '' intaglio'' family. It was the first printing process that yielded half-tones without using line- or dot-based techniques like hatching, cross-hatching or stipple. Mezzotint achieves tonal ...
s. These feature impossible constructions, explorations of
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
, architecture, visual
paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
es and
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
s. Some painters and sculptors create work distorted with the mathematical principles of
anamorphosis Anamorphosis is a distorted projection requiring the viewer to occupy a specific vantage point, use special devices, or both to view a recognizable image. It is used in painting, photography, sculpture and installation, toys, and film special e ...
, including South African sculptor Jonty Hurwitz. British constructionist artist John Ernest 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 Anthony Hill and Peter Lowe. Computer-generated art is based on mathematical
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s.


Quotes by mathematicians

Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
expressed his sense of mathematical beauty in these words:
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. 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.
Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
expressed his views on the ineffability of mathematics when he said, "Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I ''know'' numbers are beautiful. If they aren't beautiful, nothing is".


See also

*
Argument from beauty The argument from beauty (also the aesthetic argument) is an argument for the existence of a realm of immaterial ideas or, most commonly, for the existence of God, that roughly states that the elegance of the laws of physics or the elegant laws of ...
*
Cellular automaton A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tesse ...
*
Descriptive science Descriptive research is used to describe characteristics of a population or phenomenon being studied. It does not answer questions about how/when/why the characteristics occurred. Rather it addresses the "what" question (what are the characteris ...
*
Fluency heuristic In psychology, a fluency heuristic is a mental heuristic in which, if one object is processed more fluently, faster, or more smoothly than another, the mind infers that this object has the higher value with respect to the question being considered. ...
*
Golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
*
Mathematics and architecture Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define the spatial form of a build ...
*
Neuroesthetics Neuroesthetics ( or neuroaesthetics) is a relatively recent sub-discipline of empirical aesthetics. Empirical aesthetics takes a scientific approach to the study of aesthetic perceptions of art, music, or any object that can give rise to aestheti ...
*
Normative science In the applied sciences, normative science is a type of information that is developed, presented, or interpreted based on an assumed, usually unstated, preference for a particular outcome, policy or class of policies or outcomes. Regular or tradit ...
*
Philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...
*
Processing fluency theory of aesthetic pleasure The processing fluency theory of aesthetic pleasure is a theory in psychological aesthetics on how people experience beauty. Processing fluency is the ease with which information is processed in the human mind. Overview The theory is based on ...
*
Pythagoreanism Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...
*
Theory of everything A theory of everything (TOE or TOE/ToE), final theory, ultimate theory, unified field theory or master theory is a hypothetical, singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all asp ...


Notes


References

* Aigner, Martin, and Ziegler, Gunter M. (2003), ''
Proofs from THE BOOK ''Proofs from THE BOOK'' is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each math ...
,'' 3rd edition, Springer-Verlag. * Chandrasekhar, Subrahmanyan (1987), ''Truth and Beauty: Aesthetics and Motivations in Science,'' University of Chicago Press, Chicago, IL. * Hadamard, Jacques (1949), ''The Psychology of Invention in the Mathematical Field,'' 1st edition, Princeton University Press, Princeton, NJ. 2nd edition, 1949. Reprinted, Dover Publications, New York, NY, 1954. * Hardy, G.H. (1940), ''A Mathematician's Apology'', 1st published, 1940. Reprinted,
C. P. Snow Charles Percy Snow, Baron Snow, (15 October 1905 – 1 July 1980) was an English novelist and physical chemist who also served in several important positions in the British Civil Service and briefly in the UK government.''The Columbia Encyclope ...
(foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992. * Hoffman, Paul (1992), ''
The Man Who Loved Only Numbers ''The Man Who Loved Only Numbers'' is a biography of the famous mathematician Paul Erdős written by Paul Hoffman. The book was first published on July 15, 1998, by Hyperion Books as a hardcover edition. A paperback edition appeared in 1999. ...
'', Hyperion. * Huntley, H.E. (1970), ''The Divine Proportion: A Study in Mathematical Beauty'', Dover Publications, New York, NY. * Lang, Serge (1985)
''The Beauty of Doing Mathematics: Three Public Dialogues''
New York: Springer-Verlag. . * 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. . * Peitgen, H.-O., and Richter, P.H. (1986), ''The Beauty of Fractals'', Springer-Verlag. * * * Strohmeier, John, and Westbrook, Peter (1999), ''Divine Harmony, The Life and Teachings of Pythagoras'', Berkeley Hills Books, Berkeley, CA.


Further reading

* * * *


External links


Mathematics, Poetry and BeautyIs Mathematics Beautiful?
cut-the-knot.org
Justin Mullins.comEdna St. Vincent Millay (poet): ''Euclid alone has looked on beauty bare''
*
Terence Tao Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...

''What is good mathematics?''Mathbeauty Blog
*
''A Mathematical Romance''
Jim Holt December 5, 2013 issue of
The New York Review of Books ''The New York Review of Books'' (or ''NYREV'' or ''NYRB'') is a semi-monthly magazine with articles on literature, culture, economics, science and current affairs. Published in New York City, it is inspired by the idea that the discussion of i ...
review of ''Love and Math: The Heart of Hidden Reality'' by
Edward Frenkel Edward Vladimirovich Frenkel (; born May 2, 1968) is a Russian-American mathematician working in representation theory, algebraic geometry, and mathematical physics. He is a professor of mathematics at University of California, Berkeley, a member ...
{{mathematical art Aesthetic beauty Elementary mathematics Philosophy of mathematics Mathematical terminology Mathematics and art