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

Let \({\mathcal {S}}\) $\mathcal{S}$ be a finite set of integer points in \({\mathbb {R}}^d\) ${\mathbb{R}}^d$ , which we assume has many symmetries, and let \(P\in {\mathbb {R}}^d\) $P\in {\mathbb{R}}^d$ be a fixed point. We calculate the distances from P to the points in \({\mathcal {S}}\) $\mathcal{S}$ 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 \({\mathcal {S}}\) $\mathcal{S}$ is the set of vertices of a hypercube in \({\mathbb {R}}^d\) ${\mathbb{R}}^d$ and P is any point inside, then almost all triangles PAB with \(A,B\in {\mathcal {S}}\) $A,B\in \mathcal{S}$ are almost equilateral. Or, if P is close to the center of the cube, then almost all triangles PAB with \(A\in {\mathcal {S}}\) $A\in \mathcal{S}$ and B anywhere in the hypercube are almost right triangles.