Our patterns can have more symmetries than just translations.

Reflecting a piece across a horizontal mirror before translating it,

Generates a new pattern, with more symmetry than the one before.

The pattern still has translations - it can still shift over without changing.

But it also has a horizontal mirror: The entire pattern can reflect across the same mirror that transformed our first piece, and appear unchanged.

Patterns can have vertical mirrors as well.

These mirrors shift over with each repeated translation, so once a pattern has one vertical mirror, it has an infinite number of vertical mirrors.

Twice that many, really.

Even though we start with a vertical mirror on one side of each piece, as the pattern repeats, another different vertical mirror shows itself.

Check in: Can you see the mirrors in the following patterns?

All of the mirrors in our frieze patterns can be removed with color.

With color, we can reduce the patterns so that translations are their only symmetries.

Why can we do this? This brings us back to subgroups.

Our patterns with vertical mirror reflections belong to a symmetry group with translations and vertical mirrors.

vertical mirror reflection & translation:

Naturally, the group with only translations is a subgroup.

translation:

The same goes for our patterns with horizontal mirrors. Color can remove their mirrors as well, and reduce them to patterns with only translations.

Frieze patterns can also have ½ turns as symmetries.

We can see how they are generated by looking at a single piece that rotates by a ½ turn around a point,

before translating.

The entire pattern can then be rotated by a ½ turn around that rotation point.

And just as we saw for vertical mirrors, once there is one point of rotation, there are infinitely many more, on either side of each piece,

That the entire pattern can rotate around, yet remain unchanged.

There is another type of symmetry called glide reflection.

A glide reflection is a transformation that reflects across a mirror line at the same time as translating along it.

By continuing to translate or glide, a pattern with glide reflection is generated.

Glide reflections show themselves in other patterns as well. The patterns we generated with horizontal mirrors have glide reflections too,

And color can reduce them

To patterns with glide reflections only.