We have now seen patterns with each of the frieze group symmetries.

translation:

horizontal mirror reflection & translation:

vertical mirror reflection & translation:

glide reflection & translation:

½ turn rotation & translation:

They all have translations, and all but the simplest have an additional generator of either a horizontal mirror, vertical mirror, glide reflection, or ½ turn.

Let’s clarify what we’ve been talking about and coloring...

The frieze patterns illustrate the frieze groups. These groups contain symmetries, not patterns - the patterns just help us see them.

For example, vertical mirror reflections and translations are symmetries in a group that can be seen with the patterns:

And we can come up with infinitely more pattern designs to illustrate it.

This is the case for all of our pattern groups. As long as a pattern has units

Where applying the same symmetries to any unit leaves the entire pattern unchanged,

Then the pattern illustrates the same group as any other patterns with the same symmetries.

½ turn rotation & translation: