Shapes of Space: The Three-Torus

The Three-Torus

In the explanation of gluing, we looked at a square with edges glued.

If this square is glued only abstractly, it is called a flat two-dimensional torus. (4) But the attachments can also be made physically in 3-space out of a sheet of rubber. This produces a donut shape called a 2-torus.

There is a three-dimensional analog of a 2-torus called (not surprisingly) a 3-torus. A 3-torus can be constructed from a cube with top and bottom faces glued, left and right glued, and front and back glued.

The faces can only be physically glued in 4-space, so we will concentrate our thoughts on the abstractly glued cube, which is easier to imagine. Both versions of the 3-torus represent the same object, however. If our universe is a 3-torus, there is only one cube of space, but we will perceive it as a pattern of cubes repeating over and over in every direction.

Again, let us imagine that the universe is very small. It contains only you and a few tiny stars that act as landmarks. You look in front of you and see your own back. You look up and see your feet. You look down and see your own head looking down. You can try to catch yourself, but you will never succeed--as soon as you move toward yourself, you move away from yourself. The chase is completely futile, because there is only one you in the universe, although there appear to be many. As you keep moving in one direction, you notice that you keep passing the same pattern of stars over and over again. It's pretty creepy!

This image from the Shape of Space video, produced by the Geometry Center, depicts a view from inside a spaceship in a tiny 3-torus universe.

Here is Albert Einstein in a similar space:

Next: The Three-Klein Bottle

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