Dihedral symmetry

Dihedral symmetry consists of the rotations and reflections preserving a regular polygon. For an $n$-gon with $n\ge 3$, its dihedral group has $2n$ elements generated by a rotation $r$ through $2\pi /n$ and a reflection $s$, with $r^n=s^2=1$ and $srs=r^{-1}$.

Every element is uniquely $r^k$ or $r^{k}s$, for $0\le k<n$. For a square this gives four rotations and four reflections. The relation $srs=r^{-1}$ explains why changing orientation reverses the direction of rotation and why the group is generally noncommutative. Notation varies: $D_n$ and $D_{2n}$ are both used for the group of order $2n$.