Counterintuitive patterns on angles and distances between lattice points in high dimensional hypercubes

TL;DR

This paper explores surprising geometric patterns involving angles and distances between lattice points in high-dimensional hypercubes, revealing counterintuitive properties as the dimension increases.

Contribution

It provides new insights into the geometric relationships of lattice points in high-dimensional spaces, highlighting unexpected patterns in distances and angles.

Findings

Almost all triangles with a fixed interior point and hypercube vertices are nearly equilateral in high dimensions.

Triangles formed with a point near the cube's center and hypercube vertices tend to be nearly right-angled.

High-dimensional geometry exhibits counterintuitive properties not observed in low dimensions.

Abstract

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