Our Earth is, topologically, a finite world. If you go far enough in any one direction, in a “straight” line, you are actually traveling in a great circle around the globe, so you will sooner or later end up back where you started.
For most people this is a theoretical concept. It is rare that anyone has occasion to go completely around the world, so the finiteness of the Earth is in many ways disconnected from our everyday experience of life.
But suppose we lived in a universe that was truly finite, at a scale small enough for it to matter on a human level. Suppose that any time you walked, say, a mile in any direction, you found yourself back where you started. What would that be like?
Things become even more radically different as the scale gets smaller. Imagine a world that repeated on such a small scale that if you looked out into the distance, you could see the back of your own head. Where if you shone a laser beam, it would come back from the other direction. In such a world, guns would be worse than useless — if you shot off a firearm, the most likely outcome would be suicide. Now we are getting to the sorts of questions that M.C. Escher was clearly thinking about.
How would living in such a world change the way we think about things? It would certainly change the way we think about city planning and architecture, but would it also transform our aesthetics, our mathematics, our music and art?
Rudy Rucker has a great idea about the ‘dimension’ of scale being finite and circular also. This means as you travel up through the incredibly large you eventually return back to the incredibly small. So solar systems or even galaxies are actually atoms in a much larger universe. This makes sense as the you see the same structure repeating across different scales – i.e. a central node surrounded by smaller satellite entities (think atoms, planets, solar systems, galaxies). Also this idea removes the need for an infinite universe. Infinite universe lead to all sorts of logical paradoxes.
Yes, that’s a powerful idea, which shows up in all sorts of places. To mention two: (1) The aliens playing marbles with our universe in the closing animation of Men in Black, and (2) In Animal House, when Larry philosophizes while smoking pot for the first time at his drug-supplying professor’s pad. To quote Larry: