Contents Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii List of Photographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv Part I Simple Things: How Structures of Human Cognition Reveal Themselves in Mathematics 1 A Taste of Things to Come. . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Simplest possible example . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Switches and flows: some questions for cognitive psychologists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Choiceless computation. . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Polynomial time complexity . . . . . . . . . . . . . 7 1.3.2 Choiceless algorithms. . . . . . . . . . . . . . . . . . . 9 1.4 Analytic functions and the inevitability of choice . . 10 1.5 You name it—we have it . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 pleasurable than others? . . . . . . . . . . . . . . . . . . . . . . . 15 1.7 What lies ahead?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 What You See Is What You Get . . . . . . . . . . . . . . . . . . . . . 23 2.1 The starting point: mirrors and reflections . . . . . . . 23 2.2 Image processing in humans . . . . . . . . . . . . . . . . . . . . 25 2.3 A small triumph of visualization: Coxeter’s proof of Euler’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Mathematics: interiorization and reproduction . . . 30 2.5 How to draw an icosahedron on a blackboard . . . . . 33 2.6 Self-explanatory diagrams . . . . . . . . . . . . . . . . . . . . . . 38 vi Why are certain repetitive activities more
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