2023/09/26 by Emil Molnár, Molnár, Emil, Jenő Szirmai +1 · 1 citation
Chemistry · Materials Science · Mathematics · #52C17 #57M07 #57M60 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Liquid Crystal Research Advancements #Metric Geometry (math.MG) #Molecular spectroscopy and chirality
paper · pdf · doi:10.48550/arxiv.2309.15168
openalex publication_date 2023/09/26 · openalex created_date 2023/09/30 · openalex updated_date 2026/07/28
We intend to continue our previous papers (\citeMSz17 and \citeMSz18, as indicated there) on dense ball packing hyperbolic space \HYP by equal balls, but here with centres belonging to different orbits of the fundamental group Cw(2z, 3 ≤ z ∈ \bN, odd number), of our new series of \it tube or cobweb manifolds Cw = \HYP/\BCw with z-rotational symmetry. As we know, \BCw is a fixed-point-free isometry group, acting on \HYP discontinuously with appropriate tricky fundamental domain Cw, so that every point has a ball-like neighbourhood in the usual factor-topology. Our every Cw(2z) is minimal, i.e. does not cover regularly a smaller manifold. It can be derived by its general symmetry group \BW(u, v, w = u) that is a complete Coxeter orthoscheme reflection group, extended by the half-turn \Bh (0 ↔ 3, 1 ↔ 2) of the complete orthoscheme A0A1A2A3 ∼ b0b1b2b3 (Fig.~1). The vertices A0 and A3 are outer points of the \HYP, as 1/u + 1/v ≤ 1/2 is required, 3 ≤ u = w, v for the above orthoscheme parameters. 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 (bij) = (⟨ \Bbi, \Bbj ⟩) in formula (1.1) and its inverse (Ajk) = (⟨ \BAj, \BAk ⟩) in (1.3). 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 Cw(2z). As first results we concentrate only on particular constructions by computer for probable material model realizations, atoms or molekules by equal balls, for general W(u;v;w=u) as well, summarized at the end of our paper.