The Roots of Complex Problems

BOOKS & ARTS NATURE|Vol 446|1 March 2007 The roots of complex problems Fearless Symmetry: Exposing the Hidden solving some of the deepest problems in Patterns of Numbers mathematics. A notable example is the by Avner Ash & Robert Gross result, obtained by Niels Henrik Abel Princeton University Press: 2006. 302 pp. and extended by Évariste Galois, that B. KALANTARI $24.95, £15.95 there is no formula for solving general quintic equations, or at least no formula Timothy Gowers that involves just algebraic operations If you ask somebody who knows a little mathe- and taking roots. A much more recent matics what is meant by the symbol i, they will example is Andrew Wiles’ solution of probably tell you, correctly, that it stands for Fermat’s Last Theorem. the square root of −1. But there is a subtlety to Fearless Symmetry started life as the question that is easy to overlook. Suppose an expository paper intended to that you pedantically point out that there are help mathematicians in other areas two square roots of −1. The response is likely to understand Wiles’ remarkable to be that the other square root is −i. But now achievement. It then grew into a comes a much harder question: which square book, and to make it more acces- root is i and which one is −i? sible the authors added a lot of The more one thinks about this question, background material. The result is the more one realizes that it does not have an that to begin with there are plenty answer. In fact, the question doesn’t really make of gentle sentences such as: “We sense, and mathematicians even have a way of start our consideration of groups proving that it doesn’t make sense. The rough by thinking about a beautiful per- idea of the proof is this. If z is any complex fect sphere, one foot in radius, number, written in its usual form a + ib, where made of pure marble.” By the a and b are real numbers, then we define the end these are accompanied by complex conjugate of z to be a − ib, and denote sentences such as: “By the Modu- this number by z៮. It can then be proved that, for larity Conjecture, there is a cuspidal normal- any two complex numbers z and w, z៮៮៮ +៮៮៮ w = z៮ + w៮ Polynomiographs such as this one, Acrobats, are ized newform f of level N and weight 2 such and zw៮៮៮ = z៮ w៮. In mathematical terminology, the created using mathematical formulae. that for all primes w that are not factors of N, function that takes each complex number to its aw ( f ) = aw (E), and hence these pairs of inte- conjugate is an automorphism, because it ‘pre- square of 1 is itself, but the square of −1 is gers are congruent modulo p.” In between, the serves’ the basic arithmetical operations. not itself. level of sophistication rises steadily. A typical Because of this automorphism, there is no Automorphisms such as this are the ‘fear- reader, therefore, will find that the book starts true mathematical sentence about i that is not less symmetries’ of the title of Avner Ash and by covering familiar ground, then becomes equally true when all occurrences of i (both Robert Gross’s book. They can be thought of as interesting and informative, and finally implicit and explicit) are replaced by −i. This symmetries because they are transformations becomes too difficult to understand. Where is the sense in which i and −i are indistin- of a mathematical object (which happens to be these transitions take place will vary from guishable and is the reason that one cannot algebraic rather than geometrical) that leave reader to reader: I learned a lot from about the answer the question: “Which square root of its important properties unchanged. It turns middle third of the book and not much from −1 is i?” By contrast, it is possible to distin- out that understanding these symmetries the outer thirds. But that was enough to make guish between 1 and −1, for example, as the in more complicated situations is the key to it worth reading, and perhaps one day I will be ready to have another go at the later chapters. One small disappointment was a section NEW IN PAPERBACK promisingly entitled: “Digression: What is so The Evolution–Creation In brief, a vigorous turn animal cognition and the great about elliptic curves?” Anybody who has Struggle towards nuclear power will continuity between human followed the story of the proof of Fermat’s Last by Michael Ruse (Harvard be necessary to prevent the and animal minds.” Juliette Theorem will have heard that elliptic curves are University Press, £10.95, catastrophic climatic changes Clutton-Brock, Nature 434, very important, but it is not at all obvious from €14.40, $16.95) caused by an increase in 958–959 (2005). the definition here why they should be. Eager “In this book, Ruse aims not atmospheric carbon dioxide.” for a better understanding, I turned to that to attack but to understand. Tyler Volk, Nature 440, Terrors of the Table: The section only to find that the answer is that For that he wisely turns to 869–870 (2006). Curious History of Nutrition elliptic curves are incredibly useful to number history — specifically to by Walter Gratzer (Oxford theorists. There are less question-begging the history of evolutionary Thinking With Animals: University Press, £9.99, answers later in the book, but by then the going theory.” John Hedley Brooke, New Perspectives on $16.95) is rather tough. Nature 437, 815–816 (2005). Anthropomorphism “[Gratzer’s] purpose is ‘to But that was just a digression. In general, edited by Lorraine Daston astonish, to instruct and, most the authors are to be admired for taking a very The Revenge of Gaia & Gregg Mitman (Columbia especially, to entertain’. And difficult topic and making it, if not fully acces- by James Lovelock University Press, $25, £16) what could possibly be more sible, then certainly more accessible than it ■ (Penguin, £8.99) “An unusual book that will entertaining than the history was before. Timothy Gowers is at the Centre for “James Lovelock… offers his surely join the growing of nutrition?” Marion Nestle, Mathematical Sciences, University of Cambridge, take on the future of energy. literature on consciousness, Nature 438, 425–426 (2005). Wilberforce Road, Cambridge CB3 0WB, UK. 26.

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