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

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.