What is the shape of space? Before we can answer this question, we must decide what it means to ask it. "Space" here refers to the universe as a whole, but specifically to the vast vacuum lying between the stars and planets and other chunks of matter. We generally think of this space as empty, but the behavior of light suggests that space is in fact filled with a substance that nineteenth century scientists called "aether". Instead of trying to imagine what aether is made of or what its properties are, just think of aether as the "fabric of space".
As light passes near a large body, such as a planet, the path of the light bends. The Earth's gravity keeps our Moon in orbit, while the Moon's gravity affects the Earth's tides. If space were truly empty, there would be no way to explain the distortion of light rays or the influences of gravitating bodies on one another--how would gravitational attraction be transmitted through empty space? Postulating the existence of aether gives gravitational force, as well as magnetic and electrical forces, something to travel through.
To understand how matter affects the path of light, imagine a large sheet of rubber with a helium balloon trapped beneath it. As the balloon rises, it creates a large bulge in the surface of the rubber.
Now picture yourself as a Flatlander living in the surface of the sheet. (1) If you want to get from a point at the bottom of the bump to a point on the opposite side, it is shorter to go around the bulge than to go over it. You do not perceive the bump as a hill--it is a bulge into a third dimension which you cannot detect--but as an expanse of land too wide to walk across. Similarly, light rays in Flatland curve around the bump--light does not always travel in a straight line, but its path follows the straightest possible line. Like the balloon, dense bodies in our universe distort the fabric of space into a higher dimension, the fourth dimension, causing light to travel in a curved path.
The fabric of space is distorted locally by matter, but all of space may also be curved into something other than ordinary three-dimensional Euclidean space (3-space). Thinking about possible shapes of space is an important part of the science of cosmology, and it leads to speculation about fascinating scenarios which sound like plots from science fiction novels. What if you could travel billions of light years in one direction and end up back at Earth, without ever altering your course? What if when you returned, you were a mirror image of your former self? What if everything in the entire universe appeared to be within a few feet of you? We will consider shapes of space which could produce these phenomena.
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