On parallelohedra of Nil-space

Schultz, Benedek and Szirmai, Jenő (2012) On parallelohedra of Nil-space. Pollack Periodica, 7 (Supple). pp. 129-136. ISSN 1788-1994

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Abstract

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.

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