Definition:Symmetry Group of Rectangle

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Group Example

Let $\mathcal R = ABCD$ be a (non-square) rectangle.

SymmetryGroupRectangle.png

The various symmetry mappings of $\mathcal R$ are:

The identity mapping $e$
The rotation $r$ (in either direction) of $180^\circ$
The reflections $h$ and $v$ in the indicated axes.


The symmetries of $\mathcal R$ form the dihedral group $D_2$.


Cayley Table

The Cayley table of the symmetry group of the (non-square) rectangle can be written:

$\begin{array}{c|cccc} & e & r & h & v \\ \hline e & e & r & h & v \\ r & r & e & v & h \\ h & h & v & e & r \\ v & v & h & r & e \\ \end{array}$


$D_2$ acts on the vertices of $\mathcal R$ according to this table:

$\begin{array}{cccc} e & r & h & v \\ \hline A & C & D & B \\ B & D & C & A \\ C & A & B & D \\ D & B & A & C \\ \end{array}$


Also see


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