BOOK REVIEW

# The Shape of the Cosmos

*How the Universe Got Its Spots: Diary of a Finite Time in a Finite Space*. Janna Levin. x + 208 pp. Princeton University Press, 2002. $22.95.

Many of us enjoy the night sky. We like to stay outdoors and call the planets, stars and constellations by their names. After half an hour in the darkness, new stars pop up to our eyes. These bright dots multiply by thousands with the aid of even a small pair of binoculars, and looked at through a professional telescope, each tiny patch of night sky unfolds in a myriad of marvelous galaxies. Will the stars continue forever? Some may say, Of course! But then, the universe, is it infinite? At that point, we might like to stop and think twice. . . .

No infinity has ever been observed around us. Is it that nature abhors infinity?

Most cosmologists live comfortably with the idea that everything, including of course space and time, came from a "Big Bang." Could that singular event have created an infinite universe from just nothing?

Janna Levin thinks that it could not have, and in this lovely, utterly original book—which begins as a series of letters written (and never sent) to her mother and mutates into a running diary of her life—she takes us on a cosmological tour of her "small universe." She advances the idea that our universe is edgeless but finite—a huge, but not unending, cosmos.

*How the Universe Got Its Spots* is about cosmic topology, a subject that focuses on questions about the shape, volume and connectedness of the universe. This novel field of cosmological research, largely overlooked by mainstream Big Bang followers for many years, is rapidly gaining momentum.

After a delightful promenade through the topics of gravitation and infinities, Levin reminds us that general relativity is a theory of geometry. However, as such, it is incomplete: General relativity tells us how space curves locally, but it is not able to distinguish between geometries with different global properties. Those belong to the realm of topology, as does the study of the connectedness of three-dimensional space.

Topology is the branch of mathematics that describes properties of spaces that remain unchanged under smooth deformations (which allow no surface tearing or hole punching). Consider two-dimensional surfaces. You can squeeze a sphere into a dumbbell and the topology is preserved (just concentrate on the exterior "skin" of the object, not on its volume). A sphere and a dumbbell have no holes; thus both are called *simply connected spaces.* If we allow for holes, we change the topology. Both a doughnut and a coffee mug with a handle have just one hole. Thus they share the same topology: You could take a doughnut made of clay and smoothly deform it into a mug. But tear the surface of the clay to make a hole in the bottom of the mug, and you change the topology from that of a doughnut to that of an eyeglass frame.

Levin reflects on how the axioms of topology apply to three-dimensional spaces such as our universe. Could there be "holes" and "handles" in space?

Einstein himself was prone to disregard non-simply-connected hyperspaces—mainly on an aesthetic basis, rather than for scientific reasons—and he had a strong prejudice in favor of a finite universe. However, finiteness and connectedness are two different things.

The Russian-born mathematician Alexander Friedmann, one of the fathers of the Big Bang theory, was among the first to call attention to the fact that Einstein's equations don't demand that the universe have any particular topology. (By a happy coincidence, I began reading Levin's book while traveling to Brazil to attend the Fifth Alexander Friedmann International Seminar on Gravitation and Cosmology.) Friedmann's 1923 book *The World as Space and Time* was the first popular account of this idea. The finiteness or otherwise of our universe, he pointed out, does not follow from the equations Einstein used to describe it. A simple example helps show this. Einstein's equations describe the world we see around us. But in the same way that an ant traversing a gigantic, but finite, cylinder might have the impression that it is walking on a flat surface that is infinite, we don't have the perspective to tell whether our universe is finite or infinite, nor can we discern its shape. Thus Friedmann concludes that many topological spaces could equally well describe the very same solution of Einstein's equations.

But now, how to distinguish the actual form of our universe? In the case of the coffee mug we could see the shape of its two-dimensional surface from our vantage point in three dimensions. But we have no fourth dimension from which to "look down" and see what form the three-dimensional space we live in has.

Levin convincingly demonstrates that if our universe did actually have holes and handles, we could detect their peculiar footprints in the all-pervading radiation field that is the afterglow of the Big Bang. The cosmic microwave background radiation would reflect those features in a peculiar pattern of hot and cold spots, which would tell us the actual global shape of our universe.

Mixing lucid arguments with anecdotes and personal experiences, Levin makes it easy to understand seemingly complicated subjects such as transfinite arithmetic, naked singularities and compact spaces. She speaks of cosmic cacophony, Giacometti sculptures and intergalactic origami. Her descriptions allow the reader to hear the shape of a drum and to understand the struggles of cosmologists to see the shape of the universe.

In sum, this intimate account of the life and thought of a physicist is one of the nicest scientific books I have ever read—personal and honest, clear and informative, entertaining and difficult to put down. A marvelous diary that makes the reader long to meet the author . . . and ask her to tell the rest of the story.*-Alejandro Gangui, Physics, University of Buenos Aires, and Institute for Astronomy and Space Physics, Argentina
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