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REVIEWS Contents
Mathematicians: An Outer View of the Inner World
Mathematicians: An outer view of the inner world by Mariana Cook, Princeton University Press, 2009, 208 pp, £24.95, $35.00, ISBN 978-0-691-13951-7. This book features ninety-two leading mathematicians. For each mathematician a photograph is given on the right-hand page and an autobiographical account appears on the left-hand page. The black-and-white photographs are by Mariana Cook who has a high reputation for books of photographic portraits. Cook superbly captures the character of these mathematicians in this stunning collection of photographs. They are works of art. The autobiographical texts vary greatly in nature. Some mathematicians give details of their career while others concentrate on how they became interested in the subject and why they are so passionate about it. The different approaches taken by each mathematician not only gives us insight into the person but also make for a more interesting book. Many draw parallels between mathematics and composing music or writing poetry. We read comments such as "In a way, doing mathematics feels like writing a novel where your problem evolves like a live character" and "Sitting in a good mathematics lecture is like sitting in a good opera." We learn about the interaction between mathematicians, about their joys and disappointments, and most of all their deep feelings towards their subject. The book is aimed at the general public so the mathematicians try to avoid technical descriptions of their work. This produces some beautiful comments such as: "Should I explain what a lemma is? A mountain climber needs holds to get from one level to the next one. Lemmas are the holds of mathematicians." There are, however, a few statements which will have little meaning to the general reader such as one mathematician speaking about "Carlos Simpson’s development of a non-Abelian version of Hodge theory for Kähler manifolds". All the featured mathematicians are winners of major awards and honours, although no attempt has been made to list precisely the ’top’ mathematicians of today. Although these ninety-two mathematicians come from many different countries and a wide variety of backgrounds, all but one of them is based in one of four countries, the United States, Britain, France or Germany. In fact over 80% are based in the United States. It is clear that the choice has been somewhat biased by ease of obtaining the portraits, yet clearly a very high proportion of the ’top’ mathematicians are based in the United States. Let me ask the reader to ponder: does this matter? This large-format, coffee-table book, is a joy to dip into. It will certainly help everyone who reads it glimpse into the world of mathematics and begin to understand the passion that drives its creators. Edmund Robertson
Logicomix: An Epic Search for Truth by A. Doxiadis, C.H. Papadimitriou, A. Papadatos, A. Di Donna, Bloomsbury, 2009, 352 pp, £16.99, ISBN 978-0-747-59720-9. Logicomix is a full-colour ’graphic novel’ on the life of Bertrand Russell and the foundations of mathematics. Although 347 pages long, the format means that the book can be comfortably, and enjoyably, read in an afternoon. The story takes the form of Bertrand Russell giving a talk on the story his own life and his search for mathematical certainty. This use of self-reference is just one of a number of clever techniques the authors and artists use throughout the book. Another is that on a number of occasions the narrative is interrupted by sections where the authors and artists argue over the nature of the story, and how well it is being told. Visually the book is very pleasing, and the narrative moves along well with good coverage of the major characters (such as Whitehead, Wittgenstein, Gödel and Hilbert) and, given the genre of the book, appropriate explanations of mathematical ideas such as Hilbert’s Hotel, Russell’s paradox and Gödel’s Incompleteness Theorem. It must be noted however that this is a graphic, historical, novel. Thus it does take liberties with the truth. Into the story the authors weave meetings which never took place (such as Russell meeting Cantor), or events which Russell did not attend (such as a lecture by Gödel). The authors are explicit about this, and in a section at the end of the book entitled Logicomix and reality freely confess that they have simplified facts, invented events and deviated from history, and after giving some examples of where they had done so remark that "Historically keen readers can have fun locating many more such deviations from fact" (p. 345). I have to admit to initially being both irritated and suspicious of a book that played ’fast and loose’ with history. However I quickly accepted the book on its own terms, admitted that it was not aimed at a middle-aged academic such as myself, and got on with enjoying the broad sweep of the narrative. I consider this an excellent book. The format is eye-catching and novel, and the material is pitched at a level which will engage and enthuse a teenage reader. I hope that it will find its way into many school libraries and Christmas stockings. Mark McCartney
Symmetry in Chaos (2nd edition) by Michael Field and Martin Golubitsky, SIAM, 2009, 199 pp, $59.00, ISBN 978-0-898716-72-6. This beautiful book is an updated version of the original 1992 publication. In addition to a general enhancement of quality and colouring in the computer-generated images this edition includes new material on patterns on average and some additional diagrams explaining the mathematics, while the original appendix on now-outdated Basic programs has been removed. The book shows three types of pattern generated by suitably chosen dynamical systems in the plane with symmetry: the icons arising from polygonal symmetry, the quilts with planar lattice symmetry and the fractals obtained from symmetric iterated function systems. The strange forms of attractors that can arise in planar dynamical systems have been studied since the 1970s when computing power became readily available, and polygonal and planar symmetries have a long mathematical pedigree – but when the two are combined, together with careful colour-coding of orbit-densities, the results are more than the sum of the parts and quite spectacular. The images chosen for the book have been arrived at after considerable fine-tuning of the parameters and colours, and are chosen for their aesthetic qualities, in many cases bearing startling resemblances to natural objects such as flowers or snowflakes or to human artefacts like floor tiles or stained glass windows. This juxtaposition of visual art and mathematical images is purposeful, vividly making the point that the mathematical processes can themselves be powerful artistic tools. Not only have some of these images been adopted by others for decorative or symbolic use (for example the IMA, Minneapolis, uses one for its logo) but the authors have contributed to multidisciplinary conferences such as Bridges (see Newsletter 350) and Michael Field has taught courses on symmetry to art students. The authors aim to make the ideas in this book accessible to as wide an audience as possible, and therefore in the text they introduce the very basic concepts of symmetry and iteration as well as planar geometry using complex numbers. It is difficult to imagine that a reader quite unfamiliar with mathematics would be able to follow the exposition all the way through to contraction maps on a space of subsets of the plane (with symmetry thrown in), but the effort to provide a friendly introduction to these topics, motivated by the fascination of the pictures, is laudable if not heroic. For a mathematics student interested in the techniques of symmetric chaos this exposition of the key ideas would be very helpful. Whether you understand the mathematics or not, however, the pictures are wonderful and repay much contemplation: an excellent Christmas present for the more discerning relative. David Chillingworth
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