6. Regular Polygons

All symmetries can be expressed using just two generators: $r$ and $s$

Symmetry of a Regular Polygon

For a regular polygon with $n$ sides, the dihedral group $D_n$ is the group of all symmetries

• Rotations: $Id, r, r^2,..., r^{n-1}$
• Reflections: $s, rs, r^2s,..., r^{n-1}s$

The generators $r$ and $s$ satisfy

$$r^n = Id, \quad s^2 = Id, \quad srs = r^{-1}$$

Compact Notation for $D_n$

$$D_n = \langle r, s \mid r^n = Id, \; s^2 = Id, \; s r s = r^{-1} \rangle$$