We saw that the cyclic groups are commutative. The order in which we combined rotations did not matter - the result was always the same. The dihedral groups are not commutative. We can see this in our D4 shapes: rotating our D4 shapes by a ¼ turn and then reflecting across a vertical mirror,

Is not the same as reflecting across a vertical mirror and then rotating by a ¼ turn.

Challenge: Show that D3 is not commutative. Find 2 symmetries of our triangle where transforming the triangle by one symmetry and then the next is not the same as applying the transformations in the reversed order.

Challenge: We showed how the ⅓ turn and a vertical mirror could be used as generators for D3 and generate all of the other mirrors of a regular triangle. Show how the ¼ turn and a vertical mirror can be used to generate all of the other mirrors of a square.