What is the connection between the tessellation and the sphere-packing? —Dan Fred Lunnon wrote: ----- Duh --- light dawns! Unwinding Dan's n-torus into its universal covering tesselation of Euclidean space yields simply what seems to be known only as " n-space analogue of the face-centred cubic crystal tiling" or such (maybe Neil can correct this?): the unit-sphere packing where centres have evenly many odd integer Cartesian components. From the combinatorics of the hypercube it is now straightforward that the number of regions in the torus equals 2^n 3^(n-1) ; also the number of vertices equals 2^(n-1) . I might flesh this out with more detail on request. -----