Counterintuitive Patterns on Angles and Distances Between Lattice Points in High Dimensional Hypercubes

Let S be a finite set of integer points in R^d, which we assume has many symmetries, and let P∈R^d be a fixed point. We calculate the distances from P to the points in S and compare the results. In some of the most common cases, we find that they lead to unexpected conclusions if the dimension is sufficiently large. For example, if S is the set of vertices of a hypercube in R^d and P is any point inside, then almost all triangles PAB with A,B∈S are almost equilateral. Or, if P is close to the center of the cube, then almost all triangles PAB with A∈S and B anywhere in the hypercube are almost right triangles.