Visibility phenomena in hypercubes

Jayadev S. Athreya, Cristian Cobeli and Alexandru Zaharescu

Chaos, Solitons & Fractals, 2023, vol. 175, issue P1

Abstract: 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=([0,N]∩Z)d are almost equilateral having all sides almost equal to dN/6, and the sine of the typical angle between rays from the visual spectra from the origin of W is, in the limit, equal to 7/4, as d and N/d tend to infinity. We also show that there exists an interesting number theoretic constant Λd,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.

Keywords: Hypercube; Visible points; Polytope; Euclidean distance (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:175:y:2023:i:p1:s0960077923009256

DOI: 10.1016/j.chaos.2023.114024

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