Preface

Summary

Having perceived the connexions, he seeks the proof, the clean revelation in its simplest form, never doubting that somewhere writing in the chaos is the unique elegance, the precise, airy structure, defined, swift-lined, and indestructible.

Lillian Morrison Poet as Mathematician

Theorems and their proofs lie at the heart of mathematics. In speaking of the “purely aesthetic” qualities of theorems and proofs in A Mathematician's Apology [Hardy, 1969], G. H. Hardy wrote that in beautiful proofs “there is a very high degree of unexpectedness, combined with inevitability and economy.” These will be the charming proofs appearing in this book.

The aim of this book is to present a collection of remarkable proofs in elementary mathematics (numbers, geometry, inequalities, functions, origami, tilings, …) that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, we hope the charming proofs in our collection will invite readers to enjoy the beauty of mathematics, to share their discoveries with others, and to become involved in the process of creating new proofs.

The remarkable Hungarian mathematician Paul Erdős (1913–1996) was fond of saying that God has a transfinite Book that contains the best possible proofs of all mathematical theorems, proofs that are elegant and perfect. The highest compliment Erdős could pay to a colleague's work was to say “It's straight from The Book.”