[math-fun] Geometry question

This may be trivial, hard, or auropileous; I'm not sure: Suppose we are given a finite subset X of R^n such that * The dot product of any two vectors between points of X is an integer. (I.e., for all x,y,z,w in X, the real number <x-y, z-w> lies in Z.) Question: --------- Does it follow that for any sufficiently high dimension d, there is a subset X' of the integer lattice Z^d such that X' is congruent to X ??? —Dan