How Round Is Your Circle?  Where Engineering and Mathematics Meet

by John Bryant & Chris Sangwin
Princeton University Press, Princeton, NJ, 2008
352 pp., illus. 240 b/w, 30 col. Trade, $29.95 / £17.95
ISBN: 978-0-691-13118-4.

Reviewed by Phil Dyke
Professor of Applied Mathematics, School of Mathematics and Statistics
University of Plymouth

phil.dyke@plymouth.ac.uk

As a schoolboy I used to enjoy making mathematical models of polyhedra.  On a wet Saturday morning you would find me copying nets from a book such as Cundy and Rollett (Mathematical Models, Oxford University Press, 1961) out of thin card and then bending them into polyhedra; the stellated dodecahedron was a favourite.  I would even paint them so that each plane face was a different colour.  Sadly however, most often they would not quite fit together perfectly; there was always the odd small gap no matter how carefully one drew the net, folded precisely along the dotted line or cut the card.  Of course, the reason was the thickness of the card, assumed zero by the mathematicians but never zero in practice.  No doubt if any card bending was required, similar problems would occur with cutting models out of empty cereal packets, annoying children everywhere.

If such things intrigue you, you will enjoy this book written by an engineer (John Bryant) and a mathematician (Chris Sangwin).  This difference between the ideal and the practical encapsulates the kind of problem found in this book.  Here they tackle more fundamental issues such as constructing straight lines and constructing circles, both pretty simple at first glance, but full of intrigue.  For example, getting linkages to draw a straight line precisely is difficult.  There are almost always slight inaccuracies.  Of course on a practical level, often any variation from straight is dwarfed by the thickness of the pencil line itself, but that is the whole point.  When do such inaccuracies become unacceptable?  Are they ever?  Are they more acceptable if you know with precision the magnitude of the inaccuracy?

Drawing a circle gives rise to different issues; here the authors try making polygons inscribed in a circle using that doyen of mathematical commands, ruler and compasses only.  Trisecting the angle, one of the classically impossible problems alongside squaring the circle and doubling the cube is solved by cheating a bit; and here and throughout the text there is detailed and interesting historical insight.  Further into the book, the mathematics does get a little more sophisticated which might put off some readers.  Geometric progressions are required when looking at dissections, and calculus (just elementary differentiation) needed for looking at pursuit curves and catenaries.  Analytic geometry, trigonometry and integral calculus get an airing when the authors examine measuring areas, but the calculus is restricted to appear only towards the end.  There are chapters on objects such as plane areas of constant diameter (the UK 50 pence piece is one example) with applications to rotors, and computation using the slide rule, surely obscure now to anyone under 30.  Finally mechanics gets a look in through problems such as stacking dominoes in different ways such that they just do not fall.  There are links to number theory and integration that are explored.

It is not clear whom the book is really intended for.  A mathematician such as myself might be intrigued by some of the detail but irritated by harping on about the (to me) blindingly obvious.  On the other hand, just how many non mathematicians are there out there who could get to grips with the more technical parts?  Unfortunately it is analytical geometry and calculus that are required here and both topics have dwindled in school syllabi that will diminish the readership, and this is a pity. This is a curious book, beautifully written and presented, but never a best seller.  On balance, this mathematician welcomes it to the bookshelves where it sits happily alongside Cundy and Rollett mentioned earlier and Lockwood’s Book of Curves (Cambridge University Press, 1960) and given the nearly fifty year gap between these classic out-of-print texts and How Round Is Your Circle? maybe the time is ripe for such an addition to the recreational mathematics literature.  I think mechanical engineers with an interest in measurement would similarly embrace the book.  A third type of reader might be more philosophical; captured by the conflict between the ideal mathematics and the practical world, but I fear that here the technicalities do get in the way.