View More View Less
  • 1 Budapest University of Technology and Economics Department of Geometry, Institute of Mathematics Egri József u.1 H-1111 Budapest Hungary
Restricted access

Purchase article

USD  $25.00

Purchase this article

USD  $387.00

The S2×R geometry can be derived by the direct product of the spherical plane S2 and the real line R. In [1] J. Z. Farkas has classified and given the complete list its space groups. In [6] the second author has studied the geodesic balls and their volumes in S2×R space, moreover he has introduced the notion of geodesic ball packing and its density and have determined the densest geodesic ball packing for generalized Coxeter space groups of S2×R.The aim of this paper to develop a method to study and visualize the Dirichlet-Voronoi cells belonging to a given ball packing. We apply our former results on the equidistant surfaces of the S2×R geometry (see [5]) to determine the D-V cells to locally optimal ball packings belonging to S2×R space groups generated by glide reflections.E. Molnár has shown in [3], that the homogeneous 3-spaces have a unified interpretation in the real projective 3-sphere, in our work we will use this projective model of S2×R geometry. We will use the Wolfram Mathematica software for visualization of the arrangement of a locally optimal geodesic ball packing and its Dirichlet-Voronoi cell of a given glide reflection space group.

  • Farkas J. Z. The classification of S 2 ×R space groups. Beiträge zur Algebra und Geometrie Vol. 42, 2001. pp. 235–250.

    Farkas J. Z. , 'The classification of S2×R space groups ' (2001 ) 42 Beiträge zur Algebra und Geometrie : 235 -250.

    • Search Google Scholar
  • Molnár E. On projective models of Thurston geometries, some relevant notes on Nil orbifolds and manifolds. Siberian Electronic Mathematical Reports, ( http://semr.math.nsc.ru ), Vol. 7, 2010, pp. 491–498

    Molnár E. , 'On projective models of Thurston geometries, some relevant notes on Nil orbifolds and manifolds ' (2010 ) 7 Siberian Electronic Mathematical Reports : 491 -498.

    • Search Google Scholar
  • Molnár E. The projective interpretation of the eight 3-dimensional homogeneous geometries. Beiträge Alg. Geom. Vol. 38, No. 2, 1997, pp. 261–288.

    Molnár E. , 'The projective interpretation of the eight 3-dimensional homogeneous geometries ' (1997 ) 38 Beiträge Alg. Geom. : 261 -288.

    • Search Google Scholar
  • Molnár E., Szirmai J. Symmetries in the 8 homogeneous 3-geometries, Symmetry, Culture and Science, Vol. 21, No. 1–3, 2010, pp. 87–117.

    Szirmai J. , 'Symmetries in the 8 homogeneous 3-geometries ' (2010 ) 21 Symmetry, Culture and Science : 87 -117.

    • Search Google Scholar
  • Pallagi J., Schultz B., Szirmai J. Visualization of geodesic curves, spheres and equidistant surfaces in S 2 ×R space, KoG, Vol. 14, 2010, pp. 35–40.

    Szirmai J. , 'Visualization of geodesic curves, spheres and equidistant surfaces in S2×R space ' (2010 ) 14 KoG : 35 -40.

    • Search Google Scholar
  • Szirmai J. Geodesic ball packings in S 2 ×R space for generalized Coxeter space groups. Beiträge Alg. Geom. to appear 2011.

  • Szirmai J. Geodesic ball packings in H 2 ×R space for generalized Coxeter space groups, Math. Communications, to appear 2011.

  • Szirmai J. The densest geodesic ball packing by a type of Nil lattices, Beiträge Alg. Geom. Vol. 48 No. 2, 2007, pp. 383–397.

    Szirmai J. , 'The densest geodesic ball packing by a type of Nil lattices ' (2007 ) 48 Beiträge Alg. Geom. : 383 -397.

    • Search Google Scholar
  • Szirmai J. Simply transitive geodesic ball packings to S 2 ×R space groups generated by glide reflections, (Manuscript) (submitted 2011).

  • Thurston W. P., Levy S. Three-dimensional geometry and topology, Princeton University Press, Princeton, New Jersey, Vol 1. 1997.

    Levy S. , '', in Three-dimensional geometry and topology , (1997 ) -.

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Aug 2020 1 0 0
Sep 2020 0 0 0
Oct 2020 0 0 0
Nov 2020 0 0 0
Dec 2020 1 0 0
Jan 2021 0 0 0
Feb 2021 0 0 0