Frieze Groups

For millennia, people have used patterns and ornamentation to enhance the natural and human-made world around us.

I think one reason we love patterns so much is that they are a sweet spot between chaos and order, or randomness and structure. I believe that we are naturally drawn to both extremes, so structures that have elements of both have an innate tension that we find ourselves drawn to: where is the harmonious order? Where is the turmoil of chaos?

These questions lead naturally to the study of symmetry, which generally speaking looks at how similar objects combine, or are changed, in ways that obey particular rules, but nevertheless lead to complexity that we wouldn’t have guessed at beforehand.

Suppose that you start with a standard bunch of symmetry operations constrained to one direction in the plane (that is, creating a strip): translation (moving), rotation, reflection, and glide reflection (moving and reflecting in one step). Now start with a piece of an image, and apply any of these operations. Take the result and apply that operation again, or any other. Repeat as much as you please. It’s easy to prove, but still surprising, that no matter how you choose to combine these operations, you can only produce seven fundamentally distinct patterns, called the frieze groups. Here are examples of these seven types of patterns.

I drew and decorated these patterns by hand using Painter.

I wrote about these patterns in one of my columns for IEEE Computer Graphics and Applications. You can read that column here, or browse all of them here.