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

E. Molnár, J. Szirmai

We intend to continue our previous papers on dense ball packing hyperbolic space $\mathbf{H}^3$ by equal balls, but here with centres belonging to different orbits of the fundamental group $\boldsymbol{Cw}(2z, 3 \le z \in \mathbb{N}$, odd number), of our new series of {ı tube or cobweb manifolds} $Cw = \mathbf{H}^3/\boldsymbol{Cw}$ with $z$-rotational symmetry. As we know, $\boldsymbol{Cw}$ is a fixed-point-free isometry group, acting on $\mathbf{H}^3$ discontinuously with appropriate tricky fundamental domain $Cw$, so that every point has a ball-like neighbourhood in the usual factor-topology. Our every $\boldsymbol{Cw}(2z)$ is minimal, i.e. does not cover regularly a smaller manifold. It can be derived by its general symmetry group $\boldsymbol{W}(u; v; w = u)$ that is a complete Coxeter orthoscheme reflection group, extended by the half-turn $\boldsymbol{h}$ $(0 \leftrightarrow 3, 1 \leftrightarrow 2)$ of the complete orthoscheme $A_0A_1A_2A_3 \sim b_0b_1b_2b_3$ (see Figure 1). The vertices $A_0$ and $A_3$ are outer points of the (Beltrami-Cayley-Klein) B-C-K model of $\mathbf{H}^3$, as $1/u + 1/v \le 1/2$ is required, $3 \le u = w, v$ for the above orthoscheme parameters. For the above simple manifold-construction we specify $u = v = w = 2z$. Then the polar planes $a_0$ and $a_3$ of $A_0$ and $A_3$, respectively, make complete with reflections $\boldsymbol{a}_0$ and $\boldsymbol{a}_3$ the Coxeter reflection group, where the other reflections are denoted by $\boldsymbol{b}^0$, $\boldsymbol{b}^1$, $\boldsymbol{b}^2$, $\boldsymbol{b}^3$ in the sides of the orthoscheme $b^0b^1b^2b^3$. 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 $(b^{ij}) = (\langle \boldsymbol{b}^i, \boldsymbol{b}^j \rangle)$ in formula (1) and its inverse $(A_{jk}) = (\langle \boldsymbol{A}_j, \boldsymbol{A}_k \rangle)$ in (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 $\boldsymbol{Cw}(2z)$. As first results we concentrate only on particular constructions by computer for probable material model realizations, atoms or molecules by equal balls, for general $W(u;v;w=u)$ as well, summarized at the end of our paper.