This is my favorite one of the Frieze Groups. I like it so much because, after being confused about it for quite some time, I finally saw that it’s just like a sine wave.
If you are happy with the explanation above and have an adverse reaction to being confused to the point that your head feels like scrambled eggs in a pressure cooker, then skip the next paragraph.
In many places this frieze group is described as having more actions, specifically, vertical reflection lines, glide reflections, translations, 180° rotations and horizontal reflection. There may be a choice in the description that says do a vertical reflection or a horizontal reflection. Or maybe the description will describe fewer actions. So what’s going on? This is what’s going on: first, all frieze symmetries contain the translation action, which simply means that the original unit is being repeated. What’s helpful to understand is that when there’s horizontal reflection there is also glide reflection. Then here’s the next piece to decipher: Pick two actions from this list.
- Vertical Reflection
- Glide Reflection
Any two of the above actions gives you the third one, so it’s not a matter of doing either this or that or all three, it’s just choosing which way to think about it.
While the appearance of the final design doesn’t’ necessarily look just like a sine wave, the pattern of its creation is the same.
You’ll notice that some of the images above look straight, not like the way you’d think of a sine wave. If you don’t understand what happened to the waviness, please go back and look at my post about frieze glide reflections. In fact, more often than not, this frieze group presents itself without the wave-like action being obvious, so it’s a good idea to get clear about this.
The video might help.
Just for good measure, here’s a pattern block showing this frieze symmetry group.