Crystallographic and Quasicrystallographic Lattices from the Finite Groups of Quaternions

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.