The possibility of circumnavigating the globe isn't obvious to a surface denizen who sees just an outstretched plain or ocean, however. Such a small patch of Earth looks flat, and one could hypothesize that the surface must either have an edge or extend indefinitely.

Moreover, even from a few thousand meters above the ground, it isn't readily apparent to an observer, convinced that Earth is a finite object, whether the slightly curved patch below represents a portion of the surface of a sphere, a doughnut, or an irregular blob.

Scientists face similar difficulties in discerning the overall shape, or topology, of the universe -- in this case, the shape of three-dimensional space itself rather than that of a two-dimensional surface.

Mathematicians have discovered and investigated a wide variety of three-dimensional forms that could serve as models of three-dimensional physical space. Among the startling possibilities are those corresponding to a finite universe, in which a starship could blast off on a voyage of a few billion light-years in one direction and eventually return from another direction.

Astrophysicists and cosmologists are starting to pay attention to such bizarre notions. Instead of a limitless expanse of space studded with stars and dust, the universe could be finite and connected together in a complicated way.

Prompted in part by the enticing possibility of detecting the signature of the universe’s topology in detailed maps of temperature fluctuations throughout space, a group of scientists and mathematicians met last October at Case Western Reserve University in Cleveland to compare notes.

“There was a real sense of excitement at the meeting,” says astrophysicist David N. Spergel of Princeton University. “We realized that, in this field, we had a lot to learn from mathematicians, and we had some interesting new questions for them. At the same time, it was exciting for mathematicians to see their results applied in this context.”

According to current theory, the universe started out in an extremely hot, incredibly dense, highly contracted state. However, the Big Bang wasn’t so much like a firecracker exploding into an existing space as the rapid expansion of space itself.
That expansion continued as matter cooled and condensed into dust, stars, and galaxies, drawn together by the force of gravity. As space kept expanding, those galaxies moved farther apart.

In the standard view, the ultimate fate of the universe depends on the density of matter within it. If the mass density were greater than a value called the critical density, gravity would be strong enough to reverse the expansion, eventually causing the universe to collapse into what could be called the Big Crunch.

In effect, such a universe would curve back on itself to form a closed space of finite volume. That space would have a positive curvature. A starship traveling in a straight line would eventually return to its point of origin.

If the mass density were precisely equal to the critical value, the universe would go on expanding forever, though its rate of expansion would get closer and closer to zero. Its geometry would be called Euclidean, or flat, like the familiar geometry of lines and angles on a sheet of paper.

If the mass density were less than the critical value, the universe would also keep expanding forever, but at a constant rate. Such a space would be negatively curved and have a so-called hyperbolic geometry.

At present, observational data of various types suggest that the universe does not contain nearly enough matter to make it closed or even flat.

In contemplating a hyperbolic universe, cosmologists had generally assumed that such a space would have to be infinite rather than finite.

Mathematician William P. Thurston of the University of California, Davis and others, however, have demonstrated that three-dimensional hyperbolic space can take on a multitude of forms that are finite in extent. Any one of these forms could serve as a model of the universe’s basic shape.

The underlying assumption is that physical space can be described in terms of a mathematical form known as a three-dimensional manifold. “There are lots of different possible manifolds that could represent space,” notes freelance geometer Jeffrey R. Weeks of Canton, N.Y..

The surface of a ball is an example of a two-dimensional manifold. A small region of the surface looks nearly flat—like a piece of a two-dimensional plane. Earth’s surface is a two-dimensional manifold because it looks essentially flat until one gets far enough away to see that it curves into a sphere. The surface of a doughnut, or torus, is also a two-dimensional manifold.

Another way to think about the topology of a two-dimensional manifold is in terms of gluing together the sides of a rubbery rectangle. For example, a torus is simply a rectangle with opposite sides glued together. The first gluing creates a tube, and the second gluing connects the two ends of the tube to form a ring.

The same idea can be generalized to describe a three-dimensional manifold. For instance, one can try to imagine gluing together the opposite faces of a flexible cube to produce a hypertorus—the three-dimensional equivalent of a doughnut surface.

Example of a polyhedron used to represent a negatively curved, or hyperbolic, three-dimensional manifold. The polyhedron’s faces are color-coded to indicate which pairs are linked to create a multiply connected, finite space.  Thurston and Weeks were key figures in the development of a comprehensive catalog of closed three-dimensional manifolds, most of which appear to have a hyperbolic geometric structure. These weird shapes can be understood in terms of three-dimensional polyhedrons whose faces are glued together to create finite, multiply connected spaces.

The cosmological consequences are startling. If such a topology described the universe, what astronomers might think is a distant galaxy could actually be the Milky Way—seen at a much younger age because the light has taken billions of years to travel around the universe.

References:

Cornish, N.J., D.N. Spergel, and G.D. Starkman. 1998. Measuring the topology of the universe. Proceedings of the National Academy of Sciences 95(Jan. 6):82.

Levin, J., E. Scannapieco, and J. Silk. Preprint. Is the universe infinite or is it just really big?

Levin, J.J., et al. Preprint. Flat spots: Topological signatures of an open universe in COBE sky maps.