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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Mathematical beauty | 1/3 | https://en.wikipedia.org/wiki/Mathematical_beauty | reference | science, encyclopedia | 2026-05-05T07:24:17.882469+00:00 | kb-cron |
Mathematical beauty is a type of aesthetic value that is experienced in doing or contemplating mathematics. The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such as G.H. Hardy, have characterized mathematics as an art form that seeks beauty.
Beauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understanding beauty in general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstract ideas which can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than (for example) listening to music.
Furthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors identify mathematical elegance with mathematical beauty; others distinguish elegance as a separate aesthetic value, or as being, for instance, limited to the form of mathematical exposition. Beauty itself is often linked to, or thought to be dependent on, the abstractness, purity, simplicity, depth or order of mathematics.
== Examples of beautiful mathematics ==
=== Results ===
Euler's identity is often given as an example of a beautiful result:
e
i
π
+
1
=
0
.
{\displaystyle \displaystyle \mathrm {e} ^{\mathrm {i} \pi }+1=0\,.}
This expression ties together arguably the five most important mathematical constants (e, i, π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of Euler's formula, which the physicist Richard Feynman called "our jewel" and "the most remarkable formula in mathematics". Another example is Fermat's theorem on sums of two squares, which says that any prime number such that
p
≡
1
(
mod
4
)
{\displaystyle p\equiv 1{\pmod {4}}}
can be written as a sum of two square numbers (for example,
5
=
1
2
+
2
2
{\displaystyle 5=1^{2}+2^{2}}
,
13
=
2
2
+
3
2
{\displaystyle 13=2^{2}+3^{2}}
,
37
=
1
2
+
6
2
{\displaystyle 37=1^{2}+6^{2}}
), which both G.H. Hardy and E.T. Bell thought was a beautiful result. In a survey in which mathematicians were asked to evaluate 24 theorems for their beauty, the top-rated three theorems were: Euler's equation; Euler's polyhedron formula, which asserts that for a polyhedron with V vertices, E edges, and F faces,
V
−
E
+
F
=
2
{\displaystyle V-E+F=2}
; and Euclid's theorem that there are infinitely many prime numbers, which was also given by Hardy as an example of a beautiful theorem.
=== Proofs ===
Cantor's diagonal argument, which establishes that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers, has been cited by both mathematicians and philosophers as an example of a beautiful proof.
Visual proofs, such as the illustrated proof of the Pythagorean theorem, and other proofs without words generally, such as the shown proof that the sum of all positive odd numbers up to 2n − 1 is a perfect square, have been thought beautiful. The mathematician Paul Erdős spoke of The Book, an imaginary infinite book in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would proclaim it "straight from The Book!". His rhetorical device inspired the creation of Proofs from THE BOOK, a collection of such proofs, including many suggested by Erdős himself.
=== Objects === In Plato's Timaeus, the five regular convex polyhedra, called the Platonic solids for their role in this dialogue, are called the "most beautiful" ("κάλλιστα") bodies. In the Timaeus, they are described as having been used by the demiurge, or creator-craftsman who builds the cosmos, for the four classical elements plus the heavens, because of their beauty.
In his 1596 book Mysterium Cosmographicum, Johannes Kepler argued that the orbits of the then-known planets in the Solar System have been arranged by God to correspond to a concentric arrangement of the five Platonic solids, each orbit lying on the circumsphere of one polyhedron and the insphere of another. For Kepler, God had wanted to shape the universe according to the five regular solids because of their beauty, and this explained why there were six planets (according to the knowledge of the time).
A more modern example is the exceptional simple Lie group
E
8
{\displaystyle E_{8}}
, which has been called "perhaps the most beautiful structure in all of mathematics".
=== Scientific theories === The mathematical statements of scientific theories, especially in physics, are sometimes considered to be mathematically beautiful. For example, Roger Penrose thought there was a "special beauty" in Maxwell's equations of electromagnetism:
∇
⋅
E
=
ρ
ε
0
∇
⋅
B
=
0
∇
×
E
=
−
∂
B
∂
t
∇
×
B
=
μ
0
(
J
+
ε
0
∂
E
∂
t
)
{\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} \,\,\,&={\frac {\rho }{\varepsilon _{0}}}\\\nabla \cdot \mathbf {B} \,\,\,&=0\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \times \mathbf {B} &=\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\end{aligned}}}
Einstein's theory of general relativity has been characterized as a work of art, and, among other aesthetic praise, was described by Paul Dirac as having "great mathematical beauty" and by Penrose as having "supreme mathematical beauty". (There can be more to the beauty of a scientific theory than just its mathematical statement. For example, whether a theory is visualizable or deterministic might have an influence on whether it is seen as beautiful.)