Shapes of Space: Properties of Shapes

Properties of Shapes

There are many different possible shapes of space. Study of the shape of the universe is highly theoretical, for at this time we have no way to determine its true shape, and we may perhaps never be able to do so. One of the obstacles to discovering the correct shape is the vast size of the universe. Weeks explains why this is an issue:

"In theory, if the universe is a three-torus we should be able to look out into space and see ourselves. Does the fact that astronomers have not done so mean that the real universe cannot be a three-dimensional torus? Not at all! The universe is only 10 or 20 billion years old, so if it were a very large three-dimensional torus--say 60 billion light-years across at its present stage of evolution--then no light would yet have had enough time to make a complete trip across. Another possibility is that we are in fact seeing all the way across the universe, but we just don't know it: when we look off into distant space we see things as they were billions of years ago, and billions of years ago our galaxy looked different than it does now. (This effect occurs because the light which enters a telescope today left its source billions of years ago, and has spent the intervening time travelling through intergallactic space.) In any case, we don't even know exactly what our galaxy looks like now, because we are inside it!" (p. 23-4)

In spite of the difficulties of proving any theory about the shape of space, you cannot wildly guess a shape and hope it will be considered seriously. There are some conditions which any plausible shape must meet, and these criteria rule out the theory that the universe is shaped like your Aunt Sylvia's head. Every possible shape of the universe is a three-dimensional manifold (3-manifold), a space which has the same local topology as ordinary space. Most cosmologists also believe that the universe is homogeneous, isotropic, and finite, so any proposed shape should have these properties, which are defined below.

Finite, unbounded shapes can be created by a process called "gluing". A square is a bounded surface--a Flatlander on a square has to worry about falling off the edges. But if the top and bottom of the square are glued together, and the left and right are glued together, the Flatlander may walk off the top of the square and reappear at the bottom, or walk to the left and come back to the starting point without turning around.

The gluing does not have to be a physical attachment, although it can be, as we will see in a later section. A three-dimensional object may also be glued together. You can glue the faces of a cube: top to bottom, left to right, front to back.

Next: The Hypersphere

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