'The shape of inner space'

review by Marianne Freiberger

The shape of inner space

By Shing-Tung Yau and Steve Nadis

This book tells the fascinating story of strange geometric objects that have achieved some fame outside of maths: they're called Calabi-Yau manifolds. We've looked at the story in more detail in the Plus article Hidden dimensions, but here's a synopsis: inspired by an open question in geometry, the mathematician Shing-Tung Yau goes in search of a weird multi-dimensional object he thought didn't exist, finds it, and wins the Fields medal for his efforts. A year later theoretical physicists noticed that the object he found, or rather objects for there are many, are just what they needed. String theory, an attempt at a physical "theory of everything", claims that we live in a ten-dimensional Universe. Since we can only perceive four of these dimensions (three space, one time), the other six must be hiding somewhere. As it turns out, the kind of object that can harbour the six extra dimensions and cater to other requirements of string theory is a Calabi-Yau manifold. String theory claims that every point in the 4D space we can perceive is in fact a tiny little 6D world with the structure of a Calabi-Yau manifold. It's so small, we just can't see it.

The story is told by Shing-Tung Yau himself, with the help of the science writer Steve Nadis. When Plus interviewed Yau on a recent visit to London, he was adamant that maths should be brought to the masses without dumbing down or glossing over the tricky parts. And this is just what this books sets out to achieve. While aimed at a general audience, it doesn't just tell the story of Calabi-Yau manifolds, but explores their maths in detail.

The book takes a broad approach, starting with a look at the intertwined history of geometry and physics. This sets the scene to explain the question, first asked by the mathematician Eugenio Calabi, which eventually led Yau to the famous manifolds. True to Yau's conviction, the maths is at the forefront throughout the book. Every single technical term in Calabi's conjecture is explained, and there's a chapter devoted to Yau's proof as well as the mathematical machinery developed for it. Yau and Nadis explore the manifolds' relevance to string theory, but also another interesting twist to the story: while geometry boosted string theory, string theory in turn revived a nearly forgotten area of geometry by providing an answer to a century-old question. The book wraps up by exploring how and if all of this is relevant to the real world and pondering the connections between maths, beauty and truth.

The collaboration between a mathematician and a science writer has worked wonders in this book. It's crowded with beautiful metaphors that clarify complex ideas and provide a peek into higher-dimensional worlds. Personally, I already knew a little bit about some of the maths involved, yet I had several penny-dropping moments I wish I'd had when I was first studying it.

If you're a complete novice, you'll have to muster up a little stamina to get through the book. While the authors make every effort to explain all the technicalities, some flipping back through the pages to remind yourself of what it all means might be necessary. Having said that, it is perfectly possible to skim some bits and still enjoy the rest of the story. And if you're drawn to the science fiction potential rather than the actual maths, there's a chapter on how the Universe might one day unravel with the hidden dimensions blowing up to full size.

One thing that comes through on every page of this book is the beauty of the maths and its power to shed light on the secrets of our Universe. If this is the kind of thing that fascinates you, then this is a great book to while away those dark winter evenings.

Book details:
The shape of inner space
Shing-Tung Yau and Steve Nadis
hardback — 400 pages (2010)
Basic Books
ISBN: 978-0465020232
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