By looking for rotations and reflections, we can see when shapes share a symmetry group,

Or when they do not.

And now that we have groups with more symmetries, there are more interesting subgroups to find.

We can again use color to reduce the amount of symmetry a shape has. For example, a D6 shape has 6 mirrors and 6 rotations, but with color we can remove 3 of these mirrors and 3 of these rotations to reduce it to a D3 shape.

Alternatively, we could have reduced the D6 shape to a D2 shape.

This is possible because D3 and D2 are subgroups of D6. Similarly, D4 is a subgroup of D8, and D2 is a subgroup of both D4 and D8.

Check in: What are the symmetry groups for these colored shapes?

What happens when color is added to remove only mirrors and not rotations?

The dihedral groups have mirror reflections, while the cyclic groups do not. When these mirrors are removed, we can see the cyclic groups are subgroups of the dihedral groups.

Color can also take away a shape's rotations to show us subgroups with only mirror reflections.

Here is an example where a D4 shape is colored with 2 colors so that it has only 1 mirror and that mirror is horizontal.

Challenge: Find other ways to color the D4 shape with 2 colors so that its only mirror is the horizontal mirror.

After being colored in this way, the shape no longer has the symmetries of D4, instead it illustrates a subgroup of D4.

Challenge: Can you find different ways to color the D4 shape with 2 colors so that it has only 1 mirror and that mirror is diagonal?

Our D4 group has multiple subgroups that have just 2 symmetries. One of those subgroups is the group of symmetries with just the horizontal mirror and the 0 turn (the 0 turn is also known as the identity).

Challenge: Can you find the other subgroups of our D4 group that have just 2 symmetries?

There are multiple ways to color our D4 shape to reduce it to a shape with only a 0 turn, a ½ turn, and 2 mirrors. This is just our D2 group! Here is an example where we keep the 2 diagonal mirrors.

Challenge: Color the D4 shape to remove the diagonal mirrors but keep the horizontal and vertical mirrors.

Challenge: Is it possible to color a D4 shape to remove the horizontal mirror while keeping the vertical mirror and ¼ turn? (Hint: Think about group closure)

There is a relationship between the number of symmetries in our dihedral groups, the number of symmetries in their subgroups, and the maximum number of colors we can use to reduce our dihedral shapes to show those subgroups.

When we reduce our D4 shapes to D2 shapes, we reduce their number of symmetries from 8 (4 mirrors, 4 rotations) to 4 (2 mirrors, 2 rotations). This is also the case when we reduce our D4 shapes to C4 shapes: Shapes go from having 4 mirrors and 4 rotations to having just 4 rotations.

Challenge: In any of the cases where we remove half the symmetries of these dihedral shapes, what is the maximum number of colors we can use?