Frieze Groups

Frieze Symmetry, intro #3: Square Bases & Gyration Points

I am going to be writing about what’s missing from everything I’ve looked at about learning the seven frieze groups.

It  has to do with two tools: square bases and gyration points.

To get started, I emphatically want to say that the easiest way to understand the frieze groups is to make them yourself.

A good way to construct frieze patterns is to move around simple paper shapes.

A note about simple shapes:  For the purpose of learning frieze patterns, the unit shapes should not have lines of symmetry which are parallel to or perpendicular to the horizon.  I’ve created a file of shapes that can be altered so that their horizontal and vertical lines of symmetries are disrupted. This can be done by slightly tilting the shapes, cutting out bits to make regular shapes less symmetrical, or strategically adding color or other shapes to the shapes.  A standard circle paper punches is also a good tool to use for the purpose of disrupting the symmetry of a regular shape. For examples, see the image below

It’s not that using symmetrical shapes for frieze pattern is a bad thing. Just the opposite is true. It’s just now, when first learning about the different symmetry groups that it’s preferable to avoid the confusion that can come with using shapes which have their own embedded horizontal and vertical symmetries.

The way to handle the unit shapes is to mount each of them on a square base, preferably one that is transparent. Skeptical? You will be convinced. I am certain of it. Eventually you will be able see the squares in your mind’s eye, but to start out, the squares are like training wheels.

For the rest of this post my shapes will  be attached to a square base. Think about this:

  • it’s the underlying base that moves horizontally to create the frieze.
  • It’s the shapes that are attached to the base that create the design.

My squares are made from a mostly transparent drafting film.

Here are some units glued to the square bases:

The only rule with the square base is that one side always remains parallel to horizon line of the frieze. This infers that the two of the sides of the square are always perpendicular to the horizon line.

Overlaps, extreme spacing, vertical reflections, horizontal reflections, and 180 degree rotations. which are the defining characteristics of frieze patterns, will work out just fine with the square base, as all of these changes do keep a side of the square base parallel to the horizon line.

Repeated Tilted Square
Repeated Tilted Square

If a design is made with an asymmetrically tilted square, the square base square doesn’t tilt, rather just the repeating unit is tilted.

If the shape overhangs the base, that’s not a problem. Let it overhang in any direction.

If the shape is so big that it obliterates the square, use a bigger square.

Now it’s time to talk about gyration points.

As I sifted through lots of examples of friezes that didn’t seem to correspond to each other, I finally realized that recognizing the role of gyration points in frieze patterns is essential. Gyration points are like the secret that nobody talks about

If you don’t know what a gyration point is, well, it’s a term that I also just learned.  The concept of what it is, however, has been one of my favorite visual playthings for many years. I just didn’t know what it was called.

Simply put, it’s a point around which an object is rotated which is not  located in the center of the object.

For instance, the sun is a gyration point of the earth, since the earth rotates around it.

A gyration point is a hard concept to immediately internalize and accept as important. Don’t worry if it feels fuzzy right now. If you follow along with these posts it will become clear.

Here’s some visuals that are intended to get things started.

In the image above, the same shape is rotated, but each pair is rotated around a different center, each creating a look that is different than the rest.

Now here are these same pairs, each creating a frieze pattern:

Friezes symmetry with gyration point
Friezes symmetries with gyration points

Varying the gyration point creates completely different visuals. Be sure to squint when looking at the bottom frieze and you will see a zig-zag pop out.

Here’s a video, repeating everything I’ve just written. I think that seeing the transformation happen on video makes a more convincing case for using a square base, and better illustrates gyration points.


The first two posts in this introduction are :

With any luck, I’ll be posting about the seven frieze groups before too long. It’s a linear process. 🙂

Frieze Groups

Frieze Symmetry Patterns, Intro #2: Language & Notation

Horizontal reflection, vertical reflection, translation
Horizontal reflection, vertical reflection, translation of that little square on the upper left


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
vertical reflection, translation
vertical reflection, translation

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:
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.
Frieze Book Cover 2002
Here’s me in 2002. I promise I am not talking to this young man about glide reflection. What a great border design he created!






Decoration · design · Frieze Groups · Math and Book Arts

Frieze Symmetry Patterns, introduction #1

Frieze border Book Cover 2002
Frieze Border for Book Cover 2002

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.

Frieze Storybook Decoration 2009
Frieze Storybook Decoration 2009

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.

Example of Glide Pattern

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…