Original scientific paper
Geodesic ball packings in $\mathbf{H}^2\!\times\!\mathbf{R}$ space for generalized Coxeter space groups
Jenő Szirmai
orcid.org/0000-0001-9610-7993
; Department of Geometry, Budapest University of Technology and Economics Institute of Mathematics, Budapest, Hungary
Abstract
After having investigated the geodesic ball packings in
$\mathbf{S}^2\!\times\!\mathbf{R}$
space we consider the analogous problem in
$\mathbf{H}^2\!\times\!\mathbf{R}$
space from among the eight Thurston geometries.
In this paper, we determine the geodesic balls of
$\mathbf{H}^2\!\times\!\mathbf{R}$
space and compute their volume,define the notion of the geodesic ball packing and its density.
Moreover, we develop a procedure to determine the density of the geodesic ball packing for generalized Coxeter space groups of
$\mathbf{H}^2\!\times\!\mathbf{R}$
and apply this algorithm to them.
E. Molnár 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 will use this projective model of
$\mathbf{H}^2\!\times\!\mathbf{R}$
geometry and in this manner the geodesic lines, geodesic spheres can be visualized on the Euclidean screen of computer.
Keywords
Thurston geometries; geodesic ball packing; tiling; space group
Hrčak ID:
82993
URI
Publication date:
12.6.2012.
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