The Group of Symmetries of a Rectangle

# The Group of Symmetries of a Rectangle

We recently look at The Group of Symmetries of the Square. Now consider a similar shape - a rectangle, and assume that this rectangle is strictly not a square (otherwise we could induce additional symmetries). Once again, label the vertices of this rectangle $1$, $2$, $3$, and $4$.

There are only four symmetry functions that was can describe. The first two are rotational symmetries of $0^{\circ}$ and $180^{\circ}$ which transform the original rectangle to:

The other two are perpendicular axial symmetries. We can draw two perpendicular bisectors through the rectangle to get two axes that are perpendicular to the edges that they bisect:

These two symmetries transform the original rectangle to:

Let $\rho_0, \rho_1 : \{ 1, 2, 3, 4 \} \to \{ 1, 2, 3, 4 \}$ be functions that represent the rotational symmetries of rotating the rectangle $0^{\circ}$ and $180^{\circ}$ respectively. Let $\mu_1, \mu_2 : \{ 1, 2, 3, 4 \} \to \{1, 2, 3, 4 \}$ be functions that represent the axial symmetries of flipping the rectangle over the axes described respectively above. Then:

(1)
\begin{align} \quad \rho_0 = \begin{pmatrix} 1 & 2 & 3 & 4\\ 1 & 2 & 3 & 4 \end{pmatrix} \quad , \quad \rho_1 = \begin{pmatrix} 1 & 2 & 3 & 4\\ 3 & 4 & 1 & 2 \end{pmatrix} \end{align}
(2)
\begin{align} \quad \mu_1 = \begin{pmatrix} 1 & 2 & 3 & 4\\ 4 & 3 & 2 & 1 \end{pmatrix} \quad , \quad \mu_1 = \begin{pmatrix} 1 & 2 & 3 & 4\\ 2 & 1 & 4 & 3 \end{pmatrix} \end{align}

Let $G = \{ \rho_0, \rho_1, \mu_1, \mu_2 \}$ and let $\circ : G \to G$ be defined as function composition such that for all $f_1, f_2 \in G$ we have that $(f_1 \circ f_2)(x) = f_1(f_2(x))$. Then once again, $(G, \circ)$ forms a group as you should verify.