Non-periodic geodesic ball packings to infinite regular prism tilings in $\SLR$ space

Abstract

In \cite{Sz13-1} we defined and described the {\it regular infinite or bounded} $p$-gonal prism tilings in $\SLR$ space. We proved that there exist infinitely many regular infinite $p$-gonal face-to-face prism tilings $\cT^i_p(q)$ and infinitely many regular bounded $p$-gonal non-face-to-face prism tilings $\cT_p(q)$ for integer parameters $p,q;~3 \le p$, $ \frac{2p}{p-2} < q$. Moreover, in \cite{MSz14} and \cite{MSzV13} we have determined the symmetry group of $\cT_p(q)$ via its index 2 rotational subgroup, denoted by $\mathbf{pq2_1}$ and investigated the corresponding geodesic and translation ball packings. In this paper we study the structure of the regular infinite or bounded $p$-gonal prism tilings, prove that the side curves of their base figurs are arcs of Euclidean circles for each parameter. Moreover, we examine the non-periodic geodesic ball packings of congruent regular non-periodic prism tilings derived from the regular infinite $p$-gonal face-to-face prism tilings $\cT^i_p(q)$ in $\SLR$ geometry. We develop a procedure to determine the densities of the above non-periodic optimal geodesic ball packings and apply this algorithm to them. We look for those parameters $p$ and $q$ above, where the packing density large enough as possible. Now, we obtain larger density $\approx 0.626606$ for $(p, q) = (29,3)$ then the maximal density of the corresponding periodical geodesic ball packings under the groups $\mathbf{pq2_1}$. In our work we will use the projective model of $\SLR$ introduced by E. {Moln\'ar} in \cite{M97}.