A [[Dihedral Group]] is a type of [[Group]] of [[Symmetry|symmetries]] of a [[Regular Polygon]], which includes [[Rotation Transformation|rotations]] and [[Reflection Transformations|reflections]]. For some [[Regular Polygon]] with $n$ sides, there are $2n$ total symmetries: - $n$ [[Rotation Transformation|rotational symmetries]] - $n$ [[Reflection Transformations|reflection symmetries]] These symmetries form the [[Dihedral Group]] of order $n$, denoted as $D_n$. >[!example] [[Dihedral Group]] of order $8$ ($D_8$) >![[Pasted image 20251020232830.png]] If the group $D_n$ has rotation elements $r_{0}, r_{1}, \dots, r_{n-1}$ and reflection elements $s_{0},s_{1},\dots,s_{n-1}$, the following properties hold: $\huge \begin{align} r_{i} \circ r_{j} &= r_{i+j} \\ r_{i}\circ s_{j} &= s_{i+j} \\ s_{i}\circ r_{j} &= s_{i-j} \\ s_{i} \circ s_{j} &= s_{i-j} \end{align}$ Note that this assumes that the order of actions $\set{r_{0}, \dots}$ is [[Ordered Set|ordered]] from least change in [[Counter-Clockwise|counter-clockwise]] angle to greatest, and the order of actions $\set{s_{0},\dots}$ is [[Ordered Set|ordered]] such $s_0$ is reflection across the $x$-axis and successive actions $s_1, s_2, \dots$ have a axis of reflection increasing [[Counter-Clockwise|counter-clockwise]] by a constant increasing angle. >[!example] >![[Pasted image 20251020233358.png|invert_S]]