# Platonic Solids

In The Name of God, The Compassionate, The Merciful. This book is dedicated to Professor Keith Critchlow, whose teaching made it possible, and to the memory of Dr. Buckminster Fuller, who taught him. I am indebted to the many geometers, authors and artists who have explored the world of polyhedra. Thanks to my family friends for comments and contributions. Introduction 1 The Platonic Solids 2 The Tetrahedron 4 The Octahedron 6 The Icosahedron 8 The Cube 10 The Dodecahedron 12 A Short Proof 14 All Things in Pairs 16 Around the Globe 18 Round and Round 20 The Golden Ratio 22 Polyhedra within Polyhedra 24 Compound Polyhedra 26 The Kepler Polyhedra 28 The Poinsot Polyhedra 30 The Archimedean Solids 32 Five Truncations 34 The Cuboctahedron 36 A Cunning Twist 38 The Icosidodecahedron 40 Four Explosions 42 Turning 44 The Archimedean Duals 46 More Explosions 48 Flatpacked Polyhedra 50 Archimedean Symmetries 52 Threedimensional Tessellations 53 Each Embracing Every Other 54 Expansions and Formulae 55 Data Table 56