<< But this reminded me of a question I've had for a long time: QUESTION: Can there be a cubed 3-torus? Start with a cubical 3-torus: an NxNxN cube with opposite faces identified. Can this 3-torus be tiled by unequal integer cubes (each of which is a union of some of the N^3 unit cubes that constitute the original cube) ?
Here's a couple of related questions: 1. Can 3-space be tiled by unequal cubes? Assume that a) the tiling is locally finite, i.e., that any sphere contains at most finitely many cubes, and b) each face of each cube is parallel to a coordinate plane. 2. What if we drop condition a) and/or b) ? ----------------------------------------------------------------- (Just to be precise: We want a collection of closed cubes {C_j} in 3-space whose union is all of 3-space and whose interiors are disjoint.) --Dan