List of Figures 1.1 A cubic curve in tropical geometry.. . . . . . . . . . . . . . . 14 2.1 A closed system of mirrors . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Grinding a plane mirror. . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 What different nations eat and drink. . . . . . . . . . . . . 34 2.4 Vision vs. “inverse vision”. . . . . . . . . . . . . . . . . . . . . . . 35 2.5 Horses by Nadia, at age of 3 years 5 months. . . . . . 35 2.6 Horses from Chauvet Cave (Ardeche) . . . . . . . . . . . . 36 2.7 A symbolic human and a naturalistic bull. . . . . . . . 37 2.8 The icosahedron inscribed into the cube. . . . . . . . . . 38 2.9 The root system D3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.10 From Kepler’s Mysterium Cosmographicum . . . . . . 40 2.11 The root systems D2 and A1 ⊕ A1. . . . . . . . . . . . . . . . 40 3.1 The Palindrome Representation Theorem.. . . . . . . . 51 3.2 Stasheff’s associahedron. . . . . . . . . . . . . . . . . . . . . . . . 57 4.1 Electrolocation in fish.. . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2 Convex and non-convex sets. . . . . . . . . . . . . . . . . . . . . 77 4.3 An impossible object. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4 The Magpie on the Gallows . . . . . . . . . . . . . . . . . . . . . 79 4.5 Spherical stone vase, Ancient Egypt . . . . . . . . . . . . . 80 5.1 A tessellation of the sphere. . . . . . . . . . . . . . . . . . . . . . 90 5.2 A hyperbolic tessellation. . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 Euclidean tessellations . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4 A pattern of angels and devils based on M. C. Escher’s Circle Limit IV and the tessellation {4, 5}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.5 The underlying Circle Limit IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.6 The railway tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.7 Paradisio, by Gustave Dor´ . . . . . . . . . . . . . . . . . . . . . 106 xxiii tessellation of M. C. Escher’s
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