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Flexagons Inside Out

Les Pook
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Robert McGuigan
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Flexagons are hinged polygonal models that have the property that "flexing" them (manipulating the hinges in a certain way) reveals different pairs of faces. Flexagons first came to public attention in 1956 and 1957 in articles by Martin Gardner in Scientific American. They reappeared in many of his subsequent books. Numerous other recreational mathematics books and articles have treated this topic as well. The mathematics behind flexagons involves geometry and combinatorics and has some of the flavor of the study of polytopes, but it is the dynamic manipulation that is most appealing.

The present book is, as far as I know, the most extensive and thorough study of flexagons available. It emphasizes the construction, manipulation, and exploration of these fascinating objects rather than their detailed rigorous mathematical analysis. As a result the book is accessible to readers with minimal background: all that is needed is some knowledge of elementary geometry. In my opinion this is a positive feature of the book because it makes it useable for exploratory projects by students of all college levels, from prospective elementary school teachers to prospective professional mathematicians. An extensive reference list gives the reader interested in the details of proofs places to find them. A very useful feature, too, is the inclusion throughout the book of many "nets" which can be duplicated and used to construct the models discussed.

Flexagons were discovered in 1939 by Arthur H. Stone, then a graduate student at Princeton. Coming to Princeton from England, Stone found that the standard notebook paper in the United States was about an inch too large for his British binder so he trimmed the paper to fit and had many strips of paper as a result. Playing at folding these, he folded a strip diagonally three times and joined the ends so that a hexagon consisting of six triangles was formed. When two adjacent triangles were pinched together and the opposite corner pushed toward the center, "flexing" the figure, the hexagon would open out and show a completely new face. Stone showed his discovery to friends at Princeton, and this led to something of a fad. A committee consisting of Stone, Bryant Tuckerman, a graduate mathematics student, Richard Feynman, a graduate physics student, and John Tukey, a young instructor in mathematics, was formed to investigate the phenomenon. After the entry of the United States into the Second World War in 1941 the flexagon committee disbanded. They never published their research, though many of their results are presented in this book. The first serious mathematical publication on the theory of flexagons was a paper by Oakley and Wisner in the Monthly in 1957.

The first flexagons were based on triangles and hexagons but it was soon discovered that it is possible to base them on other convex polygons, such as squares, pentagons, heptagons, etc. An ideal (mathematical) flexagon consists of a band of congruent, rigid, flat two-dimensional polygonal leaves, which are hinged together at common sides. "Hinged" is interpreted as meaning that the dihedral angle between two hinged leaves may vary between 0 and 360 degrees without constraint. In most cases, paper models, though obviously only an approximation to an ideal flexagon, function well enough that it isn't necessary to make a distinction between the model and the ideal. Clearly, rigorous proofs must be based on ideal, geometrical flexagons. One of the most difficult things involving the rigorous formulation of flexagons is to describe their structure and map their dynamic behavior. Indeed, the author states that the problem of how best to do this is still not completely resolved. This is a situation where visualization and models are almost necessary for understanding abstract mathematical concepts.

The book begins with an introductory chapter on making and flexing the simplest flexagons. Detailed assembly instructions are given and suggestions are made for decorating the faces to make the dynamic behavior easier to see. A chapter on the early history is followed by one on the geometry and basic rigorous study of flexagons. Two chapters are devoted to hexaflexagons, the first type to be studied. These have equilateral triangles as leaves and a hexagonal outline when assembled. They have a large number of degrees of freedom. Serious mathematical study has been given to the question of the number of distinct types of hexaflexagons with a given number of faces.

Next is a chapter on square flexagons, the second type to be discovered. They are less well understood and more complicated than hexaflexagons, exhibiting more complex dynamical behavior. Convex polygon flexagons represent the first infinite family of flexagons. The leaves are regular convex polygons, hinged together at common sides. Our understanding of convex polygon flexagons in general is incomplete, but they are generalizations of square and triangle flexagons and so have somewhat similar behavior.

The second infinite family of flexagons is the star flexagon family. A star flexagon has the appearance of an even number of regular polygons arranged about its center, each with a vertex at the center. In these cases the dynamics of flexing become more and more interesting and less and less obvious. By now we are reaching a point in our descriptions where the 1000 to 1 efficiency ratio of pictures over words is in danger of becoming realized, so interested readers must be referred to the book itself for more details!

The final chapter is about a generalization of flexagons to four dimensions, called flexahedra. The "nets" from which the flexahedra are constructed are made of three-dimensional polyhedra "hinged" at common faces. While we may have difficulty visualizing the four-dimensional flexahedron, the nets from which they are made are easily visualized. Of course, the dynamics of flexing are four-dimensional. According to the author the study of flexahedra is a rich and largely unexplored subject.

The book is very well written, making it relatively easy for the adventurous reader to carry out the constructions and repeat the investigations described. There are no exercises, though a lot of the book is a kind of guided exercise. Pook's academic background is in mechanical engineering. He became interested in flexagons in the 1960s through Martin Gardner's books and made many paper models and carried out theoretical investigations which duplicated much of the unpublished work of the pioneers in the field. I think this book is a valuable contribution to the literature, collecting in one place various scattered and difficult-to-find results in a user-friendly way. It has a variety of uses: as a supplement in a combinatorics class, as a resource for hands-on projects for geometry classes for prospective elementary school teachers, as a source of geometry projects for math majors or activities for math clubs. I recommend the book highly for college mathematics libraries and for personal libraries of mathematicians interested in combinatorial geometry or recreational mathematics.

Robert McGuigan is Professor of Mathematics, Emeritus, at Westfield State College, Westfield, Massachusetts. He credits some stimulus toward his career as a mathematician to reading as a teenager Martin Gardner's articles on flexagons in Scientific American in the 1950's.

1. Making and flexing flexagons; 2. Early history of flexagons; 3. Geometry of flexagons; 4. Hexaflexagons; 5. Hexaflexagon variations; 6. Square flexagons; 7. Introduction to convex polygon flexagons; 8. Typical convex polygon flexagons; 9. Ring flexagons; 10. Distorted polygon flexagons; 11. Flexahedra.