## Interactive and animated Geometric Algebra with Cinderella

- Point Groups in Two Dimensions

Point groups are essential for the study of symmetries of molecules and crystalls. Here I illustrate the
complete two dimensional point groups of regular polygons with k=1...6 sides (corners). The case of k=1
corresponds to a non-symmetric object. All rotations are represented as compositions of reflections. All reflections
are represented by unit vectors r (rr=1) which point to a corner or the middle of a side of the regular polygons:
x -> rxr. These
vectors r mark the directions of the lines of reflection. This representation slightly differs from the one given by
Hestenes: x -> x'= -(1/p)xp. p represents a vector perpendicular to the line of reflection, but in the cases of k=3,5
more vectors p then the above described vectors r become necessary, thus complicating the elegance of description.

The list below refers to applets illustrating first oriented **reflections in the symmetry group 2H_k**
and second oriented **rotations in the dicyclic point symmetry group 2C_k** of each regular polygon.

Compare also 2D lattices.

**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
| vectors | outer prod. | triangle | rotations | oscillations | circ. pol. waves
| conics
| circle chain
| struc. mechanics ]

Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).