Mathematics is often taught as a tool, but here we explore it as an art form. This page is for the curious — those who want to see what makes a proof, problem, or idea elegant. The topics here are ones that I find particularly interesting, clever, and beautiful.

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.

— Henri Poincaré

Proofs

• Euler’s Formula (and Identity) How five of the most important constants in mathematics — 0, 1, π, e, and i — combine into a single line of breathtaking simplicity.
• Euclid’s Theorem: The Infinitude of Primes A 2,300-year-old proof that the primes never end, using nothing but multiplication, addition, and a single brilliant twist.
• Proof that the Square Root of 2 is Irrational An elegant argument by contradiction showing that some numbers simply cannot be written as fractions, no matter how hard you try.
• Fermat’s Theorem on Sums of Two Squares Which primes can be written as the sum of two squares? The pattern is surprising — and one modern proof of it fits in a single sentence.
• Nicomachus’s Theorem The sum of the first n cubes equals the square of their sum. A striking identity with an even more striking explanation.
• Gabriel’s Horn A surface with finite volume but infinite surface area — a paradox that reveals just how strange infinity can be.

Explorations

• The Nine-Point Circle Nine geometrically significant points of any triangle always lie on a single circle. The pattern holds for every triangle, no matter the shape.
• Tautochrone Drop two balls from different heights onto this curve and they reach the bottom at the same moment. The shape that makes this possible is one of the most elegant results in classical mechanics.

Practice

• Challenging Problems A growing collection of problems chosen for their elegance — and for the satisfaction of working them through.