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,
Vol. 7, 2010, pp. 491–498
Molnár E., 'On projective models of Thurston geometries, some relevant notes on Nil orbifolds and manifolds' (2010) 7Siberian Electronic Mathematical Reports: 491-498.
Molnár E.On projective models of Thurston geometries, some relevant notes on Nil orbifolds and manifoldsSiberian Electronic Mathematical Reports20107491498)| false