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

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 (mhunacek@iastate.edu) teaches mathematics at Iowa State University.