# Upper bound of density for packing of congruent hyperballs in hyperbolic $3-$space

## TL;DR

This paper establishes an upper bound for the density of congruent hyperball packings in hyperbolic 3-space, showing it is approximately 0.86338, and analyzes the non-monotonic relationship between hyperball height and packing density.

## Contribution

It proves the density upper bound for hyperball packings in truncated tetrahedra and demonstrates the non-monotonic behavior of density with hyperball height.

## Key findings

- Density upper bound is approximately 0.86338.
- Optimal hyperball density is achieved in regular truncated tetrahedra.
- Density does not increase monotonically with hyperball height.

## Abstract

In \cite{Sz17-2} we proved that to each saturated congruent hyperball packing exists a decomposition of $3$-dimensional hyperbolic space $\mathbb{H}^3$ 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. In this paper we prove, using the above results and the results of papers \cite{M94} and \cite{Sz14}, that the density upper bound of the saturated congruent hyperball (hypersphere) packings related to the corresponding truncated tetrahedron cells is realized in a regular truncated tetrahedra with density $\approx 0.86338$. Furthermore, we prove that the density of locally optimal congruent hyperball arrangement in regular truncated tetrahedron is not a monotonically increasing function of the height (radius) of corresponding optimal hyperball, contrary to the ball (sphere) and horoball (horosphere) packings.