Group Actions and Orbits

Review: Symmetries of a Square

Recall from chapter one that the symmetries of a square are the eight transformations that can be applied to a square in order to produce another square in the same position as the original. These transformations, known as the D4 group, are comprised of four rotations, denoted "r" and four reflection, denoted "s". In the original example, the square has been colored to make the transformation more obvious.

Click here to see the original demo of symmetry of a square.

Focus on one index in the demo and try the various transformations. If you focus on index 1, you will notice that only 7, 9, 3 and 1 ever occupy this space. If you focus on index 5 (the middle area), you will notice that it never changes. The orbit of any one of the areas is made up of the areas that can be obtained by applying elements of D4 to the square. Thus, the orbit of 3 is {1, 3, 7, 9} and the orbit of 5 is {5}.

You can also think about orbits in terms of the complete square. The orbit of the square under the action of D4 is the collection of transformed squares that are made by applying elements of D4 to the original square.