#527 How Can Space Be Flat but Finite?

Dr. craig’s response

This is a great question, Nicolás, and its answer intriguing. The question is whether space (or the universe) can be flat and yet finite. If space were like a Euclidean plane and yet finite, then you could come to the edge of space and, inexplicably, not be able to go any further, which seems bizarre. (You couldn’t fall off the edge, since there is nothing beyond the edge.) So some people infer that if space is flat, it must be infinite.

Such a conclusion is hasty. It ignores the topology of space. (Topology is the study of the deformations you can make of space without tearing it.) Mathematicians realize that whether a flat, boundless space is finite or infinite depends on the topology of space. For example, suppose we take our flat, finite space above and roll it up in the shape of a tube. Now it no longer has an edge in one direction: if you go around the tube, you just come back to your starting point without ever encountering a boundary or edge. You might think that the space is no longer flat. But you would be mistaken. Geometrically speaking, the tube has the same flat geometry as the plane. It just has a different topology.

Ah, you might say, but there is still an edge in the other direction! The tube has two ends at which it terminates. Right; so suppose we bend the tube around and join the two ends together like a doughnut. Now we have a torus-shaped space. Such a space is finite yet boundless: whichever direction you go, you’ll never encounter a boundary or edge. Yet, paradoxically it seems, the geometry of such a space is flat.

There are lots of such topological maneuvers that mathematicians can make. So neither the philosophers nor the physicists you mention are mistaken: the only person who is mistaken is he who infers that because the universe is flat, it is spatially infinite.