Regular Triangle Symmetry Group Exploration

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Objective: Understanding the finite symmetry groups.


The square

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

<math>E</math> (identity) <math>R</math> (rotation 90) <math>R^2</math> (rotation 180) <math>R^3</math> (rotation 270) <math>M1</math> (reflection) <math>M2</math> (reflection) <math>M3</math> (reflection) <math>M4</math> (reflection)
<math>E</math>
<math>R</math>
<math>R^2</math>
<math>R^3</math>
<math>M1</math>
<math>M2</math>
<math>M3</math>
<math>M4</math>

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 <math>M1</math> x <math>M1</math> = <math>M1^2</math>? , <math>M2</math> x <math>M2</math> = <math>M2^2</math>? , <math>M3</math> x <math>M3</math> = <math>M3^2</math>?
  3. What is <math>M1</math> x <math>M2</math>? , <math>M2</math> x <math>M1</math>? , <math>M3</math> x <math>M1</math>? , <math>M1</math> x <math>M3</math>? , <math>M3</math> x <math>M2</math>? , <math>M2</math> x <math>M3</math>?
  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.