The rotations we have been finding for shapes are symmetries of these shapes - they are transformations that leave the shapes unchanged. When shapes have the same symmetries, they share a symmetry group.

Giving names to the groups that our shapes share will help us talk about and play with them later. We can call the group with 2 rotations C2, and call the group with 3 rotations C3, call the group with 4 rotations C4, and so on...

These groups are called the cyclic groups.

Check in: Which shapes illustrate C6?

Our shapes help us see our groups, but the members of the groups are the rotations, not the shapes.

The rotations within each group are related to each other...

Another way to think about rotating a C4 shape by a 3/4 turn is to rotate it by a 1/4 turn and then rotate it again by a 2/4 turn.

Notice that the order in which these rotations are combined does not matter. For this reason we say the cyclic groups are commutative.

Similarly, for our C3 group, a 2/3 turn is the same as combining a 1/3 turn with another 1/3 turn.

Adding another 1/3 turn brings the shape back to its starting position - the 0 turn.