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.

## No Background Needed

Area of a triangle within a rectangle

Number of games in a tournament

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

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