New lower bound for the optimal congruent geodesic ball packing density of screw motion groups in \(\textbf{H}^2\!\times \!\textbf{R}\) space

Abstract

In this paper, we present a new record for the densest geodesic congruent ball packing configurations in \(\textbf{H}^2\!\times \!\textbf{R}\) geometry, generated by screw motion groups. These groups are derived from the direct product of rotational groups on \(\textbf{H}^2\) and some translation components on the real fibre direction \(\textbf{R}\) that can be determined by the corresponding Frobenius congruences. Moreover, we developed a procedure to determine the optimal radius for the densest geodesic ball packing configurations related to the considered screw motion groups. The highest packing density, \(\approx 0.80529\), is achieved by a multi-transitive case given by rotational parameters (2, 20, 4). E. Molnár demonstrated that homogeneous 3-spaces can be uniformly interpreted in the projective 3-sphere \(\mathcal{P}\mathcal{S}^3(\textbf{V}^4, \varvec{V}_4, \textbf{R})\). We use this projective model of \(\textbf{H}^2\!\times \!\textbf{R}\) to compute and visualize the locally optimal geodesic ball arrangements.