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.