This book is perfect for segments of mathematics courses in late middle and high school that deal with polyhedral structures and their properties. The first section begins with the folding of paper to make the five Platonic solids; the structures are introduced through a sequential series of diagrams beginning with the original cuts of the paper. After this, dual pairs of Platonic solids are made and the pairs are:
- A cube with an interior octahedron
- An icosahedron mostly contained within a dodecahedron
- A tetrahedron in combination with another tetrahedron
These shapes are all beautiful and give the students a clear and visual success once they have completed the task. Chapter 3 consists of diagrams demonstrating how to construct some basic shape faces. In chapter four, the origami techniques used to create a shape of twenty triangular pyramids are described.
Chapter five, with the title “Proof?!” is a high point of mathematical demonstration. Using what is similar to a Socratic dialog, the students are taken through a series of questions and answers designed to convince them that the wonderful Eulerian formula
V + F – E = 2
is universally true. This “proof” begins with some examples where it holds and then supposed counterexamples are put forward. The definition of the type of object the formula is to refer to is altered as the counterexamples are examined and in a step by step manner, an argument for why V + F – E = 2 is developed. While it is not a proof in the rigorous mathematical sense, it is so sound, understandable and visible that children will be able to understand it.
Origami and other paper folding activities have the enormous advantage that success is easily seen and visually understood. I strongly recommend that this book be considered as a text in the curriculum designed to prepare middle and high school teachers of mathematics.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.