You are here

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

Claudi Alsina and Roger B. Nelsen
MAA Press
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions 50
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Mark Hunacek
, on

I must start with a confession: for my entire mathematical life, I have strongly preferred two dimensions to three. I think this preference was born in my (poorly taught) undergraduate multivariable calculus course, which left me with two firm conclusions: first, that dealing with triple integrals was an absolutely horrible experience, and second, that I had little no or no intuition about how to visualize things in three dimensions.

My preference for “keeping things planar” has persisted to this day, and affects the way I do certain things. For the last several years, for example, I have taught a two-semester course at the senior undergraduate level in geometry, and although in those two semesters I teach geometry from a variety of perspectives (analytic and synthetic, Euclidean and non-Euclidean, transformational, etc.) I have never given any real thought to doing solid geometry at all. As a result of looking at this book, though, that may change. If I had had access to this text in my formative years, my entire attitude about multidimensional geometry might well be different now.

The book offers a smorgasbord of topics in solid geometry, arranged in chapters that (except for the first one, which is introductory) each center on a useful technique: enumeration, representation (of numbers as volumes), dissection, taking plane sections, intersection, iteration, motion, projection and folding/unfolding. Each chapter provides several examples of the technique that is the subject of that chapter; some are quite modern but others are classical. To give a sense of what is discussed, I list below a topic or problem from each of these nine chapters:

  • What is the maximum number of parts that three-dimensional space can be divided into by n planes?
  • A geometric proof of the arithmetic mean/geometric mean inequality for three numbers, followed by some examples of its use in geometric optimization problems
  • Establishing formulas for the volume of a number of different polyhedra
  • The theorem of de Gua (a three-dimensional version of the Pythagorean theorem for right tetrahedra)
  • A discussion of Prince Rupert’s Cube, the largest cube that can pass through a hole in the unit cube (amazingly, this cube has edge length greater than 1)
  • The Schwarz Lantern (a polyhedral surface, depending on two parameters, that can be inscribed in a cylinder; depending on the parameters, however, the Schwarz Lantern can have an area varying from that of the cylinder to any arbitrarily large positive number)
  • Viviani’s theorem for a regular tetrahedron (involving a formula for the sum of the perpendicular distances from an interior point of the tetrahedron to the four faces)
  • A projection-based proof of Euler’s formula \(V - E + F = 2\)
  • Solving the classical Greek problem of doubling the cube (which is impossible using only compass and straightedge) by paper folding

The foregoing selection of topics represents only a small sample of the vignettes that are discussed in the book. One interesting feature of the text that should be noted is that on several occasions the authors use three-dimensional theory to prove results in the plane: for example, on pages 119–120, there is a brief discussion of Desargues’ theorem (although I was surprised that the name wasn’t used) and, on pages 185–186, proofs of two little-known results of Euclidean plane geometry (for example, if three circles with the same radius are all drawn through a point, then the other three points of intersection determine a circle with the same radius).

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. Each chapter contains lots of photographs and drawings; hardly a page goes by without some kind of picture on it. Many of these are drawings that illustrate the mathematics, but there are also a lot of photographs of works of art and architecture around the world, accompanied by useful discussion. In addition, each chapter ends with quite a few (ranging from about ten to twenty) exercises, called “challenges”; solutions or hints to all of these are provided in the back of the book, which may enhance its value for self-study but make it less convenient for use as a text. There is a good six-page list of references, including both books and journal articles, including a few that are not in English.

Not many university mathematics departments offer courses on solid geometry, so I don’t imagine that this book will find a lot of use as a text for a regularly offered course. However, I can certainly see this book being used with profit as a text for an honors seminar or as supplemental reading in a geometry course. 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 ( teaches mathematics at Iowa State University. 

1. Introduction
2. Enumeration
3. Representation
4. Dissection
5. Plane sections
6. Intersection
7. Iteration
8. Motion
9. Projection
10. Folding and Unfolding
Solutions to the Challenges
About the Authors