Category:Dihedral Group D3

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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.


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.