Crystallographic and Quasicrystallographic Lattices from the Finite Groups of Quaternions

• Published in: Croatica Chemica Acta Vol. 69; no. 3; p. 805
• Main Author: King, R. Bruce
• Format: Paper
• Language: English
• Published: Hrvatsko kemijsko društvo 01.11.1996
• ISSN: 0011-1643; 1334-417X

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.