## 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).