Visibility phenomena in hypercubes

TL;DR

This paper explores the geometric and probabilistic properties of visible lattice points in high-dimensional hypercubes, revealing typical shapes, angles, and a new number theoretic constant related to mutual visibility.

Contribution

It introduces new results on the shape and angle distributions of visible lattice points and defines a novel constant for the probability of mutual visibility among polytope vertices.

Findings

Most self-visible triangles are nearly equilateral with sides close to rac{d N}{sqrt{6}}

The typical angle between rays from the origin approaches rac{sqrt{7}}{4} as dimensions grow

Existence of a constant rac{Lambda_{d,K}} that limits the probability of mutual visibility in K-polytopes

Abstract

We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic tones. For example, we prove that almost all self-visible triangles with vertices in the lattice of points with integer coordinates in $\mathcal W=[0,N]^d$ are almost equilateral having all sides almost equal to $\sqrt{d}N/\sqrt{6}$, and the sine of the typical angle between rays from the visual spectra from the origin of $\mathcal W$ is, in the limit, equal to $\sqrt{7}/4$, as $d$ and $N/d$ tend to infinity. We also show that there exists an interesting number theoretic constant $\Lambda_{d,K}$, which is the limit probability of the chance that a $K$-polytope with vertices in the lattice $\mathcal W$ has all vertices visible from each other.