[math-fun] Old 2D geometry enigma asked in 3D or nD

An old geometry problem discussed here asks for a finite set of points in the plane such that the perpendicular bisector of any 2 of the points contains exactly 2 of the points. At last check there was just one known solution (up to rotation and scaling). In case anyone feels like looking for a solution, I won't mention the answer unless asked to. * * * OK, what about a solution in R^3 or R^n ? We would of course want it to be more than just the planar solution, so we'll require that "an n-dimensional solution" to this question must not lie in any affine Euclidean space R^k in R^n. I know a little, but not much, about the answers. —Dan