Yearning for the Impossible by Stillwell, John -- Read -- Imperial Library of Trantor

Index

Cover Halftitle Page Title Page Copyright Contents Page

Preface to the Second Edition Preface 1 The Irrational

1.1 The Pythagorean Dream 1.2 The Pythagorean Theorem 1.3 Irrational Triangles 1.4 The Pythagorean Nightmare 1.5 Explaining the Irrational 1.6 The Continued Fraction for 2 1.7 Equal Temperament

2 The Imaginary

2.1 Negative Numbers 2.2 Imaginary Numbers 2.3 Solving Cubic Equations 2.4 Real Solutions via Imaginary Numbers 2.5 Where Were Imaginary Numbers before 1572? 2.6 Geometry of Multiplication 2.7 Complex Numbers Give More than We Asked for 2.8 Why Call Them”Complex”Numbers?

3 The Horizon

3.1 Parallel Lines 3.2 Coordinates 3.3 Parallel Lines and Vision 3.4 Drawing without Measurement 3.5 The Theorems of Pappus and Desargues 3.6 The Little Desargues Theorem 3.7 What Are the Laws ofAlgebra? 3.8 Projective Addition and Multiplication

4 The Infinitesimal

4.1 Length and Area 4.2 Volume 4.3 Volume of a Tetrahedron 4.4 The Circle 4.5 The Parabola 4.6 The Slopes of Other Curves 4.7 Slope and Area 4.8 The Value of π 4.9 Ghosts of Departed Quantities

5 Curved Space

5.1 Flat Space and Medieval Space 5.2 The 2‐Sphere and the 3‐Sphere 5.3 Flat Surfaces and the ParallelAxiom 5.4 The Sphere and the ParallelAxiom 5.5 Non‐Euclidean Geometry 5.6 Negative Curvature 5.7 The Hyperbolic Plane 5.8 Hyperbolic Space 5.9 Mathematical Space and Actual Space

6 The Fourth Dimension

6.1 Arithmetic of Pairs 6.2 Searching for an Arithmetic of Triples 6.3 Whyn-tuples Are Unlike Numbers when n ≥ 3 6.4 Quaternions 6.5 The Four-Square Theorem 6.6 Quaternions and Space Rotations 6.7 Symmetry in Three Dimensions 6.8 Tetrahedral Symmetry and the 24-Cell 6.9 The Regular Polytopes

7 The Ideal

7.1 Discovery and Invention 7.2 Division with Remainder 7.3 The Euclidean Algorithm 7.4 Unique Prime Factorization 7.5 Gaussian Integers 7.6 Gaussian Primes 7.7 Rational Slopes and Rational Angles 7.8 Unique Prime Factorization Lost 7.9 Ideals—Unique Prime Factorization Regained

8 Periodic Space

8.1 The Impossible Tribar 8.2 The Cylinder and the Plane 8.3 Where the Wild Things Are 8.4 Periodic Worlds 8.5 Periodicity and Topology 8.6 A Brief History of Periodicity 8.7 Non‐Euclidean Periodicity

9 The Infinite

9.1 Finite and Infinite 9.2 Potential and Actual Infinity 9.3 The Uncountable 9.4 The Diagonal Argument 9.5 The Transcendental 9.6 Yearning for Completeness

Epilogue References Index