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.

• Local topology of ordinary space: Topology refers to the characteristics of a surface or space which do not change when the shape is distorted. Remember the rubber Flatland deformed by the Weeks, p. 267) All 3-manifolds are both homogeneous and isotropic.
• Finite: Although the universe is finite, it is not bounded--that is, it doesn't have edges. You cannot travel to the end of the universe and fall into nothingness. It may at first seem strange to speak of a space which is both unbounded and finite, but we shall see that many such shapes exist. Consider a spherical Flatland. A Flatlander can travel forever and never fall off the edge of the universe, even though this Flatland has a finite area. (2) In the next section, we will encounter analogous shapes for our space.

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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