# Frieze Symmetry Patterns, introduction #1

I call them borders. For decades I’ve been creating lessons for young kids on ways of creating geometric borders in the books that I make with them in the classroom. Kids love these lessons. They sit quietly, raptly attentive, and can’t wait to get to work.

Long overdue, I thought I’d take a closer look at these linear repeat patterns. Thought I’d have it all figured out in an afternoon. That was a couple of weeks ago. Now, deep in the rabbit hole, I’m reporting back. What was going to be one post will be many posts. It’s not that any of this is difficult, but there’s much going on that’s not evident with a cursory look or a single example.

What’s just as challenging as deciphering the patterns one can make is deciphering the notation that describes them. There are three separate systems of notations that I will be listing, though these aren’t the only systems. Notation will be filling up my next post.

Here’s the first amazing fact about a pattern that grow along a horizontal strip, which I will henceforth refer to as a Frieze as in Frieze Groups or Frieze Patterns, or Frieze Symmetry:

#### There are only seven possible ways to create a frieze pattern.

Any frieze pattern you see will be some configuration of only one of seven ways of manipulating a base unit.

Doesn’t seem like this could be true, and if it is, doesn’t seem like it would be too hard to figure out.

It is true, there are only seven possible ways that frieze symmetry happen, and it is not easy to grasp. Some symmetries are easier than others, but each of the seven ways have their quirks that need to be addressed, which is something that I will do in one post after the other until I am done.

Here’s a list of the main resources I have been using:

### My resources:

Beautiful Symmetry by Alex Berke

Frieze Group, Wikipedia

Talk: Frieze Group, Wikipedia

Gait Sequence Analysis Using Frieze Patterns, Table 1, Yanxi Liu

Gait Sequence Analysis Using Frieze Patterns, Table 5, Yanxi Liu

Geogebra Apps by Steven Phelps:

To be continued…