symmetry group (thing)

If X denotes a subset of Rⁿ then its symmetry group Symm(X) is the group of isometries of Rⁿ that leave the subset X stable. For examples of isometries see the write-ups on isometries of the plane and the orthogonal group.

It may be that every symmetry of X is forced to fix the origin and so be a linear transformation. This is the case for a regular polygon centred at the origin in R², for example, and for also for a Platonic solid centred at the origin in 3-space.

In that case Symm(X) will be a subgroup of the orthogonal group O(n) and we can also consider its direct (or rotational) symmetry group Rot(X) which is the intersection of Symm(X) with the special orthogonal group SO(n).

We are mostly interested in the case of n=2 and n=3. In that case Rot(X) consists of rotations, as its name suggests.

Examples

• The symmetry group of a regular polygon is the dihedral group.
• See also the symmetry groups of the Platonic solids.
• The crystallographic groups are the symmetry groups of crystal lattices and of two-dimensional wallpaper patterns.