Single Digits by Chamberland, Marc -- Read -- Imperial Library of Trantor

Index

Cover Page Title Page Copyright Page Contents Preface Chapter 1 The Number One

Sliced Origami Fibonacci Numbers and the Golden Ratio Representing Numbers Uniquely Factoring Knots Counting and the Stern Sequence Fractals Gilbreath’s Conjecture Benford’s Law The Brouwer Fixed-Point Theorem Inverse Problems Perfect Squares The Bohr–Mollerup Theorem The Picard Theorems

Chapter 2 The Number Two

The Jordan Curve Theorem and Parity Arguments Aspect Ratio How Symmetric Are You? The Pythagorean Theorem Beatty Sequences Euler’s Formula Matters of Prime Importance The Ham Sandwich Theorem Power Sets and Powers of Two The Sylvester–Gallai Theorem Formulas for π Multiplication The Thue–Morse Sequence Duals Apollonian Circle Packings Perfect Numbers and Mersenne Primes Pythagorean Tuning and the Square Root of 2 Inverse Square Laws The Arithmetic-Geometric Mean Inequality Positive Polynomials Newton’s Method for Root Finding More Division via Multiplication The Allure of π²/6 Jacobian Conjectures

Chapter 3 The Number Three

The 3x + 1 Problem Triangular Numbers and Bulgarian Solitaire Rock-Paper-Scissors and Borromean Rings Random Walks Trisecting an Angle The Three-Body Problem The Lorenz Attractor and Chaos Period Three Implies Chaos Patterns among the Stars Fermat’s Last Theorem Leftovers Anyone? Egyptian Fractions Arrow’s Impossibility Theorem Mapping Surfaces Guarding an Art Gallery The Poincaré Conjecture Monge’s Three-Circle Theorem Marden’s Theorem The Reuleaux Triangle The Third Critical Point Sums of Cubes Approximating Decay

Chapter 4 The Number Four

The Four Color Theorem The Tennis Ball Theorem Sum of Squares Identities Rearranging Four Pieces Ducci Sequences Euler’s Sum of Powers Conjecture Villarceau Circles The Inscribed Square Problem Regular Polygons on a Computer Screen The Four Travelers Problem The Four Exponentials Conjecture Concentric Quadrilaterals The Four Hats Problem

Chapter 5 The Number Five

The Miquel Five Circles Theorem The Platonic Solids Solving Polynomial Equations Diophantine Approximation The Petersen Graph The Happy Ending Problem Tessellations Of Balls and Sausages Knight’s Tours on Rectangular Boards Magic with Five Cards Soccer Balls and Domes Recycling ad Infinitum The Rogers–Ramanujan Identities

Chapter 6 The Number Six

Optimal Packing Of Friends and Strangers Six Degrees of Separation A Necklace of Spheres Hexagons in Pascal’s Triangle The Game of Hex The Wendt Determinant Six Lengths in Geometry

Chapter 7 The Number Seven

The Seven Circles Theorem Digits of 1/7 and Ellipses Strassen’s Matrix Multiplication The Fano Plane Border Patterns The Szilassi Polyhedron and the Heawood Graph The Kuratowski Closure–Complement Theorem Can You Hear the Shape of a Drum? Barker Codes Recreational Mathematics Experiments with Integrals

Chapter 8 The Number Eight

The Pizza Theorem Shuffling Cards The Game of Life Repetition in Pascal’s Triangle The Sierpiński Carpet Quaternions and Octonions The Summit of E8

Chapter 9 The Number Nine

Nine Points and Collinearity Casting Out Nines Primes and Nines The Fifteen Theorem Circle Packings with Two Sizes Catalan’s Conjecture The Heegner Numbers

Chapter 10 Solutions

Rearranging Four Pieces (Chapter 4) The Four Hats Problem (Chapter 4) The Kuratowski Closure–Complement Theorem (Chapter 7) Recreational Mathematics (Chapter 7)

Further Reading Credits for Illustrations Index