Combining the frieze group symmetries yields even more groups of patterns. For example, we can make patterns with glide reflection, vertical mirror reflection, ½ turn rotation, translation:

And color can again reduce the symmetry in these patterns so that they share the same symmetry groups as the simpler patterns we already colored.

½ turn rotation & translation:

glide reflection & translation:

vertical mirror reflection & translation:

Patterns illustrating the frieze group with all possible symmetries (glide reflection, horizontal mirror reflection, vertical mirror reflection, ½ turn rotation, translation)

Can be reduced to each of the pattern groups we have already seen.

½ turn rotation & translation:

horizontal mirror reflection & translation:

You can find the rest!

Check in: Can you see ½ turns in these patterns? What about vertical mirrors?

There are 7 frieze groups, and we have now colored all of them. There are no other ways to combine our symmetries to generate patterns that repeat forever in one direction. Surprised? Then try to generate patterns with different groups of symmetries by again starting with a single piece

Or use color to reduce a pattern to one with a combination of symmetries that we did not yet see, like a pattern with just horizontal mirror reflection, vertical mirror reflection, translation.

You will have to give up - it's not possible for a pattern to have just those symmetries because combining a horizontal mirror with a vertical mirror brings about a ½ turn rotation. This is just one example of how combining symmetries results in other symmetries, and brings us back to pattern groups we already saw.

Yet we can still find more repeating patterns. Frieze patterns are limited to repetition along one dimension, but wallpaper patterns do not have that limit.

When that limit is removed for the wallpaper patterns, the number of possible patterns and amount of symmetry within them grows beyond what we have colored.

Challenge: What happens when you start with a single piece and then transform it with both rotation and glide reflection? What other symmetries emerge?

Aside from our simplest frieze pattern group that has just translation, we can see how different types of symmetries can be used to generate the same pattern.

See, we can reflect across one mirror,

And then across another different mirror,

And keep reflecting across these alternating mirrors,

To generate a pattern that can also be generated by just one mirror and a translation.

This is an example of how various sets of generators - two different mirrors versus one mirror and a translation - can be used to generate the same pattern group.

Challenge: For each of the frieze pattern groups, what are the various sets of symmetries that can be used to generate the entire pattern group?

translation:

horizontal mirror reflection & translation:

vertical mirror reflection & translation:

glide reflection & translation:

½ turn rotation & translation:

vertical mirror reflection, glide reflection, ½ turn rotation & translation:

horizontal and vertical mirror reflection, ½ turn rotation & translation:

(click on the patterns to regenerate them)