The Equation That Couldn’t Be Solved: How Mathematical Genius Discovered the Language of Symmetry, Mario Livio Simon & Schuster, New York, 2005. $26.95 (353 pp.). ISBN 0-7432-5820-7
Hiding in the Mirror: The Mysterious Allure of Extra Dimensions, from Plato to String Theory and Beyond, Lawrence M. Krauss Viking, New York, 2005. $24.95 (276 pp.). ISBN 0-670-03395-2
In hindsight one could judge these books by their covers. With its parchment-like background, multiple typefaces, butterfly photograph, and sprawling layout, Mario Livio’s The Equation That Couldn’t Be Solved: How Mathematical Genius Discovered the Language of Symmetry promises a richly detailed story linking life, history, and mathematics. Lawrence Krauss’s Hiding in the Mirror: The Mysterious Allure of Extra Dimensions, from Plato to String Theory and Beyond, with its geometric graphics, single visual focal point, and subtle literary allusion, hints at the more streamlined story and argument within.
Both volumes introduce sophisticated technical topics to the general public, but their aims and expository styles diverge. Livio, an astrophysicist and also author of The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number (Broadway Books, 2002), attempts to make the abstract topics of algebraic equations, symmetry, and group theory appealing and relevant to readers who have no mathematical background. His varied and detailed examples are well calculated to engage such readers and make them feel capable of following his narrative of mathematical challenges and progress. Capsule biographies of mathematicians woven throughout the story depict mathematics as a fundamentally human endeavor, while puzzles in the appendices urge active engagement.
Krauss, a professor of physics at Case Western Reserve University in Cleveland, Ohio, and also author of Atom: An Odyssey from the Big Bang to Life on Earth (Little, Brown and Co, 2001), examines theories of extra spatial dimensions, an idea with which many readers will have a superficial acquaintance. He tries to remove pervasive misconceptions, display connections to modern physics, and reveal how speculative those extradimensional theories remain after decades of study. The author’s ability to convey, with minimal detail, the essence of a scientific idea allows him to show the links among strands of thought in science, philosophy, and science fiction. He pulls the reader into contemplating the relative values of beautiful and useful theories.
In the first seven chapters of The Equation That Couldn’t Be Solved, Livio masterfully develops the broad theme of symmetry through a suite of vignettes, just as medieval tapestries or classical Chinese scrolls display an epic as a mosaic of smaller scenes. He first introduces the concept of symmetry as encountered in art, music, science, and mathematics and then begins the central story by tracing the origins of algebraic equations, from early Babylonian land records through 16th-century European mathematical competitions. The sadly short lives of Niels Henrik Abel and évariste Galois provide background for an exploration of attempts first to solve the general quintic equation and then to prove it could not be solved in closed form. Livio takes the reader carefully through the essence of Galois’s development of group theory and its consequences; it was disappointing that he did not include a similar exposition of Abel’s accomplishments.
The story closes with a whirlwind tour of symmetry’s crucial role in various topics in modern physics. Apart from a few oddities—for example, Livio asks the reader to imagine the inverse of a deformation of a piece of Play-Doh (page 164) and alludes to Dan Brown’s description of CERN in Angels and Demons (Pocket Books, 2000) as though it were serious rather than parody (page 219)—his narrative of mathematical progress through symmetry is quite satisfying.
In contrast, the final two chapters of Livio’s book are rather frustrating. Without the underpinning of a dramatic story line, the author’s style becomes a liability, making the chapters seem like a jumble of barely related facts, assertions, and surmises. In particular, the discussions of the role of symmetry in evolutionary psychology and the nature of creativity feel superficial and unconvincing.
Hiding in the Mirror smoothly recounts the history of extra dimensions as an element or even a driving factor in scientific and humanistic explorations of the nature of the universe. Much of the book is devoted to an engrossing overview of theories of gravity, particle physics, cosmology, and string theory. It also contains significant discussion of developments in science fiction, art, and spiritualism in which extra dimensions have taken a role. Krauss plays these parallel histories against one another, specifically contrasting the types of extra dimensions most prevalent in science and science fiction, and noting the common impulse by scientists and humanists to employ new spacetime geometries as a tool for opening the mind to new insights. In the closing chapters, he reminds the reader that, in the past, data have falsified seemingly plausible scientific invocations of extra dimensions. He argues that current incarnations of extra dimensions within string theory remain physically speculative, albeit mathematically beautiful, because no firm connection with our physical universe has been established.
Krauss uses humor and a broad palette of cultural references—from Passover to the X-Files—to engage the reader and carry the narrative forward. The colloquialism softens his precise physics descriptions and gives the reader a welcoming sense of chatting with a colleague rather than listening to an expert. A few of Krauss’s allusions, however, might make some readers feel like baffled outsiders. For example, on page 182, the sequence of puns about the heterotic string is undermined by his not defining heterosis or giving any hint to the lay reader about why “kinky” might be a pun in this context. Likewise, the remark on page 132 that “1 + 2 + 3 + 4 + 5 + … can be shown to equal not infinity, but rather −1/12” prompts readers to wonder whether they are being subtly mocked, which is a shame because the remark follows a very accessible account of Hilbert’s Hotel and the pitfalls of infinities.
I recommend both books to a general audience seeking a mix of entertainment and education. The Equation That Couldn’t Be Solved is perhaps best read in short chunks, leaving aside time to digest the mathematical details or solve the puzzles in the appendices. Readers of Hiding in the Mirror may profit from more sustained reading sessions to appreciate the full arc of each chapter. Each publisher should consider embellishing its book with accompanying websites that offer interactive puzzles and resources for interested readers.
