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 $D_6$.