# Regular Triangle Symmetry Group Exploration

From EscherMath

**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:

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

**Handin:**
A sheet with answers to all questions.