Definition:Symmetry Group of Equilateral Triangle

Group Example

Let $\triangle ABC$ be an equilateral triangle.

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.