Original scientific paper

Crystallographic and Quasicrystallographic Lattices from the Finite Groups of Quaternions

R. Bruce King ; Department of Chemistry, University of Georgia, Athens, Georgia 30602, USA

Abstract

Quaternions are ordered quadruples of four numbers subject to
specified rules of addition and multiplication, which can represent
points in four-dimensional (4D) space and which form finite groups
under multiplication isomorphic to polyhedral groups. Projection of
the 8 quaternions of the dihedral group D2h, with only two-fold
symmetry, into 3D space provides a basis for crystal lattices up to
orthorhombic symmetry (a "* b "* c). Addition of three-fold symmetry
to D2h gives the tetrahedral group Td with 24 quaternions, whose
projection into 3D space provides a basis for more symmetrical
crystal lattices including the cubic lattice (a = b = c). Addition of
five fold symmetry to Td gives the icosahedral group Ih with 120
quaternions, whose projection into 3D space introduces the --J5 irrationality and thus cannot provide the basis for a 3D crystal lattice.
However, this projection of Ih can provide a basis for a 6D lattice
which can be divided into two orthogonal 3D subspaces, one
representing rational coordinates and the other representing COOI'-
dinates containing the --J5 irrationality similar to some standard
models for icosahedral quasicrystals.

Keywords

Hrčak ID:

177112

URI

https://hrcak.srce.hr/177112

Publication date:

1.11.1996.

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