Gareth wrote: << (I wrote):

Consider the cubical 3-torus T^3 := R^3/Z^3 -- 3-space factored out by the group of integer translations.

Endow it with the quotient metric. (The distance between [p] := p + Z^3 and [q]:= q + Z^3 for p,q in R^3 is the least distance of the form |x-y| where x is in [p] and y is in [q].)

PUZZLE: Consider two points P, Q of T^3 that are the maximum distance apart, namely sqrt(3/4). What is the topology of the locus of points of T^3 equidistant from P and Q ?

Prosaic solution follows after spoiler space. [omitted]

...

Gareth's solution is indeed the one I had in mind. (I admit that the answer to such a prosaic problem suprised me.) N.B. Scott Huddleston also came up with the same picture, in private e-mail.) --Dan