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In The Shape of Inner Space, Shing-Tung Yau and Steve Nadis fight string theory鈥檚 corner by explaining multidimensional geometry

HERE鈥橲 a sentence you don鈥檛 read every day: 鈥溾�楳y pleasure,鈥� she said, smiling coquettishly and curling up into a Calabi-Yau shape.鈥�

It鈥檚 safe to say that few of The New Yorker鈥榮 readers would have known what Woody Allen was on about when he wrote that in July 2003. Allen probably didn鈥檛 know either. In case you鈥檙e wondering, the geometric form known as a Calabi-Yau space (see picture) exists in multiple dimensions and is the pedestal on which string theory has been built. And in case you want to know more than that, one of the inventors (discoverers?) of the Calabi-Yau space, the geometer Shing-Tung Yau, has written a book that explains it all in exquisite detail.

The Shape of Inner Space is a hymn to geometry. Without geometry, Yau points out, we cannot account for the forces of nature. Einstein鈥檚 general theory of relativity is, essentially, nothing but geometry. Yet geometry is the poor relation of modern science.

Yau is aiming to put that right. 鈥淚 would go so far as to say that geometry not only deserves a place at the table alongside physics and cosmology, but in many ways it is the table,鈥� he writes. A measure of his obsession is his admission that discovering the Calabi-Yau space gave him the same feeling he experienced when he first laid eyes on his wife.

In many ways, this book is a hard sell. After all, string theory has already been 鈥渄one鈥� in books by Brian Greene, Yau鈥檚 prot茅g茅. Then the subject was written off as a waste of physicists鈥� time in Peter Woit鈥檚 Not Even Wrong and Lee Smolin鈥檚 The Trouble With Physics. No wonder that when Yau told his colleagues he wanted to write a book about the mathematics behind string theory, they told him it was like drawing attention to having worked on the blueprints for the Titanic.

Yau pressed on anyway, and we should be glad he did. It is a testimony to his careful prose (and no doubt to the skills of co-author Steve Nadis) that this book so compellingly captures the essence of what pushes string theorists forward in the face of formidable obstacles. It gives us a rare glimpse into a world as alien as the moons of Jupiter, and just as fascinating.

The Shape of Inner Space is not always an easy read. Neither are the explanations always perfect: are we really supposed to cope with sentences like, 鈥淎 compact Kahler manifold with a vanishing first Chern class will admit a metric that is Ricci flat鈥�? Nevertheless, Yau and Nadis have produced a strangely mesmerising account of geometry鈥檚 role in the universe. And if you ever want to curl up into a multidimensional shape, this book surely offers the best guide there is.