2014/03/13 by Jenő Szirmai, Jenö Szirmai, Szirmai, Jenö
Engineering · Materials Science · Mathematics · #51M20 #52B15 #52C17 #52C22 #53A35 #Advanced Materials and Mechanics #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG) #Quasicrystal Structures and Properties #math.MG #msc:51M20 #msc:52B15 #msc:52C17 #msc:52C22 #msc:53A35
paper · pdf · doi:10.48550/arxiv.1403.3192
16 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1304.0546
arxiv created 2014/03/13 · openalex publication_date 2014/03/13 · arxiv updated 2014/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In \citeSz13-1 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 \cTip(q) and infinitely many regular bounded p-gonal non-face-to-face prism tilings \cTp(q) for integer parameters p,q;~3 ≤ p, (2p)/(p-2) < q. Moreover, in \citeMSz14 and \citeMSzV13 we have determined the symmetry group of \cTp(q) via its index 2 rotational subgroup, denoted by pq21 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 \cTip(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 ≈ 0.626606 for (p, q) = (29,3) then the maximal density of the corresponding periodical geodesic ball packings under the groups pq21. In our work we will use the projective model of \SLR introduced by E. Molnár in \citeM97.