Interactive and animated Geometric Algebra with Cinderella
- Two Dimensional Lattice Symmetries

Any fixed-point symmetry of a lattice transforms lattice points into new lattice points. Its matrix elements with respect to a basis of lattice point vectors must therefore all be integers. As a consequence also the trace

trace = 2 cos(theta) + n-2

of any lattice rotation must be integer, with n the dimension of the space. Together with the cyclic condition theta = 2Pi/p for a rotation generating a rotation subgroup we get:

p = 1,2,3,4,6

but not 5,7,8, etc.

Below two applets show graphically why no two-dimensional lattices exist with five (p=5) and seven (p=7) fold rotation symmetries. In contrast to this the hexagonal (p=6) honeycomb lattice is well known to exist.

Reference
D. Hestenes, Point Groups and Space Groups in Geometric Algebra in L. Dorst, C. Doran, J. Lasenby (eds.), Applications of Geometric Algebra in Computer Science and Engineering, Birkhaeuser, Boston, 2002, pp. 3-34.

[ GA with Cinderella | 2D point groups ]

Soli Deo Gloria. Created with Cinderella v2.0beta and v1.2 by Eckhard Hitzer (Fukui).