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´ e . . . . . . . . . . . . . . . . . . . . . 106
xxiii
tessellation of M. C. Escher’s
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