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A Panoply of Polygons

Claudi Alsina and Roger B. Nelsen
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions
[Reviewed by
John D. Cook
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MAA Press published A Panoply of Polygons by Claudi Alsina and Roger B. Nelsen on January 30, 2023. A few weeks later, on March 20, 2023, David Smith et al caused quite a stir announcing their discovery of an aperiodic monotile. The timing of the publication of Panoply was fortunate in that the book came out just as there is heightened interest in polygonal tilings. The timing is unfortunate in that it could have included a discussion of the aperiodic monotile if the book had been published a little later.
Panoply looks at polygons from many angles, pardon the pun, including examining their tiling properties. The cover of Panoply shows a periodic tiling of the plane by a five-sided polygon. I imagine if the book had come out a few weeks later the cover would have featured the aperiodic monotile, also known as the “hat” or “einstein” tile (the name is not capitalized in the literature). The latter name comes from the connection to the so-called Einstein problem.
The recent resolution to the long-standing question about the existence of aperiodic monotiles shows that there are deep questions one can ask about simple shapes. Panoply is a more advanced book than you might expect if your last exposure to polygons was in a high school geometry class, though the book is accessible to readers without much more than a high school level background mathematics.
In addition to presenting and proving geometric theorems as one would expect, Panoply includes photos of where various polygons appear in the wild, such as the regular heptagon over the high alter in the Basilica de la Sagrada Familia in Barcelona and the irregular 23-gon at the base of the Statue of Liberty in New York. The book also has interesting historical diversions, such as a brief discussion of how Albrecht Dürer approximated the construction of a regular 11-gon (a hendecagon) with a compass and straight edge. (The Gauss-Wantzel theorem proves this cannot be done exactly, though Dürer came close.)
A Panoply of Polygons is a fun read. The topics are nontrivial, but accessible, and can be read one or two pages at a time. It is not a textbook, at least not a conventional textbook, though it does have challenges at the end of each chapter that could be used as exercises. I think of it more as a book to keep on your nightstand, if, like the reviewer, you are the kind of person who keeps mathematics books on your nightstand.
John D. Cook is an applied mathematician and the founder of Kingwood Data Privacy.