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In this paper, we present a new record for the densest geodesic congruent ball packing configurations in $\mathbf{H}^2\!\times\!\mathbf{R}$ geometry, generated by screw motion groups. These groups are derived from the direct product of rotational groups on $\mathbf{H}^2$ and some translation components on the real fibre direction $\mathbf{R}$ 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, $\approx0.80529$, is achieved by a multi-transitive case given by rotational parameters $(2,20,4)$. E. Moln\'{a}r demonstrated that homogeneous 3-spaces can be uniformly interpreted in the projective 3-sphere $\mathcal{PS}^3(\mathbf{V}^4, \boldsymbol{V}_4, \mathbf{R})$. We use this projective model of $\mathbf{H}^2\!\times\!\mathbf{R}$ to compute and visualize the locally optimal geodesic ball arrangements., Comment: 27 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:2311.12260

After having investigated several types of geodesic ball packings in $\mathbf{S}^2 \times \mathbf{R}$ space, in this paper we study the locally optimal geodesic of simply and multiply transitive ball packings with equal balls to the space groups generated by rotations in $\mathbf{H}^2 \times \mathbf{R}$ geometry. These groups can be derived by direct product of the isometries on hyperbolic plane $\mathbf{H}^2$ and the real line $\mathbf{R}$. Moreover, we develop a procedure to determine the densities of the above locally densest geodesic ball packing configurations. Additionally, we examine the monotonicity properties of the densities within infinite series of the considered space groups. E. {Moln\'ar} showed, that the homogeneous 3-spaces have a unified interpretation in the projective 3-sphere $\mathcal{PS}^3(\mathbf{V}^4,\boldsymbol{V}_4, \mathbf{R})$. In our work, we use this projective model of $\mathbf{H}^2 \times \mathbf{R}$ to visualize the locally optimal ball arrangements., Comment: 5 Figures in EPS format. arXiv admin note: text overlap with arXiv:1206.0566, arXiv:1210.2202

In this paper we study the regular prism tilings and construct ball packings by geodesic balls related to the above tilings in the projective model of $\mathbf{Nil}$ geometry. Packings are generated by action of the discrete prism groups $\mathbf{pq2_{1}}$. We prove that these groups are realized by prism tilings in $\mathbf{Nil}$ space if $(p,q)=(3,6), (4,4), (6,3)$ and determine packing density formulae for geodesic ball packings generated by the above prism groups. Moreover, studying these formulae we determine the conjectured maximal dense packing arrangements and their densities and visualize them in the projective model of $\mathbf{Nil}$ geometry. We get a dense (conjectured locally densest) geodesic ball arrangement related to the parameters $(p,q)=(6,3)$ where the kissing number of the packing is $14$, similarly to the densest lattice-like $\mathbf{Nil}$ geodesic ball arrangement investigated by the second author ., Comment: arXiv admin note: text overlap with arXiv:1105.1986

In this paper we study the locally optimal geodesic ball packings with equal balls to the $\mathbf{S}^2\!\times\!\mathbf{R}$ space groups having rotation point groups and their generators are screw motions. We determine and visualize the densest simply transitive geodesic ball arrangements for the above space groups, moreover we compute their optimal densities and radii. The densest packing is derived from the $\mathbf{S}^2\!\times\!\mathbf{R}$ space group $\mathbf{3qe.~I.~3}$ with packing density $\approx 0.7278$. E. Moln\'ar has shown, that the Thurston geometries have an unified interpretation in the real projective 3-sphere $\mathcal{PS}^3$. In our work we shall use this projective model of $\mathbf{S}^2\!\times\!\mathbf{R}$ geometry., Comment: arXiv admin note: substantial text overlap with arXiv:1206.0566, arXiv:1210.2202

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., Comment: 15 pages and 6 figures