Visibility phenomena in hypercubes

We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic themes. For example, we prove that almost all self-visible triangles with vertices in the lattice of points with integer coordinates in W = (left perpendicular0, Nright left perpendicular boolean AND Z)(d) are almost equilateral having all sides almost equal to root dN/root 6, and the sine of the typical angle between rays from the visual spectra from the origin of. is, in the limit, equal to root 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 W has all vertices visible from each other.

Automated summary

This article investigates the set of visible lattice points in multidimensional hypercubes, combining geometric, probabilistic, and number-theoretic themes. The research demonstrates that nearly all vertices, under certain conditions, form almost equilateral triangles with sides nearly equal to root dN/root 6, while the typical angle between rays from the visual spectra approaches root 7/4 as d and N/d approach infinity. Additionally, the article introduces a number-theoretic constant, Lambda(d,k), representing the limit probability of visibility between vertices of a K-polytope in lattice W.