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