Mathematics (like programming) is often called an art form. This makes mathematics the art of argument, not in the sense that anything is debatable, but in the sense that the beauty of a proof lies in its method. Analogously, a painting is not necessarily beautiful only for what it depicts or the medium used, but rather the way aspects like brushstroke, inspiration, and contemporary social attitudes combine in the artist’s self-expression. In mathematics as well, an aesthetic proof is an elegant, inspired, concise, and beautiful expression. Here we present a collection of aesthetic proofs of more or less simple facts we’ve come across over the years. As time goes on, we may add more advanced proofs. For now, we’ll stick to common problems, pushing some well known solutions to the next level, and providing some proofs of our own discovery.

No Background Needed

Area of a triangle within a rectangle
Number of games in a tournament
Computing percentages easier
The party problem
Tiling chessboards with dominoes
A rook game
The handshake lemma (with double counting)

Number Theory & Combinatorics

Sums of k powers
Sum of the first n numbers, sum of the first n squares
An arithmetic expression for $ \binom{n}{2}$
There are infinitely many primes (a lower bound on $ \pi(n)$)
Ramsey number lower bound $ R(m,m)$
Learning a single variable polynomial
A parlor trick for SET

Linear Algebra

A spectral analysis of Moore graphs
A parlor trick for SET

Geometry

Double Angle Trigonometric Formulas
Geometric series with geometric proofs
Three circles and collinear centers of dilation
Mobius transformations are isometries of a sphere
The square root of two is irrational (geometric proof)

Logic

n-Colorability is equivalent to finite n-colorability

Analysis

Cauchy-Schwarz inequality (by amplification)

Abstract Algebra

The smallest non-cyclic simple group has order 60
$ \mathbb{Z}[\sqrt{2}]$ has infinitely many units

Complexity Theory

Classic Nintendo games are NP-hard
Testing Polynomial Equality
Encoding boolean logic in polynomials

Topology

There are infinitely many primes

The Fundamental Theorem of Algebra

With Liouville’s Theorem (complex analysis)
With fundamental groups (algebraic topology)
With Galois theory (group theory & field theory)
With Picard’s Little Theorem (complex analysis)

False Proofs

1 = 2 (with algebra)
1 = 2 (with calculus)
2 = 4 (with infinite power towers)
There are finitely many primes
31.5 = 32.5
All horses are the same color
The reals are countable
Every number can be described in fewer than twenty words