MATEMATIČKI VESNIK
МАТЕМАТИЧКИ ВЕСНИК



MATEMATIČKI VESNIK
DENSE BALL PACKINGS BY TUBE MANIFOLDS AS NEW MODELS FOR HYPERBOLIC CRYSTALLOGRAPHY
E. Molnár, J. Szirmai

Abstract

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 {\it 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.

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Keywords: Infinite series of hyperbolic space forms; cobweb or tube manifold derived by an extended complete Coxeter orthoscheme reflection group; ball packing by group orbits; optimal dense packing; hyperbolic crystallography.

MSC: 57M07, 57M60, 52C17

DOI: 10.57016/MV-H6RCD277

Pages:  118--135     

Volume  76 ,  Issue  1-2 ,  2024