Simply transitive geodesic ball packings to $S^2 \times R$ space groups generated by glide reflections

TL;DR

This paper determines the densest simply transitive geodesic ball packings in $S^2 imes R$ space groups generated by glide reflections, computing their optimal densities and radii, revealing densities higher than Euclidean packings.

Contribution

It introduces the first analysis of optimal geodesic ball packings for $S^2 imes R$ space groups with glide reflection generators, including explicit density calculations.

Findings

Densest packing density approximately 0.8041

Optimal radii and arrangements identified

Higher density than Euclidean packing result

Abstract

The $S^2 \times R$ geometry can be derived by the direct product of the spherical plane $\bS^2$ and the real line $\bR$. J. Z. Farkas has classified and given the complete list of the space groups of $S^2 \times R$. The $S^2 \times R$ manifolds were classified by E. Moln\'ar and J. Z. Farkas by similarity and diffeomorphism. In Szirmai we have studied the geodesic balls and their volumes in $S^2 \times R$ space, moreover we have introduced the notion of geodesic ball packing and its density and have determined the densest geodesic ball packing for generalized Coxeter space groups of $S^2 \times R$. In this paper we study the locally optimal ball packings to the $S^2 \times R$ space groups having Coxeter point groups and at least one of the generators is a glide reflection. We determine the densest simply transitive geodesic ball arrangements for the above space groups, moreover we compute their optimal densities and radii. The density of the densest packing is $\approx 0.80407553$, may be surprising enough in comparison with the Euclidean result $\frac{\pi}{\sqrt{18}} \approx 0.74048$. E. Moln\'ar has shown, that the homogeneous 3-spaces have a unified interpretation in the real projective 3-sphere $\mathcal{PS}^3(\bV^4,\BV_4, \mathbb{R})$. In our work we shall use this projective model of $S^2 \times R$ geometry.