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.

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