Frieze Groups

The frieze groups can be seen in patterns that repeat infinitely in opposite directions.

A page cannot do these patterns justice. It cuts them off when really they continue repeating forever...

Consider the smallest repeating piece of this pattern as a unit.

We can see the entire pattern can shift over by this unit. Each piece shifts on to an identical piece and there is always more behind to replace what was shifted,

So that the shift leaves the entire pattern unchanged. Such is the nature of infinite repetition...

This shift is a symmetry called translation.

Translations are the only symmetries in our simplest group of frieze patterns, so this group can be generated by translations alone.

We can see this by starting with a single piece

That is copied and then translated

Again and again...

...An infinite number of times...

To result in a pattern with translation as a symmetry that leaves the entire pattern unchanged.

Check in: Can you see the translations in these patterns? Can you extend your imagination to see these as infinitely repeating patterns that repeat beyond the page borders?

Like any of the symmetries we have seen, translations can be combined and the result of the combination will still be a symmetry. We can see this by combining a translation with another translation so that in the same way a pattern can shift over by 1 unit and remain unchanged, it can also shift over by 2 units and remain unchanged.

We can keep combining translations to see larger and larger shifts...

Or we can use color to take them away.

By coloring every other unit in this pattern, we can double the shortest possible distance of translation in the pattern from 1 unit to 2.

Now only shifting by an even number of units leaves the pattern unchanged in appearance. The pattern still repeats infinitely, and there are still an infinite number of translations that will leave it unchanged. By adding color, we took away ½ of its translations, but ½ of infinity is still infinity.

Challenge: What is the inverse of a translation that shifts our pattern a unit to the right?