Decomposition method related to saturated hyperball packings

TL;DR

This paper introduces a new approach to analyze hyperball packings in 3D hyperbolic space by decomposing space into truncated tetrahedra, aiming to establish density upper bounds.

Contribution

It proposes a novel definition of non-compact saturated hyperball packings and a procedure to decompose hyperbolic space into truncated tetrahedra for density analysis.

Findings

Decomposition of hyperbolic space into truncated tetrahedra.

A method to determine density upper bounds in these simplices.

Framework for future density optimization studies.

Abstract

In this paper we study the problem of hyperball (hypersphere) packings in $3$-dimensional hyperbolic space. We introduce a new definition of the non-compact saturated ball packings and describe to each saturated hyperball packing, a new procedure to get a decomposition of 3-dimensional hyperbolic space $\HYP$ into truncated tetrahedra. Therefore, in order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices.