Geometric origami. With a foreword by Andrew Jobbings (Q2883157)

It is well known that certain angles, such as \(60^{\circ}\), cannot be trisected, and that some regular polygons, such as the heptagon, cannot be constructed. Less well known is Hippocrates' theorem stating that every angle can be trisected and Archimedes' theorem stating that a regular heptagon can be constructed. The apparent contradiction is removed once we know that the first statements refer to the so-called Euclidean constructions that use a straightedge and a compass only, while the constructions of Archimedes and Hippocrates allow the use of tools that draw other conic sections. Several other methods of constructions using other tools have appeared throughout history, and are described in [\textit{G. E. Martin}, Geometric constructions. Undergraduate Texts in Mathematics. New York, NY: Springer (1998; Zbl 0890.51015)]. These include constructions using a straightedge only, a compass only, a ruler and dividers, a ruler and a rusty compass, a marked ruler, and constructions using the fascinating old Japanese art of paper folding known as origami. The book under review is about constructions using origami.NEWLINENEWLINEPart I of the book covers the mathematical theory of origami constructions. After mathematicizing the origami procedures, it proves that all Euclidean constructions can be achieved by origami. These include the bisection of an arbitrary angle, constructing a regular pentagon, and constructing the roots of a quadratic equation. It then proves that using origami, we can also trisect arbitrary angles, construct regular heptagons and regular \(n\)-gons for other values of \(n\) (such as \(n=9, 13, 19\)), and construct the roots of cubic and quartic equations. None of these can be achieved using Euclidean tools.NEWLINENEWLINEPart II of the book is devoted to easy-to-follow step-by-step instructions on how to fold fifteen regular \(n\)-gons. These include \(n\)-gons for \(n = 5, 7, 8, 9, 13, 17, 19\). The book also pays attention to approximate constructions of some regular \(n\)-gons, an issue that is important from the practical point of view.NEWLINENEWLINEThose who are interested in mathematics will surely find Part I of the book very interesting. Beside being extremely well written, very easy and pleasant to read, and mathematically sound and rigorous, the wealth of material that it contains is very hard to find anywhere else. In fact, much of this material is based on discoveries, both published and unpublished, made by the author himself. The book also contains a detailed discussion of the interesting problem of finding the maximum regular \(n\)-gon, for a given \(n\), that is contained in a given square. Origamists, too, are expected to find the book invaluable for folding certain models, and especially animals. An origamist interested in making a camel with four legs, a head, a tail, and one or two humps may need to start with a regular heptagon or octagon. These polygons, as well as other polygons needed for folding other models, are what Part II of this book is designed for. However, the book may be a little disappointing to some of those amateurs for whom the phrase ``geometric origami'' may mean little, or nothing at all, beyond geometric three dimensional models made of certain origami units. This book is not concerned at all with such solids.NEWLINENEWLINEThe reviewer cannot but pay tribute to the late great writer Martin Gardner whose July 1959 \textit{Scientific American} article had a great role in popularizing origami in the mathematical community. The article can be found in his book [\textit{M. Gardner}, Origami, Eleusis, and the Soma cube. Martin Gardner's mathematical diversions. Cambridge: Cambridge University Press; Washington, DC: Mathematical Association of America (MAA) (2008; Zbl 1153.00007)].