Non-periodic geodesic ball packings generated by 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 $p,q$, $3 \le p$, $ {2p}/({p-2}) \lt q$. Moreover, in \cite {MSz14, 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 and we prove that the side curves of their base figures are arcs of Euclidean circles for each parameter. Furthermore, 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 search for values of parameters $p$ and $q$ that provide the largest packing density. In this paper, we obtain greater density $0.626606\ldots $ for $(p, q) = (29,3)$ than the maximum density of the corresponding periodic geodesic ball packings under the groups $\mathbf {pq2_1}$.

In our work we use the projective model of $\SLR $ introduced by {Moln\'ar} in \cite {M97}.