Mathematical beauty is the aesthetic
pleasure typically derived from the abstractness, purity, simplicity, depth or orderliness of mathematics
. Mathematicians often express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as ''beautiful''. They might also 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 often made with music and poetry.
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.
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".
Beauty in method
Mathematicians describe an especially pleasing method of proof
''. 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
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
, 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 fact, Carl Friedrich Gauss
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
s or previous results are usually not considered to be elegant, and may be even referred to as ''ugly'' or ''clumsy''.
Beauty 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
Euler's identity is a special case of Euler's formula
, which the physicist Richard Feynman
called "our jewel" and "the most remarkable formula in mathematics". Modern examples include the modularity theorem
, which establishes an important connection between elliptic curve
s and modular form
s (work on which led to the awarding of the Wolf Prize
to Andrew Wiles
and Robert Langlands
), and "monstrous moonshine
", which connects the Monster group
to modular function
s via string theory
(for which Richard Borcherds
was awarded the Fields Medal
Other examples of deep results include unexpected insights into mathematical structures. For example, Gauss's Theorema Egregium
is a deep theorem which relates a local phenomenon (curvature
) to a global phenomenon (area
) 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
(and its vector versions including Green's theorem
and Stokes' theorem
The opposite of ''deep'' is ''trivial''. 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 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 ''A Mathematician's Apology
suggests that a beautiful proof or result possesses "inevitability", "unexpectedness", and "economy".
, however, disagrees with unexpectedness as a necessary condition for beauty and proposes a counterexample:
In contrast, Monastyrsky writes:
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.
Beauty in experience
Interest in pure mathematics
that is separate from empirical
study has been part of the experience of various civilizations
, including that of the ancient Greeks
, who "did mathematics for the beauty of it". The aesthetic pleasure that mathematical physicist
s tend to experience in Einstein's theory of general relativity
has been attributed (by Paul Dirac
, among others) to its "great mathematical beauty". The beauty of mathematics is experienced when the physical reality
of objects are represented by mathematical models
. Group theory
, developed in the early 1800s for the sole purpose of solving polynomial
equations, became a fruitful way of categorizing elementary particle
s—the building blocks of matter. Similarly, the study of knots
provides important insights into string theory
and loop quantum gravity
Some believe that in order to appreciate mathematics, one must engage in doing mathematics.
For example, Math Circle
is an after-school enrichment program where students do mathematics through games and activities; there are also some teachers that encourage student engagement
by teaching mathematics in a kinesthetic way (see kinesthetic learning
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
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
to present mathematics in an aesthetically pleasing way. Examples of a manipulative include algebra tiles
, cuisenaire rods
, and pattern blocks
. For example, one can teach the method of completing the square
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
. Origami, the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the mathematics of paper folding
by observing the crease pattern
on unfolded origami pieces.
, the study of counting, has artistic representations that some find mathematically beautiful. There are many visual examples that illustrate combinatorial concepts. Some of the topics and objects seen in combinatorics courses with visual representations include, among others:
* Four color theorem
* Young tableau
* Graph theory
* Partition of a set
Beauty and 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
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
'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.
mathematician Paul Erdős
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
claims that ontology
is mathematics. Badiou also believes in deep connections between mathematics, poetry and philosophy.
In some 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, Johannes Kepler
believed that the proportions of the orbits of the then-known planets in the Solar System
have been arranged by God
to correspond to a concentric arrangement of the five Platonic solid
s, each orbit lying on the circumsphere
of one polyhedron
and the insphere
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
Beauty and mathematical information theory
In the 1970s, Abraham Moles
and Frieder Nake
analyzed links between beauty, information processing
, and information theory
. In the 1990s, Jürgen Schmidhuber
formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory
: the most beautiful objects among subjectively comparable objects have short algorithmic
descriptions (i.e., Kolmogorov complexity
) relative to what the observer already knows. 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
of the observations by discovering regularities such as repetitions and symmetries
and fractal self-similarity
. Whenever the observer's learning process (possibly a predictive artificial neural network
) leads to improved data compression such that the observation sequence can be described by fewer bit
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.
Mathematics and the arts
Examples of the use of mathematics in music include the stochastic music
of Iannis Xenakis
, counterpoint of Johann Sebastian Bach
ic structures (as in Igor Stravinsky
's ''The Rite of Spring
''), the Metric modulation
of Elliott Carter
theory in serialism
beginning with Arnold Schoenberg
, and application of Shepard tones in Karlheinz Stockhausen
Examples of the use of mathematics in the visual arts include applications of chaos theory
and fractal geometry
to computer-generated art
studies of Leonardo da Vinci
, projective geometries
in development of the perspective
theory of Renaissance
in Op art
, optical geometry in the camera obscura
of Giambattista della Porta
, and multiple perspective in analytic cubism
The Dutch graphic designer M. C. Escher
created mathematically inspired woodcut
s, and mezzotint
s. These feature impossible constructions, explorations of infinity
, visual paradox
es and tessellation
s. 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
* Argument from beauty
* Cellular automaton
* Descriptive science
* Fluency heuristic
* Golden ratio
* Mathematics and architecture
* Normative science
* Philosophy of mathematics
* Processing fluency theory of aesthetic pleasure
* Theory of everything
* Aigner, Martin
, and Ziegler, Gunter M.
(2003), ''Proofs from THE BOOK
,'' 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
(foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992.
* Hoffman, Paul
(1992), ''The Man Who Loved Only Numbers
* Huntley, H.E. (1970), ''The Divine Proportion: A Study in Mathematical Beauty'', Dover Publications, New York, NY.
* Loomis, Elisha Scott
(1968), ''The Pythagorean Proposition'', The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem.
* Lang, Serge
(1985)''The Beauty of Doing Mathematics: Three Public Dialogues''
New York: Springer-Verlag. .
* 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.
External linksMathematics, Poetry and BeautyIs Mathematics Beautiful?Justin MullinsEdna St. Vincent Millay (poet): ''Euclid alone has looked on beauty bare''
*Terence Tao''What is good mathematics?''Mathbeauty Blog
*''A Mathematical Romance''Jim Holt
December 5, 2013 issue of The New York Review of Books
review of ''Love and Math: The Heart of Hidden Reality'' by Edward Frenkel
Category:Philosophy of mathematics
Category:Mathematics and art