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Is space finite or infinite? This question has perplexed scientists and driven astronomical observations for generations.

When found to be incompatible with observations and consequent new understanding of physical law, early models of a finite cosmos were replaced with that of a static, infinite universe. Such a model dominated scientific thinking until the twentieth century, when Milton Humason and Edwin Hubble discovered that extragalactic objects were receding away from us in every direction at speeds proportional to their distance. After this discovery, a radical notion began to take hold, in which the Universe was better explained by the non-static, expanding solutions of general relativity found by Alexander Friedman, Georges Lemaître, Howard Percy Robertson and Arthur Geoffrey Walker. The discovery of the cosmic microwave background radiation by Arno Penzias and Robert Wilson in 1965 provided further evidence that our Universe had expanded from an initial hot dense state. Recent observations of the cosmic microwave background by the Wilkinson Microwave Anisotropy Probe have culminated in the choice of a certain expanding solution, called the concordance model, as the one describing our Universe. Although its parameters are set by astronomical observations, the concordance model incorporates certain assumptions. In particular, although it is manifestly non-static, the Universe is still taken to be spatially infinite, an assumption tied to the hypothesis that it underwent an extended period of exponential expansion, called inflation. But is this assumption supported by observation?

In The Wraparound Universe, Jean-Pierre Luminet presents an alternative view: the Universe, although appearing spatially infinite, is actually spatially

finite. This book, the English translation of L’Universe Chiffonée, first published in 2001, is not a survey of the successes of the concordance model and other achievements of contemporary cosmology, but rather is an elegant monograph on an intriguing and potentially observable variation — that the Universe is topologically non-trivial. Seasoned with historical perspective and personal tales, the book clearly communicates key ideas, discusses tests of their viability through astronomical observations, and places them in the broader context of philosophical questions and artistic vision in European culture.

But how could something finite appear to be infinite? The answer to this question lies in the difference between what is and what is observed; it is possible to see an infinite number of images that give the appearance of an infinite space from one that is finite. An everyday example comes from mirrors. A mirrored wall on one side of a room appears to you not as a wall, but rather as a reflected copy of the room. Suppose the room contains an object, say a vase of roses. On entering the room, however, you perceive not one, but two vases of red roses: one in the room and another in its reflected copy. More precisely, you see two distinct images: one from light that travels directly from the roses to you and one from light that travels first to the mirror, then, after reflection, to you. These images are not exactly the same; one is reversed. It also appears to be farther away as its light has travelled a longer path. The image is also that of slightly younger roses, given that light has a constant finite speed. As both images are perceived simultaneously, the light forming the reflected image must have left the roses earlier than that forming the direct image. This difference is, of course, not detectable in ordinary rooms. The speed of light is so great that the roses in the reflected image are only infinitesimally younger than those in the direct one. However, if the room were big enough such that it took days for light to cross it, this difference would be striking: a direct image of wilted roses and a reflected image of fresh flowers would be seen simultaneously.

A room with one mirrored wall produces only one extra image. However, if the room has two or more mirrored walls, you would see not one, but an infinity of images extending in different directions. Thus a room covered in mirrors, although in

itself finite, appears infinite. Furthermore, the reflected images are younger than the direct one in proportion to the distances travelled by the multiply reflected paths. Consequently, it becomes surprisingly difficult to distinguish images that are, in fact, just reflections and not distinct objects.

Topology, although further removed from everyday experience, is a more physical way to achieve an infinite set of images from a finite space in the context of our Universe. Mirrored walls produce extra images because they allow light to travel from an object to you on two or more different paths. Therefore, if the room itself is connected in a way that it allows more than one path, you will see more than one image, even if no mirrors are present. For example, imagine first replacing the mirrored walls with transparent ones, then connecting them in opposite pairs, much like a higher-dimensional version of taping two opposite edges of a rectangular sheet together to form a cylinder. Such a room has non-trivial topology. In it, light could still travel directly from an object to you. But, there are now additional paths, for example the one that passes through the wall on your left, which emerges from that on your right on its journey to you. This path could be described as one that wraps around the room; hence this room is a wraparound universe. In fact, there will be an infinite number of such paths, each wrapping around the room a different number of times and, consequently, producing an infinite number of images. Again, as for the mirrored room, the images seen along such wrapping paths will appear farther away and younger (although unlike the mirrored case, the images are no longer sometimes reversed). Thus, as a wraparound universe becomes larger, it becomes more difficult to detect its finite nature. There are a myriad of ways to build wraparound universes of different topologies. Heuristically, they also can be formed by identification of transparent walls, but the rooms are no longer rectangular. As a consequence, the pattern of images produced also changes, further complicating their detection.

