# Dihedral Group/Examples

## Examples of Dihedral Groups

### Dihedral Group $D_1$

The dihedral group $D_1$ is the symmetry group of the line segment:

Let $AB$ be a line segment.

The symmetry mappings of $AB$ are:

The identity mapping $e$
The rotation $r$ of $180 \degrees$ about the midpoint of $AB$.

This group is known as the symmetry group of the line segment.

### Dihedral Group $D_2$

The dihedral group $D_2$ is the symmetry group of the rectangle:

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

The various symmetry mappings of $\RR$ 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 $\RR$ form the dihedral group $D_2$.

### Dihedral Group $D_3$

The dihedral group $D_3$ is the symmetry group of the equilateral triangle:

Let $\triangle ABC$ be an equilateral triangle.

We define in cycle notation the following symmetry mappings on $\triangle ABC$:

 $\ds e$ $:$ $\ds \tuple A \tuple B \tuple C$ Identity mapping $\ds p$ $:$ $\ds \tuple {ABC}$ Rotation of $120 \degrees$ anticlockwise about center $\ds q$ $:$ $\ds \tuple {ACB}$ Rotation of $120 \degrees$ clockwise about center $\ds r$ $:$ $\ds \tuple {BC}$ Reflection in line $r$ $\ds s$ $:$ $\ds \tuple {AC}$ Reflection in line $s$ $\ds t$ $:$ $\ds \tuple {AB}$ Reflection in line $t$

Note that $r, s, t$ can equally well be considered as a rotation of $180 \degrees$ (in $3$ dimensions) about the axes $r, s, t$.

Then these six operations form a group.

This group is known as the symmetry group of the equilateral triangle.

### Dihedral Group $D_4$

The dihedral group $D_4$ is the symmetry group of the square:

Let $\SS = ABCD$ be a square.

The various symmetry mappings of $\SS$ are:

the identity mapping $e$
the rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ around the center of $\SS$ anticlockwise respectively
the reflections $t_x$ and $t_y$ are reflections in the $x$ and $y$ axis respectively
the reflection $t_{AC}$ in the diagonal through vertices $A$ and $C$
the reflection $t_{BD}$ in the diagonal through vertices $B$ and $D$.

This group is known as the symmetry group of the square, and can be denoted $D_4$.

### Dihedral Group $D_5$

The dihedral group $D_5$ is the symmetry group of the regular pentagon:

Let $\PP = ABCDE$ be a regular pentagon.

The various symmetry mappings of $\PP$ are:

the identity mapping $e$
the rotations $r, r^2, r^3, r^4$ of $72^\circ, 144^\circ, 216^\circ, 288^\circ$ around the center of $\PP$ anticlockwise respectively
the reflections $t_A, t_B, t_C, t_D, t_E$ in the lines through the center of $\PP$ and the vertices $A$ to $E$ respectively.

This group is known as the symmetry group of the regular pentagon.

### Dihedral Group $D_6$

The dihedral group $D_6$ is the symmetry group of the regular hexagon:

Let $\HH = ABCDEF$ be a regular hexagon.

The various symmetry mappings of $\HH$ are:

The identity mapping $e$
The rotations through multiples of $60 \degrees$ anticlockwise about the center of $\HH$
The reflections in the indicated axes.

Let $\alpha$ denote rotation of $\HH$ anticlockwise through $\dfrac \pi 3$ radians ($60 \degrees$).

Let $\beta$ denote reflection of $\HH$ in the $AD$ axis.

The symmetries of $\HH$ form the dihedral group $D_6$.