The Secret World of Flexagons - Scott Sherman, Yossi Elran and Ann Schwartz ***

A great book for the right audience. When I was a teenager I loved Martin Gardner's Mathematical Puzzles and Diversions books. They combined mathematical oddities, such as there being more than one size of infinity, with numerical tricks and some physical maths-based objects, most notably flexagons. As someone who is physically inept, I never got much beyond a basic hexaflexagon, but these paper-based three dimensional structures that could change shape when manipulated, a bit like more sophisticated versions of paper fortune tellers, were fascinating.

Now we have a chunky, large format title exploring flexagons in far more depth. The first part of the book takes us into different flexagon structures and the range of possibilities enabling flexing. Some of these require quite sophisticated manipulation of the paper structure (which I did struggle with somewhat, being clumsy). Each flexagon type comes with a flat template to reproduce. These are sometimes quite small and for me benefited from blowing up a bit to make them more practical to manipulate, though many are reproduced larger later in the book.

Having got past the basics, the authors introduce more and more complex flexes (who is for a Möbius flip flex?), before exploring the culture that has built up around flexagons, such as naming conventions and 'pinch state diagrams', which use network diagrams to show how the different transformations are linked. We also get an exploration of the relationship between flexagons and group theory, some maths theory specifically built around flexagons, and the links between flexagons and topology. The final section, 'fun with flexagons' looks at combining flexagon structures with art work to give a more impressive visual impact.

This book will be hugely appealing if this is your thing. I'm giving it three stars, which my rating system describes as 'good solid book, worth reading if you are interested in the topic', not to say that it's a mid-ranking book, but rather that I suspect there are many popular science/maths readers it won't appeal to. I confess I did lose my interest in much of recreational maths as I got older, but if you find the concepts here get you excited, you absolutely need this book.

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