The regular p-gonal prism tilings and their optimal hyperball packings in the hyperbolic 3-space

Abstract

We investigate the regular p-gonal prism tilings (mosaics) in the hyperbolic 3-space that were classified by I. Vermes in [12] and [13]. The optimal hyperball packings of these tilings are generated by the "inscribed hyperspheres" whose metric data can be calculated by our method -based on the projective interpretation of the hyperbolic geometry -by the volume formulas of J. Bolyai and R. Kellerhals, respectively. We summarize in some tables the data and the densities of the optimal hyperball packings to each prism tiling in the hyperbolic space H 3 .

Decomposition method related to saturated hyperball packings

“…However, these hyperball packing configurations are only locally optimal, and cannot be extended to the whole space H 3 . Moreover, we showed that the densest known hyperball packing, dually related to the regular prism tilings, introduced in [12], can be realized by a regular truncated tetrahedron tiling with density ≈ 0.82251. [18] we discussed the problem of congruent and non-congruent hyperball (hypersphere) packings to each truncated regular tetrahedron tiling.…”

“…The above procedure is illustrated for regular octahedron tilings derived by the regular prism tilings with Coxeter-Schläfli symbol {p, 3, 4}, 6 < p ∈ N. These Coxeter tilings and the corresponding hyperball packings are investigated in [12]. In this situation the convex polyhedron D(P ) is a truncated octahedron (see Fig.…”

“…Recently, (to the best of author's knowledge) the candidates for the densest hyperball (hypersphere) packings in the 3, 4 and 5-dimensional hyperbolic space H n are derived by the regular prism tilings which have been in papers [12], [13] and [14].…”

Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic 3-space

“…Similarly to the former cases (see [21], [22], [24], [14], [16], [17]) it is interesting to study and to construct locally optimal congruent and non-congruent hyperball packings relating to suitable truncated polyhedron tilings in 3-and higher dimensions as well. This study fits into our program to look for the upper bound density of the congruent and non-congruent hyperball packings in H n .…”

“…In [21] and [22] we analysed the regular prism tilings (simply truncated Coxeter orthoscheme tilings) and the corresponding optimal hyperball packings in H n (n = 3, 4) and we extended the method -developed in the former paper [22] -to 5-dimensional hyperbolic space (see [23]). In paper [24] we studied the n-dimensional hyperbolic regular prism honeycombs and the corresponding coverings by congruent hyperballs and we determined their least dense covering densities.…”