Proof Gallery

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 is 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.

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

Topology

There are infinitely many primes