It's interesting, though very easy to prove, that the set G of rational points QxQ \int S^1 on the unit circle forms a group under ordinary complex multiplication. (This has been discussed here many years ago, but:) Whereas QxQ seems, like ZxZ, to have a squarish structure, the subset G of QxQ is rotationally homogeneous: Given any two elements of G, there is (of course) some group element that carries one onto the other. Also interesting is its group structure. I carelessly used to think G was isomorphic to Q/Z, which is easily seen to be the direct sum of the groups H_p := {k/p^n : (k,n) in Z^2}/Z for each prime p. But instead, G is unexpectedly isomorphic to the direct sum of a countable number of infinite cyclic groups and one copy of Z/4Z. --Dan