Peter Hilton and Jean Pedersen
CAMBRIDGE UNIVERSITY PRESS 2010, 290 PAGES PRICE (PAPERBACK) £19.99 ISBN 978-0-521-12821-6
A Mathematical Tapestry combines some practical recreational mathematics, in particular paper folding constructions, with deep theoretical ideas in geometry, number theory and group theory. This more than justifies the subtitle Demonstrating the Beautiful Unity of Mathematics.
The opening chapter on flexagons tells the story of four graduate students who meet at Princeton University in 1939 (Arthur H. Stone, Bryant Tuckerman, Richard P. Feynman, John W. Tukey). Each went on to different distinguished academic careers but together they played a game of recreational mathematics using folded strips of paper called flexagons. Remarkably these allow the construction of a regular polygon with any number of sides – albeit not exact, but to an arbitrary degree of precision, and with a rapid rate of convergence. This is compared to the more limited, but exact constructions available using the traditional Euclidean tools: straightedge and compass. Starting with the triangle, the algorithm used, ‘Fold-and-Twist’ or FAT, is shown to be more generally applicable to all regular polygons and some other two-dimensional shapes such as ‘star polygons’. The book goes on to describe more sophisticated constructions such as regular polyhedra.
The simple practical exercises, which have detailed instructions and diagrams, are of intrinsic interest and would be valuable in the classroom even at an elementary level. In this book they are used to demonstrate ideas in higher mathematics. Analysis of the FAT algorithm, particularly the way it converges toward the required angle, leads naturally to a question in number theory – in what circumstances is (ta −1)(tb −1) an integer? Although this is an elementary result involving greatest common divisors, it is proved (by considering numbers in different bases) in a way that sheds light on the problem. This philosophy is explicitly stated: ‘The purpose of a displayed proof is to convey the meaning of the statement of a theorem, and its significance.’ (page 52), and is practiced throughout the text.
Further chapters include more advanced number theory, including a method of checking the factorisation of large Fermat numbers which can be implemented easily on a spreadsheet. One practical activity, ‘Jennifer’s puzzle’ was designed by a high school student, a daughter of one of the authors. It is used to demonstrate several mathematical applications including finding the volume of a regular tetrahedron. Group theory is introduced in the context of symmetry in geometry. Chapter 15 offers some historical perspective, invoking René Descartes and George Pólya, the latter being well known to the authors. A philosophical reference is made to Marxist dialectic, the context being the development of Euler’s formula for polyhedra.
There are a few minor criticisms – the notation used includes tables of numbers which resemble matrices but have specific meanings. These symbols are defined in the text but an overall glossary would be useful. The book is clearly illustrated, but only in black and white. For some of the paper folding exercises, the available patterns and grey shades are used to exhaustion in the diagrams, so some colour in the body of the book, as used on the front cover, would be welcome.
In summary the book fulfils its purpose – it does bring out unexpected connections between different branches of mathematics, and bridges the gap between the practical and recreational side and the more abstract theory. However a casual reader may skip over some of the proofs, so even the amateur mathematician will find it entertaining, readable and well-referenced. There is also something here for the mathematics teacher: elementary students may simply enjoy the tactile experience of creating artefacts, older students will gain an insight into fundamental concepts such as accuracy and proof. Finally for the number theorist or algebraist it offers a fresh approach to help visualise and explain their material.
Francis McGonigal CMath MIMA
Birmingham City University
Mathematics Today December 2011
A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics can be purchased at Amazon.co.uk