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) | R2 (rotation 180) | R3 (rotation 270) | M1 (reflection) | M2 (reflection) | M3 (reflection) | M4 (reflection) | |
|---|---|---|---|---|---|---|---|---|
| E | ||||||||
| R | ||||||||
| R2 | ||||||||
| R3 | ||||||||
| 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 = M12? , M2 x M2 = M22? , M3 x M3 = M32?
- 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.
