Combinatorial cube packings in cube and torus

TL;DR

This paper investigates the properties of random cube packings in high-dimensional cubes and tori, introducing combinatorial packings to analyze their density and extensibility, revealing differences between cube and torus cases.

Contribution

It introduces the concept of combinatorial cube packings, derives density expansions, and characterizes extensibility and minimal non-extensible packings in high dimensions.

Findings

In the cube case, packings reduce to a single cube as N→∞.

In the torus case, non-extensible packings occur with positive probability for n≥3.

The number of parameters in torus packings is at least n(n+1)/2, conjectured to be at most 2^n-1.

Abstract

We consider sequential random packing of cubes $z+[0,1]^n$ with $z\in \frac{1}{N}\ZZ^n$ into the cube $[0,2]^n$ and the torus $\QuotS{\RR^n}{2\ZZ^n}$ as $N\to\infty$. In the cube case $[0,2]^n$ as $N\to\infty$ the random cube packings thus obtained are reduced to a single cube with probability $1-O(\frac{1}{N})$. In the torus case the situation is different: for $n\leq 2$, sequential random cube packing yields cube tilings, but for $n\geq 3$ with strictly positive probability, one obtains non-extensible cube packings. So, we introduce the notion of combinatorial cube packing, which instead of depending on $N$ depend on some parameters. We use use them to derive an expansion of the packing density in powers of $\frac{1}{N}$. The explicit computation is done in the cube case. In the torus case, the situation is more complicate and we restrict ourselves to the case $N\to\infty$ of strictly positive probability. We prove the following results for torus combinatorial cube packings: We give a general Cartesian product construction. We prove that the number of parameters is at least $\frac{n(n+1)}{2}$ and we conjecture it to be at most $2^n-1$. We prove that cube packings with at least $2^n-3$ cubes are extensible. We find the minimal number of cubes in non-extensible cube packings for $n$ odd and $n\leq 6$.