Dense ball packings by tube manifolds as new models for hyperbolic crystallography

TL;DR

This paper explores dense ball packings in hyperbolic space using tube manifolds with specific symmetries, providing new models for hyperbolic crystallography and analyzing their geometric properties and packing densities.

Contribution

It introduces a novel class of tube and cobweb manifolds in hyperbolic space, derived from Coxeter orthoscheme reflection groups, and studies their dense ball packings and symmetries.

Findings

Construction of new hyperbolic manifolds with specific symmetry groups.

Initial computer-based results on ball packing densities.

Descriptions of geometric metrics and potential material models.

Abstract

We intend to continue our previous papers (\cite{MSz17} and \cite{MSz18}, as indicated there) on dense ball packing hyperbolic space $\HYP$ by equal balls, but here with centres belonging to different orbits of the fundamental group $Cw(2z, 3 \le z \in \bN$, odd number), of our new series of {\it tube or cobweb manifolds} $Cw = \HYP/\BCw$ with $z$-rotational symmetry. As we know, $\BCw$ is a fixed-point-free isometry group, acting on $\HYP$ discontinuously with appropriate tricky fundamental domain $Cw$, so that every point has a ball-like neighbourhood in the usual factor-topology. Our every $Cw(2z)$ is minimal, i.e. does not cover regularly a smaller manifold. It can be derived by its general symmetry group $\BW(u, v, w = u)$ that is a complete Coxeter orthoscheme reflection group, extended by the half-turn $\Bh$ $(0 \leftrightarrow 3, 1 \leftrightarrow 2)$ of the complete orthoscheme $A_0A_1A_2A_3 \sim b_0b_1b_2b_3$ (Fig.~1). The vertices $A_0$ and $A_3$ are outer points of the $\HYP$, as $1/u + 1/v \le 1/2$ is required, $3 \le u = w, v$ for the above orthoscheme parameters. The situation is described first in Figure 1 of the half trunc-orthoscheme $W$ and its usual extended Coxeter diagram, moreover, by the scalar product matrix $(b^{ij}) = (\langle \Bb^i, \Bb^j \rangle)$ in formula (1.1) and its inverse $(A_{jk}) = (\langle \BA_j, \BA_k \rangle)$ in (1.3). These will describe the hyperbolic angle and distance metric of the half trunc-orthoscheme $W$, then its ball packings, densities, then those of the manifolds $Cw(2z)$. As first results we concentrate only on particular constructions by computer for probable material model realizations, atoms or molekules by equal balls, for general $W(u;v;w=u)$ as well, summarized at the end of our paper.