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  • 1 Budapest University of Technology and Economics Institute of Mathematics, Department of Geometry Budapest Hungary
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The parallelohedron is one of basic concepts in the Euclidean geometry and in the 3-dimensional crystallography, has been introduced by the crystallographer E.S. Fedorov (1889). The 3-dimensional parallelohedron can be defined as a convex 3-dimensional polyhedron whose parallel copies tile the 3-dimensional Euclidean space in a face to face manner. This paral-lelohedron presents a fundamental domain of a discrete translation group. The 3-parallelepiped is the most trivial and obvious example of a 3-parallelohedron. Fedorov was the first to succeed in classifying the parallelohedra of the 3-dimensional Euclidean space, while in some non-Euclidean geometries it is still an open problem.In this paper we consider the Nil geometry introduced by Heisenberg’s real matrix group. We introduce the notion of the Nil-parallelohedra, outline the concept of parallelohedra classes analogous to the Euclidean geometry. We also study and visualize some special classes of Nil-parallelohedra.

  • 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 zur Algebra und Geometrie, Vol. 38, No. 2, 1997, pp. 261–288.

    Molnár E. , 'The projective interpretation of the eight 3-dimensional homogeneous geometries ' (1997 ) 38 Beiträge zur Algebra und Geometrie : 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
  • J. Szirmai The densest geodesic ball packing by a type of Nil lattices, Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry). 48 No. 2 (2007), 383–397.

    Szirmai J. , 'The densest geodesic ball packing by a type of Nil lattices ' (2007 ) 48 Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) : 383 -397.

    • Search Google Scholar
  • 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 ) -.

  • Schultz B., Szirmai J. Interesting Surfaces in Nil space, Proceedings of 8th International Conference on Applied Informatics (2010), Eger, 2010, (to appear).

  • Fedorov E.S. The symmetry of regular systems of figures (in Russian), Zap. Miner. Obshch. Vol. 21, 1885, pp. 1–279.

    Fedorov E.S. , 'The symmetry of regular systems of figures (in Russian) ' (1885 ) 21 Zap. Miner. Obshch. : 1 -279.

    • Search Google Scholar

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