Cogwheels of the Mind: the Story of Venn Diagrams

  • A. W. F. Edwards
(foreword by Ian Stewart) Johns Hopkins University Press: 2004. 128 pp. $25, 17 pound 0801874343. | ISBN: 0-801-87434-3

Most of us know what a Venn diagram is, and many of us will have used them to represent families of objects with overlapping characteristics. Some of us will have noticed that what works for three sets seems to work less well for four, and might break down entirely for five or more sets, when perhaps our enthusiasm for classification begins to fade. Is there an elegant way of drawing a Venn diagram showing all possible intersections of, say, seven sets, and is it systematic — does it build routinely to an arbitrary number of sets?

What Cogwheels of the Mind triumphantly shows is that the answer to both questions is ‘yes’. A. W. F. Edwards not only shows how it can be done, but discusses a variety of different solutions that connect Venn diagrams to significant questions in coding theory. He also provides a fascinating glimpse into how research into mathematics gets done, and tells a satisfying historical story. And he provides what may be the most important reason for continuing to care about these diagrams: they illuminate the problem of finding maximal embeddings of subgraphs of a hypercube. All this and a great number of mathematically beautiful figures mean that the book deserves to become a minor classic and may well go on to make many friends for mathematics.