Every now and then you read a book on a world that you had already decided was not for you and discover that it in fact opens up that world in front of you. It had been years since I had given up on the visual and concrete facets of mathematics, thinking that I was not capable of appreciating them to the extent that would allow me to actually do something with them. This book gave me some hope that I too could make my own polyhedra and play around with them and make conjectures and have something solid and beautiful in my hands while talking about them.
Shaping Space is Marjorie Senechal’s third volume with a similar title. Some might recall the anthology The Shape of Content, Creative Writing in Mathematics and Science, edited by Senechal in collaboration with Chandler Davis and Jan Zwicky. The book reviewed here is more closely related to an earlier book. In 1988, Senechal, together with George Fleck, edited the original volume Shaping Space: A Polyhedral Approach. This was a volume of essays and articles about polyhedra, mainly originating from a three-day conference held in 1984 at Smith College. Arthur Loeb, the series editor for that volume, wrote in 1988:
Shaping Space is a polyhedral anthology. Like the conference of the same name which inspired it, it is polyglot and polydisciplinary. It is unlikely that as many scholars and artists actively involved with polyhedra will be together again in the near future; it was therefore deemed important to leave a physical imprint of the event by means of publication of this book, and to share its concerns with a broader audience.
The 1988 volume was eclectic and clearly reflected its source. It was in essence a conference proceedings volume, though much more exciting and colorful than most.
This 2013 edition is much different. A careful comparison of the tables of contents of the two volumes reveals both connections and departures. First of all many of the famous contributors in the first volume still are here, but the presentation has changed drastically. This new book seems to have taken the best parts of the first volume and shuffled them around and placed each individual contribution at the right place, where it now would belong within the framework of a coherent storyline. The storyline is an exploration of the Polyhedron Kingdom, a fictitious land where the rulers are the regular polyhedra and the official language is that of mathematics. But this really is not as cheesy as it may sound at first. Many loose threads of the first volume fit well into this overarching theme of a guidebook. Senechal’s own introductions to the three parts (I. First Steps, II. Polyhedra in Nature and Art, III. Polyhedra in the Geometrical Imagination) bring the whole book together into an exquisite tapestry.
There is much that is new in this volume. Besides the introductions by Senechal herself, there are new sections and chapters written by Robert Connelly, Eric Demaine (with Martin Demaine and Vi Hart), George Hart, Joseph O’Rourke, Ileana Streina, and Gunter Ziegler (with Moritz Schmidt). In fact one of my favorites in the whole book is a new addition: the essay “Balloon Polyhedra” by Eric Demaine, Martin Demaine and Vi Hart, is one of the now six recipes for making polyhedra right at the start, getting the reader all excited about jumping in to play.
And the title is new of course. The “Polyhedral Approach” is not really abandoned, but the new subtitle (“Exploring Polyhedra in Nature, Art, and the Geometrical Imagination”) explains much better the content and the attitude of the book. Two other significant new features are the two final sections of the book. “Notes and References” is for the reader who always wants to know more: here her curiosity will be quelled, whether it relate to a passing remark in Coxeter’s contribution on “Regular and Semiregular Polyhedra” or to the background and current status of any of the problems listed in the ultimate chapter by Moritz Schmidt and Gunter Ziegler (“Ten Problems in Geometry”). Then there is the section on “Sources and Acknowledgments”, which puts all the backmatter from each individual contribution together so you can read the essays seamlessly from one to the other, but can locate detailed source information you need as soon as you need it.
Reading chapter after chapter of this beautifully written, edited, and printed book, I was continually impressed by the variety of the content included as well as the diversity of ways one could approach this one topic. The images, several in full color and always big enough, added much to the written content. On a personal level, I was thrilled that I could once again get excited about making models and playing with them.
If readers have the original volume and have enjoyed it, I promise that this new one will not disappoint. You will find that even the repeated contributions find their place in this new edition; many of those have been updated to include recent results, and occasionally, new ideas. The new contributions introduce several beautiful new horizons, and the new volume is much more coherent and attractive. There are one or two essays I will most likely not read again, but I know that most of the rest will probably come up again on my personal reading list. And I will definitely read and reread the six recipes (“Six Recipes for Making Polyhedra”) many times in the near future, and I expect that both my students and my children will be better off for it.
Gizem Karaali is associate professor of mathematics at Pomona College and a founding editor of the Journal of Humanistic Mathematics. Since May 2013, she is also the associate editor of The Mathematical Intelligencer.