On geodesic balls packing generated by rotational and screw motion groups in -\mathbf{H}^2 \times \mathbf{R}- space. Arnasli Yahya (a)(b), Jeno Szirmai (a)
(a) Budapest University of Technology and Economics, Hungary
(b) Bandung Institute of Technology, Indonesia
Abstract
We present a new record for the densest ball packing configurations in \(\mathbf{H}^2 \times \mathbf{R}\), achieved through geodesic ball packings generated by screw motion groups. These groups are formed of rotational groups in \(\mathbf{H}^2\) and a translation component in the real fiber \(\mathbf{R}\). We provide exact solutions for Frobenius congruence to determine the translation part and develop a procedure to determine the optimal radius for the geodesic ball. The highest packing density, \(0.80529\cdots\), is achieved by a multi-transitive case of the rotational point group of order \((2,20,4)\). E. Molnar demonstrated that homogeneous 3-spaces can be uniformly interpreted in the projective 3-sphere \(\mathcal{PS}^3(\textbf{V}^4, \mathbf{V}_4, \mathbf{R})\). We use this projective model of \(\mathbf{H}^2 \times \mathbf{R}\) to visualize the locally optimal ball arrangements.
