New results on torus cube packings and tilings

TL;DR

This paper explores new developments in the combinatorial structures of torus cube packings and tilings, focusing on discrete and continuous cases with integral translations, expanding understanding of their properties.

Contribution

It introduces new results on the structure and properties of torus cube packings and tilings for both discrete and continuous cases, connecting to prior combinatorial studies.

Findings

Analysis of the case N=2 related to discrete tilings

Investigation of the case N=∞ related to continuous tilings

Description of new combinatorial structures in torus packings

Abstract

We consider sequential random packing of integral translate of cubes $[0,N]^n$ into the torus $Z^n / 2NZ^n$. Two special cases are of special interest: (i) The case $N=2$ which corresponds to a discrete case of tilings (considered in \cite{cubetiling,book}) (ii) The case $N=\infty$ corresponds to a case of continuous tilings (considered in \cite{combincubepack,book}) Both cases correspond to some special combinatorial structure and we describe here new developments.