Dihedral Group/Examples

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Examples of Dihedral Groups

Dihedral Group $D_1$

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

Let $\triangle AB$ be a line segment.

SymmetryGroupLineSegment.png

The various symmetry mappings of $AB$ are:

The identity mapping $e$
The rotation $r$ of $180^\circ$ about the center 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 $\mathcal R = ABCD$ 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.


Dihedral Group $D_3$

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

Let $\triangle ABC$ be an equilateral triangle.

SymmetryGroupEqTriangle.png

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

\(\displaystyle e\) \(:\) \(\displaystyle \tuple A \tuple B \tuple C\) Identity mapping
\(\displaystyle p\) \(:\) \(\displaystyle \tuple {ABC}\) Rotation of $120 \degrees$ anticlockwise about center
\(\displaystyle q\) \(:\) \(\displaystyle \tuple {ACB}\) Rotation of $120 \degrees$ clockwise about center
\(\displaystyle r\) \(:\) \(\displaystyle \tuple {BC}\) Reflection in line $r$
\(\displaystyle s\) \(:\) \(\displaystyle \tuple {AC}\) Reflection in line $s$
\(\displaystyle t\) \(:\) \(\displaystyle \tuple {AB}\) 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 $\mathcal S = ABCD$ be a square.

SymmetryGroupSquare.png

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

The identity mapping $e$
The rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ counterclockwise respectively about the center of $\mathcal S$.
The reflections $t_x$ and $t_y$ are reflections about the $x$ and $y$ axis respectively.
The reflection $t_{AC}$ is a reflection about the diagonal through vertices $A$ and $C$.
The reflection $t_{BD}$ is a reflection about the diagonal through vertices $B$ and $D$.

This group is known as the symmetry group of the square.


Dihedral Group $D_6$

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

Let $\mathcal H = ABCDEF$ be a regular hexagon.

SymmetryGroupRegularHexagon.png

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

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

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

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


The symmetries of $\mathcal H$ form the dihedral group $D_6$.