Geodesic ball packings generated by regular prism tilings in $\mathbf{Nil}$ geometry

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 .