The Dihedral Group

Snow flakes are one of nature’s beauties which are easy to appreciate even for the more mathematically or technically minded. Kenneth G. Libbrecht produces wonderful photographs of them; some are available online.

The symmetry of snow flakes is described by the dihedral group. This is one of the first groups described in Felix Klein’s book I have advertised before. Here is my illustration of how Klein described this group geometrically.

dihedra

The dihedron is the degenerate case of a regular polyhedron with only two faces. Thus a dihedron is a regular polygon embedded in three dimensional space where the two sides of the polygon are taken to be different faces. The dihedral group is its symmetry group. In fact, there is one such group for each integer n where n is the number of vertices of the regular polygon.

The yellow rotation axis in the picture visualises the subgroup of the dihedral group which is isomorphic to the cyclic group of order n. This subgroup acts on the dihedron by rotation around the yellow axis by angles which are integer multiples of 2\pi/n.

Depending on whether n is even or odd, there are different axes around which the dihedron can be flipped upside down. If n is odd, these are the green axes given in the left part of the figure above. They are defined as the lines through one vertex and the middle of the opposite side of the polygon. If n is even, there are two sets of such axes. One set (right part, green) is given by the lines through the mid-points of opposite sides of the polygon, the other set (right part, blue) is given by lines through opposite vertices.

Using this geometric model, Klein makes it easy to see that the cyclic subgroup of the dihedral group given by rotations of the polygon is a normal subgroup. Any element STS^{-1} of the dihedral group, where T is a rotation around the yellow axis and S represents flipping around one of the blue or green axes, is again a rotation around the yellow axis.

A degenerate case of the already degenerate case of the dihedron is the case n=2 of a dihedron with only two vertices. Here is an illustration of an action of the resulting symmetry group (vertices in blue, midpoints of arcs connecting the vertices in green).

kleinfour

The result is a group of four elements given by flipping the configuration around the yellow axis, the blue axis, or the green axis. It is known as the Klein four-group and its name comes from Klein’s discussion of the group in his book. This group is different from the cyclic group of order four and thus the other finite group of order four.

Stay tuned for upcoming posts on symmetry groups of the non-degenerate regular polyhedra.

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