Frieze Groups

# Frieze Symmetry Patterns, Intro #2: Language & Notation

Symmetry Groups, of which Frieze groups are a subset, are spoken about using certain words. This second post of intro (here’s the first) will begin to explain the words that are used to describe the seven frieze groups. There are only 5 terms to learn. There’s any number of places where these are written out and described. I have nothing new to add, but identifying the words that describe the way symmetries grow seems like the only way to start.

In future posts I’ll draw back the curtain on these terms in ways that will make them seem richer than their definitions, and less confusing than their examples.

Here are the five terms that describe the ways that frieze groups grow:

• translation
• vertical reflection
• horizontal reflection
• glide reflection
• 180 degree rotation

Translation simply means repeating something without any changes. If I write the letter “R” like this “RRRRRRR” that’s translation.

Vertical reflection reflects a shape across a vertical line. A “W” can be seen as a vertical reflection of a “V.” This means when the shape is vertically reflected, the pattern grows horizontally. Sorry. That’s just how it is.

Horizontal reflection reflects a shape across a horizontal axis, like tree reflecting in water. Which means that when a shape is horizontally reflected, the first action of this pattern grows  it vertically. This takes some getting used to.

Glide reflection makes a copy of the original unit, slides it over, then horizontally reflects it. There’s no need to try to wrap you mind around this one until you see examples.

180 degree rotation is a half of a full rotation. If you are facing one way and you turn around to face what was behind you, that’s a 180 degree rotation. There’s more to know about this one, so much more that, before I getting to examples, I will writing yet one more introductory post focussing on 180 degree rotations. If you don’t understand how a 180 degree rotation can happen around different points, you will soon be lost, so read the next post.

Each of the seven symmetry groups contain translations, which is to say there is repetition of some or no variation of the original tile. The rest of the frieze symmetry groups are made up of combinations of the rest of the ways frieze symmetries grow.

There are many different systems of notations around symmetry groups.

I’ve put the main ones I’ve been seeing together on the page above, though there are more notational systems then just these. For awhile it drove me nuts not knowing how each of these notations corresponded to each other.

The first column has F’s, which I am just going to assume stand for frieze. There is nothing inherently descriptive about this notation, but it is kind of nice to be able to put the groups in a numerical order.

The second column is the Hermann–Mauguin notation (or IUC notation),. The “p” stands for plane, “g” is glide reflection, the 1’s are placeholders, and sometimes aren’t written out, “m” references mirroring, and I’m not sure what the 2 means. I did know but I forgot. I’ll make an edit when I figure that out again.

The third column appears to be the most scholarly method, call Orbifold notations. I wrote to Alex Berke, whose book Beautiful Symmetries will be hot off the MIT presses in March 2020, to demystify this notation for me. This was part of the response I got back:

“Here are my notes on the notation: http://www.beautifulsymmetry.onl/?pageName=notation.
The notation is nice because it can tell you which symmetries are in the pattern.
If you’d like to more deeply understand it, I recommend The Symmetries of Things, authored by the notation’s inventors.”

Each symbol corresponds to a distinct transformation:

• an integer n to the left of an asterisk indicates a rotation of order n around a gyration point
• an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a kaleidoscopic point and reflects through a line (or plane)
• an  indicates a glide reflection
• the symbol  indicates infinite rotational symmetry around a line; it can only occur for bold face groups. By abuse of language, we might say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation; the frieze groups occur in this way.
• the exceptional symbol o indicates that there are precisely two linearly independent translations.
That’s all I have to say tonight. I think it’s important to see these notations together. It was incredibly helpful to have the above rosetta chart of notations handy. The tough thing, though, is that there are variations that aren’t obvious within these terms, the most confusing of which I will be writing about in my third, and definitely last, introductory post to frieze groups. After the next post, I will be writing about the different groups, probably one at a time.

## 5 thoughts on “Frieze Symmetry Patterns, Intro #2: Language & Notation”

1. Hi Paula
What a fascinating topic. I’m really looking forward to reading the rest of your posts on this.
I love playing with symmetry. My great passion is calligraphy and I have created lots of pieces based on symmetry … playing with letters, numbers and symbols to make glide, rotational and mirror patterns (mostly in fabric and paper, but also ceramic, metal, etc).
It’s so lovely to be on the same wavelength as you. Your ideas are a wonderful inspiration.
Cheers from Tricia

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2. I’m focusing on the art not the math, she says. Great overview of the notation.

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1. There are so many notations that I left out. It’s wild how many systems of notation people have developed.

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3. Kevin Rowney says:

Very much appreciate you wading into the different notational systems. Feels like a lot of attempts to explain this material just presumptively assume a notational convention without really acknowledging that there are several different valid names for the same complex abstraction.

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