The book is presented in two sections. The first is the main narrative in which Luminet clearly and elegantly explains the topology and properties of wraparound universes. He includes numerous figures and diagrams that greatly help to communicate the essential concepts. He

Mirror mirror on the wallThe Wraparound Universe

By Jean-Pierre Luminet

AK PEtERs: 2008. 400 PP. $39

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ends with a discussion of known methods for observationally detecting a wraparound universe: matching identifiable sources in different directions on the celestial sphere, cosmic crystallography and matched circles in the sky. The second part consists of short chapters with extra details and side remarks, including an afterword chapter added to the 2001 edition, in which Luminet discusses the constraints placed by the Wilkinson Microwave Anisotropy Probe on the topology and size of wraparound universes and passionately presents his arguments for their continued viability on larger distance scales.

Like a wraparound universe, there are multiple paths for the reader to follow through this book. Passages from the main narrative are linked to related passages in the second section by arrows in the

margins. These, in turn, link both to other passages in the second section and back to the main narrative. A linear progression through the two sections provides an accessible and enjoyable introduction to wraparound universes for general readers. However, those already familiar with aspects of cosmology and/or topology may prefer to skip from one chapter to another, either following an indicated path or not, as desired.

Yet the question remains, is our Universe spatially finite or infinite? As pointed out by Luminet, observational data conclusively rule out small wraparound universes, but cannot rule out certain larger ones, such as the Poincaré dodecahedral universe. In fact, a wraparound universe might explain features of the cosmic microwave background radiation observed at large

angles. Fortunately, the European Space Agency’s cosmic microwave observatory, Planck, launched on 3 May 2009, will soon provide new data of higher sensitivity and resolution than previous observations. Thus, we may soon learn whether or not wraparound universes are observationally ruled out. Nonetheless, this book will remain an entertaining and informative discussion of an important aspect of contemporary cosmology from an expert in the field. ❐

revieWed by KrisTin schleich

Kristin Schleich is in the Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, V6T 1Z1, Canada. e-mail: schleich@physics.ubc.ca

dance yourself dizzyI almost don’t want to say anything about it. David Bintley’s new work for Birmingham Royal Ballet has such impact that I wouldn’t want to spoil the surprise.

This amazing creation is inspired by the work of — who else? — Albert Einstein. Specifically, it was David Bodanis’ book E=mc2:

A Biography of the World’s Most Famous Equation that was the source of Bintley’s desire to develop “movement patterns based on gravity, apparent chaos, extremes of speed and slowness”. But these mere words don’t do it justice — it is a truly exhilarating achievement, and extremely moving, with the beauty of the dance matched exquisitely to Matthew Hindson’s engrossing score.

The ballet indeed shares its name, E=mc2, with that world-famous equation. The cosmic frenzy of the opening movement climaxes and gives way to a serene, confined second, more suggestive of atomic order. The final exuberant section is performed in front of a bank of spotlights, and as the array of dancers spin and tilt like tops, they create a terrific optical effect. It’s brilliant and it makes you feel good to be a physicist.

But what’s haunting me is the interlude performed by a lone Japanese dancer, emerging from darkness in a white kimono and moving so gently, then suddenly to the accompaniment of a blast of sound that rumbles and crackles and hisses. E=mc2 indeed. But the delicate, wistful dance goes on.

Birmingham Royal Ballet, which is directed by Bintley, offers E=mc2 as part of a treble bill called Quantum Leaps, sandwiched between Stanton Welch’s Powder (to Mozart’s clarinet concerto) and Garry Stewart’s The Centre and its Opposite. This last is set to an electronic soundscape devised by Huey Benjamin, and the contrast from one end of the programme to the other couldn’t be

greater: Bintley has described Stewart’s harsh, contemporary work as “probably the most extreme piece that we have ever done.” I’d aver that E=mc2 is probably one of the best. ❐

revieWed by AlisOn WriGhT

Quantum Leaps ■■ was performed at sadler’s Wells, London, on 10–11 November 2009.

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