## 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 Λ_{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.

Original language | English (US) |
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Article number | 114024 |

Journal | Chaos, solitons and fractals |

Volume | 175 |

DOIs | |

State | Published - Oct 2023 |

## Keywords

- Euclidean distance
- Hypercube
- Polytope
- Visible points

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics