Hey Bill, that makes a lot of sense. If I read you right, you're saying that just as we can expand a circle into a set of epsilon-sized circles which together make up a 2-torus (the surface of a donut whose "donut hole" is almost as big as the original circle) we can do the same thing in 4 dimensions to get a 3-torus surface, and so on. I still like to embed the 2-torus in 4 dimensions, because then nothing gets stretched. Probably just the way my mind works. Bill Thurston wrote:
The standard name for an n-cube with opposite sides identified via translation = (S^1)^n is the n-torus, T^n. It's also equivalent to the quotient of the additive group of vectors in R^n by the integer lattice subgroup, T^n = R^n/Z^n.
In general, the n-torus can be embedded in R^{n+1}. For the circle, this is immediate. Everything else can be done by induction: if the n-torus is smoothly embedded in R^{n+1}, then the boundary of an epsilon neighborhood in R^{n+2} gives a smooth embedding of T^{n+1} in R^{n+2}.
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