the top 10 most beautiful theorems

A 1988 poll of readers of the Mathematical Intelligencer ranked some of the most well-known theorems in mathematics thus:

1. Euler’s identity, $e^{i\pi }=-1$

2. Euler’s formula for a polyhedron, $V+F=E+2$

3. There are infinitely many prime numbers. See Euclid’s proof that there are infinitely many primes.

4. There are only 5 regular polyhedra

5. The sum of the reciprocals of the squares of the positive integers is $\frac{{\pi }^2}{6}$. See the Basel problem.

6. A continuous mapping of a closed unit disk into itself has a fixed point

7. The square root of 2 is irrational

8. $\pi$ is a transcendental number

9. Every plane map can be colored with just 4 colors

10. Every prime number of the form $4n+1$ is the sum of two square integers in only one way