**Dolciani Mathematical Expositions**

Volume: 50;
2015;
271 pp;
Hardcover

**Print ISBN: 978-0-88385-358-0
Product Code: DOL/50**

List Price: $55.00

AMS Member Price: $41.25

MAA Member Price: $41.25

**Electronic ISBN: 978-1-61444-216-5
Product Code: DOL/50.E**

List Price: $55.00

AMS Member Price: $41.25

MAA Member Price: $41.25

# A Mathematical Space Odyssey: Solid Geometry in the 21st Century

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*Claudi Alsina; Roger B. Nelsen*

MAA Press: An Imprint of the American Mathematical Society

Solid geometry is the traditional name for what
we call today the geometry of three-dimensional Euclidean space. This
book presents techniques for proving a variety of geometric results in
three dimensions. Special attention is given to prisms, pyramids,
platonic solids, cones, cylinders and spheres, as well as many new and
classical results. A chapter is devoted to each of the following basic
techniques for exploring space and proving theorems: enumeration,
representation, dissection, plane sections, intersection, iteration,
motion, projection, and folding and unfolding.

The book includes a selection of challenges for each chapter with
solutions, references and a complete index. The text is aimed at
secondary school and college and university teachers as an
introduction to solid geometry, as a supplement in problem solving
sessions, as enrichment material in a course on proofs and
mathematical reasoning, or in a mathematics course for liberal arts
students.

#### Reviews & Endorsements

… If I had had access to this text in my formative years, my entire attitude about multidimensional geometry might well be different now. … The writing seemed to me to be clear, though succinct and efficient. The authors have managed to keep the prerequisites for reading this book to a minimum; certainly a year of calculus is sufficient for just about everything done here, and even a college freshman without any calculus would find almost all of the text comprehensible, since calculus is used in only a handful of circumstances. The book employs a number of nice pedagogical features. … It's certainly something that anybody interested in geometry would want to take a look at—even if, like me, you didn't think you were all that interested in solid geometry.

-- Mark Hunacek, MAA Reviews

…This book provides a valuable resource for the study of solid geometry that could be used by teachers all the way from late elementary school through college. …

-- Charles Ashbacher

“A Mathematical Space Odyssey” is an excellent and thorough introduction to the basic ideas of solid geometry. … The liberal use of diagrams and figures provides good support for the mathematics presented. The book is written in such a way that it is suitable for anyone with an understanding of basic algebra, with plenty of figures to support the mathematics presented. … It would be an excellent textbook for an introductory college course in solid geometry. …

-- Hilary Smith Risser, Mathematics Teacher

# Table of Contents

## A Mathematical Space Odyssey: Solid Geometry in the 21st Century

- Cover cov11
- Half title i3
- Copyright ii4
- Title iii5
- Epigraph iv6
- Series v7
- Dedication viii10
- Preface ix11
- Contents xi13
- 1 Introduction 117
- 2 Enumeration 2743
- 3 Representation 4561
- 4 Dissection 6581
- 5 Plane sections 8399
- 5.1 The hexagonal section of a cube 8399
- 5.2 Prismatoids and the prismoidal formula 85101
- 5.3 Cavalieri's principle and its consequences 89105
- 5.4 The right tetrahedron and de Gua's theorem 93109
- 5.5 Inequalities for isosceles tetrahedra 96112
- 5.6 Commandino's theorem 97113
- 5.7 Conic sections 99115
- 5.8 Inscribing the Platonic solids in a sphere 104120
- 5.9 The radius of a sphere 107123
- 5.10 The parallelepiped law 108124
- 5.11 Challenges 110126

- 6 Intersection 117133
- 6.1 Skew lines 118134
- 6.2 Concurrent lines in the plane 119135
- 6.3 Three intersecting cylinders 120136
- 6.4 The area of a spherical triangle 121137
- 6.5 The angles of a tetrahedron 124140
- 6.6 The circumsphere of a tetrahedron 126142
- 6.7 The radius of a sphere, revisited 127143
- 6.8 The sphere as a locus of points 129145
- 6.9 Prince Rupert's cube 130146
- 6.10 Challenges 131147

- 7 Iteration 133149
- 8 Motion 147163
- 8.1 A million points in space 148164
- 8.2 Viviani's theorem for a regular tetrahedron 149165
- 8.3 Dissecting a cube into identical smaller cubes 152168
- 8.4 Fair division of a cake 153169
- 8.5 From the golden ratio to the plastic number 153169
- 8.6 Hinged dissections and rotations 154170
- 8.7 Euler's rotation theorem 156172
- 8.8 The conic sections, revisited 157173
- 8.9 Instant Insanity 158174
- 8.10 Challenges 161177

- 9 Projection 165181
- 9.1 Classical projections and their applications 165181
- 9.2 Mapping the earth 169185
- 9.3 Euler's polyhedral formula 177193
- 9.4 Pythagoras and the sphere 178194
- 9.5 Pythagoras and parallelograms in space 180196
- 9.6 The Loomis-Whitney inequality 182198
- 9.7 An upper bound for the volume of a tetrahedron 184200
- 9.8 Projections in reverse 185201
- 9.9 Hamiltonian cycles in polyhedra 187203
- 9.10 Challenges 189205

- 10 Folding and Unfolding 193209
- 10.1 Polyhedral nets 194210
- 10.2 Deltahedra 196212
- 10.3 Folding a regular pentagon 200216
- 10.4 The Delian problem: duplicating the cube 201217
- 10.5 Surface areas of cylinders, cones, and spheres 203219
- 10.6 Helices 209225
- 10.7 Surface areas of the bicylinder and tricylinder 211227
- 10.8 Folding strange and exotic polyhedra 214230
- 10.9 The spider and the fly 217233
- 10.10 The vertex angles of a tetrahedron 219235
- 10.11 Folding paper in half twelve times 219235
- 10.12 Challenges 222238

- Solutions to the Challenges 227243
- References 259275
- Index 265281
- About the Authors 272288
- Back cover 273289