Two Dimensions
Up to conjugacy the discrete point groups in 2 dimensional space are the following classes:
- cyclic groups C1, C2, C3, C4,... where Cn consists of all rotations about a fixed point by multiples of the angle 360°/n
- dihedral groups D1, D2, D3, D4,... where Dn (of order 2n) consists of the rotations in Cn together with reflections in n axes that pass through the fixed point.
C1 is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, for example the letter F. C2 is the symmetry group of the letter Z, C3 that of a triskelion, C4 of a swastika, and C5, C6 etc. are the symmetry groups of similar swastika-like figures with five, six etc. arms instead of four.
D1 is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry, for example the letter A. D2, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle, and D3, D4 etc. are the symmetry groups of the regular polygons.
The actual symmetry groups in each of these cases have two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors.
The remaining isometry groups in 2D with a fixed point, where for all points the set of images under the isometries is topologically closed are:
- the special orthogonal group SO(2) consisting of all rotations about a fixed point; it is also called the circle group S1, the multiplicative group of complex numbers of absolute value 1. It is the proper symmetry group of a circle and the continuous equivalent of Cn. There is no figure which has as full symmetry group the circle group, but for a vector field it may apply (see the 3D case below).
- the orthogonal group O(2) consisting of all rotations about a fixed point and reflections in any axis through that fixed point. This is the symmetry group of a circle. It is also called Dih(S1) as it is the generalized dihedral group of S1.
For non-bounded figures, the additional isometry groups can include translations; the closed ones are:
- the 7 frieze groups
- the 17 wallpaper groups
- for each of the symmetry groups in 1D, the combination of all symmetries in that group in one direction, and the group of all translations in the perpendicular direction
- ditto with also reflections in a line in the first direction
Read more about this topic: Symmetry Group
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