Shapes of Space: The Hypersphere

# The Hypersphere

So, what are the possible shapes of space? In the last section we discussed a Flatland in the shape of a sphere. A Flatlander could travel around the Flatuniverse in any direction and come back to the starting place.

A hypersphere is the four-dimensional analog of a sphere. Although a sphere exists in 3-space, its surface is two-dimensional. Similarly, a hypersphere has a three-dimensional surface which curves into 4-space. Our universe could be the hypersurface of a hypersphere. If it is, and if you had some extra time on your hands, you could fly a spaceship in a fixed direction and eventually arrive back at earth. According to one estimate of the circumference of the universe, if you could travel at the speed of light, this trip would only take around eighty billion years!

There are several interesting phenomena which occur in a hyperspherical universe. Pretend the universe is very small and only contains you, fortunately wearing a well-equipped spacesuit. It's peaceful, isn't it? But it's also dark and lonely. You light a flare and are surprised to see a figure in the distance. It's another person, floating upside down and also holding a flare! You drop your flare in surprise (it floats, since you are in outer space) and rocket-propel yourself over to meet the other figure, who, to your chagrin, has also let go of its flare and is fleeing from you. You increase your speed, but the figure manages to keep the distance between you the same. You arrive at the spot where the other person's light was abandoned, and you try to pick it up, but your hand goes right through it. The flare is intangible, like a ghost or a hologram. What's going on?

Since Flatland is easier than 3-space to visualize and draw, we will illustrate the same event taking place in a small spherical Flatuniverse.

Imagine that you are the Flatlander in the illustration. Light rays on a sphere travel in the straightest possible path, along the great circles (3) of the sphere. You look straight ahead of you, and light from your back reaches your eye. You can see your own back, and in fact your whole body, upside down and reflected. The image of yourself appears on the opposite side of the sphere (of course, you do not notice that the space is spherical) because that is where all the light rays from your body cross. When you move, your ghostly twin moves away from you, always remaining opposite you. The flare also has a ghost twin on the other side of the sphere, and this is the intangible flare you encounter.

Another strange event can occur in the small hypersurface. Rucker, who also wrote about the previous scenario, describes what happens next:

"After returning to your flare you decide that it's getting too cramped in your spacesuit. You take out a tremendous rubber balloon, crawl inside it, and begin filling it up with air from your tank. You've brought the flare along, so the balloon is all lit up inside. It's nice to be inside the balloon and not have to see the weird ghost images of the flare and yourself. You take off your suit and loll against the balloon's gently curving wall. The tank next to you is hissing away, and the balloon is growing.

"Suddenly, something is wrong. The balloon wall goes flat and starts curving away from you. You are outside the balloon! The tank is still next to you, but the air seems thinner. The balloon rapidly shrinks away from you, down to beachball size, and collapses. What happened?" (p. 100-101)

It's very difficult to understand what is going on, but a picture of the same event one dimension lower should clarify the situation.

Next: The Three-Torus

Send comments or questions to Lisa Eckstein