Densest geodesic ball packings to ${\mathbf{S}}^2\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}\mathbf{R}$ space groups generated by screw motions

In this paper we study the locally optimal geodesic ball packings with equal balls to the ${\mathbf{S}}^2\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}\mathbf{R}$ 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 ${\mathbf{S}}^2\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}\mathbf{R}$ space group $\mathbf{3}\mathbf{q}\mathbf{e}\mathbf{.}\mathbf{I}\mathbf{.}\mathbf{3}$ 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 ${\mathcal{P}\mathcal{S}}^3$. In our work we shall use this projective model of ${\mathbf{S}}^2\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}\mathbf{R}$ geometry.