Which Math Theorem Is the Most Beautiful? A Ranked Survey

Which mathematical theorem is the most beautiful? This article presents a ranking of 24 famous theorems based on a 1988 survey published in “Mathematics Enthusiast” where 68 readers scored each theorem from 0 to 10.

The list, ordered by average score, includes brief explanations of why each theorem is admired.

Unnamed theorem combining five fundamental constants and 0 – Average score: 7.7. A concise equation that unites the five most important constants in mathematics.

Euler’s polyhedron formula – Average score: 7.5. Relates the number of vertices, edges, and faces of a convex polyhedron (V‑E+F=2), showcasing geometric symmetry.

Infinitude of primes – Average score: 7.5. Euclid’s proof that there are infinitely many prime numbers, highlighting the depth of number theory.

Existence of five regular polyhedra – Average score: 7.0. Proves that only five Platonic solids exist, celebrated for their perfect symmetry.

Series linking π and infinite sums – Average score: 7.0. Demonstrates a surprising connection between an infinite series and the constant π.

Brouwer fixed‑point theorem (unit disc case) – Average score: 6.8. States that any continuous map from a closed unit disc to itself has at least one fixed point.

No rational number squares to 2 – Average score: 6.7. Shows √2 is irrational, one of the earliest known irrational numbers.

Lindemann’s proof of the transcendence of e – Average score: 6.5. Establishes that e is not a root of any algebraic equation, a milestone in number theory.

Four‑color theorem – Average score: 6.2. Asserts that any planar map can be colored with four colors so that adjacent regions differ.

Fermat’s theorem on sums of two squares – Average score: 6.0. Describes which primes can be expressed uniquely as the sum of two squares.

Lagrange’s theorem on subgroup order – Average score: 5.3. In group theory, the order of any subgroup divides the order of the whole group.

Cayley‑Hamilton theorem – Average score: 5.2. Every square matrix satisfies its own characteristic equation, linking linear algebra and polynomial theory.

Golden‑ratio edge division in the icosahedron – Average score: 5.0. Shows how a regular icosahedron inscribed in a regular octahedron yields edges in the golden ratio.

Series relating π to an infinite sum (variant) – Average score: 4.8.

Unit distance problem in a coloured plane – Average score: 4.7. Guarantees a pair of equally coloured points at distance 1 in any red‑yellow‑blue colouring.

Partition of odd integers equals partition into distinct parts – Average score: 4.7.

Every integer >77 can be written as a sum of integers whose reciprocals sum to 1 – Average score: 4.7.

Number of representations of an odd number as four squares is eight times its divisor count; for even numbers, twenty‑four times the odd‑divisor count – Average score: 4.7.

No equilateral triangle with vertices on lattice points – Average score: 4.7.

In any party, two people share the same number of friends – Average score: 4.7.

Two‑line puzzle involving √2 multiples – Average score: 4.2.

Word problem for groups is unsolvable – Average score: 4.1.

Maximum area of a quadrilateral given its semiperimeter – Average score: 3.9.

Relation between partition numbers and an infinite series – Average score: 4.0.

The perception of beauty in mathematics varies, with criteria such as simplicity, symmetry, surprise, universality, depth, elegance of proof, and unifying power.

Reference: Wells, D. (1990). “Are these the most beautiful?” Matmedia.it .