Book list
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Mlodinow's spry account of geometry stresses the stature of the greatest math book of all time, Euclid's Elements. Although the three-dimensional space he described in it doesn't truly represent the shape of nature, Euclid compensated by codifying an attitude essential to rational thinking--to wit, distrust intuition and therefore don't accept unjustified assumptions. Unfortunately, Euclid himself made one unjustified assumption, the parallel postulate, which worked fine in the flat-Earth mathematical world that existed until Carl Friedrich Gauss dismantled it in the nineteenth century. Gauss invented a new geometry of curved or hyperbolic space, a feat that Mlodinow honors in such amusing asides as his remark on Kant's defense of Euclid: "Gauss did not dismiss Kant's work out of hand. He read it, then dismissed it." Such japes lighten and popularize Mlodinow's approach to the further demolition of Euclid by Gauss' student Georg Riemann, whose work critically contributed to the theory of general relativity. Mlodinow's lively exposition concludes with string theorists' claim that geometry possesses no fewer than 11 dimensions. Gilbert Taylor

Kirkus
Copyright © Kirkus Reviews, used with permission.

Halfway through this articulate and droll history of math and physics, you wonder: Who is this guy with the unpronounceable name you want to recommend to all your friends? And so you discover that Mlodinow has gone from Cal-tech professor to Star Trek scriptwriter to developmental VP for an educational software firm. That would account for his knack of presenting the evolution of mathematical thought with an insider’s insight, quirky humor, and titillating facts about the great and near-great. Describing the work of Boethius (who abridged Euclid’s Elements ), Mlodinow writes, “his translations might be entitled ‘Euclid for Dummies’ or sold in TV ads imploring, call 1-800-NOPROOFS.” On the foundations of quantum mechanics he quotes Erwin Schrodinger (“It has never happened that a woman has slept with me and did not wish, as a consequence, to live with me all her life”). These little fillips engage the reader as the author chronicles how our views of the universe have been informed by concepts of space. Much of the text (and human history) does indeed reflect the view from Euclid’s window—in which space is flat, filled with points, lines, and figures (like triangles) whose angles add up to 180 degrees. That works only as long as you accept as an axiom that through a point outside a line one and only one line can be drawn parallel to a given line. Much ink was poured unsuccessfully over the years to derive this “axiom” from the other Euclidean axioms as a theorem. Then during the 19th and 20th centuries, in the works of Gauss, Riemann, and Lobachevsky, the notion of non-Euclidean curved space took root and revolutionized physics. Einstein demonstrated the curvature of space in general relativity, and he needed four-dimensional space-time to develop special relativity. Mlodinow concludes with the latest wrinkles on the geometry of space, from the early formulations of string theory and now M-theory to the abstruse work of mathematicians like John Schwarz and Ed Witten. Splendid exposition, accessible to the mathematically challenged as well as the mathematically inclined.

Library Journal
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"Euclid's work [is] a work of beauty whose impact rivaled that of the Bible, whose ideas were as radical as those of Marx and Engels. For with his book Elements Euclid opened a window through which the nature of our universe has been revealed." Strong words, but Mlodinow backs them up with this surprisingly exciting history of how mathematicians and physicists discovered geometric space beyond Euclid's three dimensions. Each advance in mathematical geometry has been followed by unexpected discoveries proving that the strange mathematics actually describe measurable physical properties. Mlodinow, a physicist and a former faculty member at the California Institute of Technology, has also written TV screenplays for Star Trek: The Next Generation and other shows. He has a good sense of popular science writing, and he personalizes geometric abstractions by endowing them with the personalities of his adolescent sons, Alexei and Nicholai. Euclid, Descartes, Gauss, Einstein, and Witten are among the mathematicians profiled, and each of them also emerges with a distinct personality based on the style of their writing and historical anecdotes. This engaging history does an excellent job of explaining the importance of the study of geometry without making the reader learn any geometry. For all math and science collections. Amy Brunvand, Univ. of Utah Lib., Salt Lake City (c) Copyright 2010. Library Journals LLC, a wholly owned subsidiary of Media Source, Inc. No redistribution permitted.

Publishers Weekly
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Mlodinow's background in physics and educational CD-ROMs fails to gel in this episodic history of five "revolutions in geometry," each presented around a central figure. The first four Euclid, Descartes, Gauss and Einstein are landmarks, while the fifth, Edward Witten, should join their ranks if and when his M-theory produces its promised grand unification of all fundamental forces and particles. Mlodinow conveys a sense of excitement about geometry's importance in human thought, but sloppiness and distracting patter combine with slipshod presentation to bestow a feel for, rather than a grasp of, the subject. Certain misses are peripheral but annoying nonetheless confusing Keats with Blake, repeating a discredited account of Georg Cantor's depression, etc. Some of them, however, undermine the heart of the book's argument. Strictly speaking, Descartes, Einstein and Witten didn't produce revolutions in geometry but rather in how it's related to other subjects, while Gauss arguably produced two revolutions, one of which non-Euclidean geometry is featured, while the other differential geometry though equally necessary for Einstein's subsequent breakthrough, is barely developed. Mlodinow completely ignores another revolution in geometry, the development of topology, despite its crucial role in Witten's work. Occasionally Mlodinow delivers succinct explanations that convey key insights in easily graspable form, but far more often he tells jokes and avoids the issue, giving the false, probably unintentional impression that the subject itself is dull or inaccessible. More substance and less speculation about the Greeks could have laid the foundations for an equally spirited but far more informative book. 11 figures, two not seen by PW. (Apr.) Forecast: The Free Press may be looking for a math popularizer in the mold of Amir Aczel, but Mlodinow falls short. Don't look for big sales here. (c) Copyright PWxyz, LLC. All rights reserved