Imagine sitting in a flat 3-torus that’s no larger than a typical bedroom. Look straight ahead and you’ll see your back, look to the right and you’ll see your left side, look up and you’ll see your feet. Next suppose that you’re invisible, but sharing the space with a floating teapot. Look in many different directions and you’ll see the teapot again and again, out to infinity. There’s only one teapot, but your vision within this space wraps around repeatedly, yielding repeated views of the one teapot.1
Any movement through the space also wraps around. You fly through a space that appears to go on forever but it really has only finite volume, maybe even small volume.
Regarding program implementation, the space is represented by an n×n×n block of cubes, each containing an instance of the teapot. As you fly through space, the block gets translated as necessary to keep you inside or near the center cube. There’s really no exit. Everything seems to goes on forever. Fog helps us accept this fact.
- The teapot appears to repeat along infinitely long lines not only in the six cardinal directions but in infinitely many different directions. To see why, we can assign integer-coordinate addresses to the apparent boxes. Assign the address (0, 0, 0) to the current box from which we view, and assign the address (a, b, c) to the apparent box that lies a positions to the right, b positions forward, and c positions above. The box at (a, b, c) belongs to the infinite line of apparent boxes with address (na, nb, nc) as n ranges over all integers. Since there are infinitely many distinct directions (a, b, c), there are infinitely many lines of teapots out to infinity. ↩