Category:Dihedral Group D3

From ProofWiki
Jump to navigation Jump to search

This category contains examples of Dihedral Group D3.

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.