Our regular triangle has 3 unique rotations and 3 unique reflections, a square has 4, and we can find shapes with 5, 6, 7, and keep going...

Shapes that are not regular polygons can have these same symmetries.

We already saw how shapes that share the same set of symmetries share a symmetry group, but then we only considered rotations. Symmetry groups can have both rotations and reflections.

We’ll call the symmetry group that contains the 3 rotations and 3 reflections of a regular triangle D3. And we’ll call the symmetry group with the 4 rotations and 4 reflections of a square D4, while we call the symmetry group with 5 rotations and 5 reflections D5, and so on...

This series of groups is called the dihedral groups. Again, these groups contain symmetries, not shapes - the shapes just help us see them.

These shapes that share a symmetry group may look different, but when viewed through the lens of group theory, they look the same. Only their symmetries - the rotations and reflections that leave them unchanged - are seen.

Check in: Which of these shapes have 8 mirrors?

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.

Challenge: What is the result of combining two different mirrors?

Is the result a reflection or a rotation?

Is this always the case?

(go ahead and draw more shapes)

For our D4 group, we can see that the result of applying a vertical mirror and then a horizontal mirror is a ½ turn.

Challenge: Can you find 2 mirrors where applying one and then the other results in a ¼ turn in our D4 group?

Is it possible to use 2 mirrors to generate all of the symmetries of our D4 group? What about our other dihedral groups?

Here are some shapes for you to puzzle over.