Regular Triangle Symmetry Group Exploration

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Time-40.svg

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:

Isometries-triangle.png
  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.