Contents Preface ix 1 Introduction 1 1.1Tenexamples......................... 1 1.2 Inhabitants of space . ..................... 9 1.3 Challenges .......................... 22 2 Enumeration 27 2.1Hexnumbers......................... 27 2.2Countingcalissons...................... 28 2.3 Using cubes to sum integers . ................ 29 2.4 Counting cannonballs ..................... 35 2.5 Partitioning space with planes ................ 37 2.6 Challenges .......................... 40 3 Representation 45 3.1 Numeric cubes as geometric cubes . ........... 45 3.2 The inclusion principle and the AM-GM inequality for threenumbers......................... 49 3.3Applicationstooptimizationproblems............ 52 3.4 Inequalities for rectangular boxes . ........... 55 3.5 Means for three numbers . ................ 59 3.6 Challenges .......................... 60 4 Dissection 65 4.1 Parallelepipeds, prisms, and pyramids . ........... 65 4.2 The regular tetrahedron and octahedron ........... 67 4.3 The regular dodecahedron . ................ 71 4.4Thefrustumofapyramid................... 72 4.5 The rhombic dodecahedron . ................ 74 4.6 The isosceles tetrahedron . ................ 76 4.7TheHadwigerproblem.................... 77 4.8 Challenges .......................... 79 xi xii Contents 5 Plane sections 83 5.1 The hexagonal section of a cube ............... 83 5.2Prismatoidsandtheprismoidalformula........... 85 5.3 Cavalieri’s principle and its consequences . ....... 89 5.4 The right tetrahedron and de Gua’s theorem . ....... 93 5.5 Inequalities for isosceles tetrahedra . ........... 96 5.6 Commandino’s theorem . ................ 97 5.7Conicsections........................ 99 5.8 Inscribing the Platonic solids in a sphere ...........104 5.9 The radius of a sphere ....................107 5.10Theparallelepipedlaw....................108 5.11 Challenges ..........................110 6 Intersection 117 6.1Skewlines..........................118 6.2 Concurrent lines in the plane . ................119 6.3 Three intersecting cylinders . ................120 6.4 The area of a spherical triangle ................121 6.5 The angles of a tetrahedron . ................124 6.6 The circumsphere of a tetrahedron . ...........126 6.7 The radius of a sphere, revisited ...............127 6.8 The sphere as a locus of points ................129 6.9 Prince Rupert’s cube .....................130 6.10 Challenges ..........................131 7 Iteration 133 7.1 Is there a four color theorem in space? . ...........133 7.2 Squaring squares and cubing cubes . ...........134 7.3 The Menger sponge and Platonic fractals . .......136 7.4Self-similarityanditeration.................139 7.5 The Schwarz lantern and the cylinder area paradox . 140 7.6 Challenges ..........................143 8Motion 147 8.1 A million points in space . ................148 8.2 Viviani’s theorem for a regular tetrahedron . ........149 8.3 Dissecting a cube into identical smaller cubes . .......152 8.4Fairdivisionofacake....................153 8.5Fromthegoldenratiototheplasticnumber.........153 8.6 Hinged dissections and rotations ...............154 Contents xiii 8.7 Euler’s rotation theorem . ................156 8.8 The conic sections, revisited . ................157 8.9InstantInsanity........................158 8.10 Challenges ..........................161 9 Projection 165 9.1Classicalprojectionsandtheirapplications.........165 9.2 Mapping the earth . .....................169 9.3 Euler’s polyhedral formula . ................177 9.4 Pythagoras and the sphere . ................178 9.5 Pythagoras and parallelograms in space ...........180 9.6 The Loomis-Whitney inequality ...............182 9.7 An upper bound for the volume of a tetrahedron .......184 9.8Projectionsinreverse.....................185 9.9 Hamiltonian cycles in polyhedra ...............187 9.10 Challenges ..........................189 10 Folding and Unfolding 193 10.1 Polyhedral nets . .....................194 10.2 Deltahedra ..........................196 10.3 Folding a regular pentagon . ................200 10.4 The Delian problem: duplicating the cube . .......201 10.5 Surface areas of cylinders, cones, and spheres .......203 10.6Helices............................209 10.7 Surface areas of the bicylinder and tricylinder .......211 10.8 Folding strange and exotic polyhedra . ...........214 10.9Thespiderandthefly....................217 10.10 The vertex angles of a tetrahedron . ...........219 10.11 Folding paper in half twelve times . ...........219 10.12 Challenges ..........................222 Solutions to the Challenges 227 Chapter 1 . ..........................227 Chapter 2 . ..........................230 Chapter 3 . ..........................234 Chapter 4 . ..........................237 Chapter 5 . ..........................239 Chapter 6 . ..........................243 Chapter 7 . ..........................245 Chapter 8 . ..........................246 xiv Contents Chapter 9 . ..........................249 Chapter 10 ..........................253 References 259 Index 265 About the Authors 272.