Definition:Symmetry Group of Equilateral Triangle

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Group Example

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.


Cayley Table

$\begin{array}{c|ccc|ccc} \circ & e & p & q & r & s & t \\ \hline e & e & p & q & r & s & t \\ p & p & q & e & s & t & r \\ q & q & e & p & t & r & s \\ \hline r & r & t & s & e & q & p \\ s & s & r & t & p & e & q \\ t & t & s & r & q & p & e \\ \end{array}$


Cycle Notation

It can also be presented in cycle notation as:

\(\displaystyle e\) \(:=\) \(\displaystyle \text { the identity mapping}\)
\(\displaystyle p\) \(:=\) \(\displaystyle \tuple {1 2 3}\)
\(\displaystyle q\) \(:=\) \(\displaystyle \tuple {1 3 2}\)


\(\displaystyle r\) \(:=\) \(\displaystyle \tuple {2 3}\)
\(\displaystyle s\) \(:=\) \(\displaystyle \tuple {1 3}\)
\(\displaystyle t\) \(:=\) \(\displaystyle \tuple {1 2}\)


Also see

  • Results about the symmetry group of the equilateral triangle can be found here.


Sources