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Symmetry 2013, 5(1), 54-80; doi:10.3390/sym5010054

Article

Non-Crystallographic Symmetry in Packing Spaces

Valery G. Rau ^1,* , Leonty A. Lomtev ²  and Tamara F. Rau ¹

¹ Vladimir State University, Gorkogo St., 87, Vladimir 600000, Russia ² Joint Stock Company "Magneton", Kuibyshev St., 26, Vladimir 600026, Russia

* Author to whom correspondence should be addressed.

Received: 25 November 2012 / Accepted: 5 December 2012 / Published: 9 January 2013

(This article belongs to the Special Issue Polyhedra)

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Abstract: In the following, isomorphism of an arbitrary finite group of symmetry, non-crystallographic symmetry (quaternion groups, Pauli matrices groups, and other abstract subgroups), in addition to the permutation group, are considered. Application of finite groups of permutations to the packing space determines space tilings by policubes (polyominoes) and forms a structure. Such an approach establishes the computer design of abstract groups of symmetry. Every finite discrete model of the real structure is an element of symmetry groups, including non-crystallographic ones. The set packing spaces of the same order N characterizes discrete deformation transformations of the structure.

Keywords: tilings; finite groups of permutations; packing spaces; polyominoes; quaternion group; cayley tables; Pauli matrices; dirac matrices

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