# Regular Triangle Symmetry Group Exploration

Objective: Understanding the finite symmetry groups.

## The square

Complete the multiplication table for D4, the symmetry group of the square.

$E$ (identity) $R$ (rotation 90) $R^2$ (rotation 180) $R^3$ (rotation 270) $M1$ (reflection) $M2$ (reflection) $M3$ (reflection) $M4$ (reflection)
$E$
$R$
$R^2$
$R^3$
$M1$
$M2$
$M3$
$M4$

## The equilateral triangle

Analyze the symmetry group D3 of the equilateral triangle:

1. How many elements are in this group?
2. What is $M1$ x $M1$ = $M1^2$? , $M2$ x $M2$ = $M2^2$? , $M3$ x $M3$ = $M3^2$?
3. What is $M1$ x $M2$? , $M2$ x $M1$? , $M3$ x $M1$? , $M1$ x $M3$? , $M3$ x $M2$? , $M2$ x $M3$?
4. How do rotations behave?
5. Can you spot C3 as a subgroup of D3? What is it?
6. Find all subgroups.
7. Write out a multiplication table for D3.

Handin: A sheet with answers to all questions.