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 “I” like this “IIIIIIIII” 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 it.

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.

Frieze-NotationsI’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…

Book Art · Book Artists

Other People’s Beautiful Books, created, designed and collected

Embroidered Book Cover
Embroidered Book Cover

A beautiful handmade book arrived in my mailbox yesterday.

It came all the way from Greece. Had lovely postage stamps on the package.

Can you believe how extraordinary this stamp is?
Can you believe how extraordinary this stamp is?

I wasn’t looking to purchase a book, but I stumbled across an image of this one on Pinterest and was completely drawn in. I could see that the binding was hand stitched

but the straight lines and the long curve on the cover also looked like the were stitched. I suppose I could have asked the seller about the stitching but I chose to buy the book and see for myself. Yes, all those line are embroidered with a cotton thread.

This book, and as it turns out, as well as many others, are created by Chara Dimopoulou. There’s a wonderful interview of Chara on the Etsy Greek Street Team site, which is written in both in Greek and English.

Chara Dimopoulou, images and desk
Chara Dimopoulou, images and desk

She mentions using a bodkin to make her holes. Hadn’t heard of that tool. Would love to know more. Did some on-line search, but there’s lots of bodkins so I’m still not quite sure what she’s using to make holes, and which holes she is using the bodkin to make. I can’t help wonder about tools.

~Addendum: Chara emailed me and told me that what she uses is a tiny awl, not a bodkin, as was translated. Glad this error showed up, though, as now I know what a bodkin is…and isn’t. ~

I was so excited about receiving this book that I made a little video showing the unpacking:


This will be the second book that I have with sewing featured on the cover that is now in my collection. The first one was given to me by my daughter for Christmas 2018.

Post-it note holder by Stefani Tadio
Post-it note holder by Stefani Tadio, Rochester NY

I spent a year admiring this bit of embroidery. This year I’ve been using it, allowing it to get dog-eared. Actually using it is a completely acceptable way of enjoying it, but it’s something I sometimes have a hard time doing.

As I was thinking of writing this post I saw that Mindell Dubansky put out an announcement about the beautiful book covers of Margaret Armstrong

Book Design by Margaret Armstrong
Book Design by Margaret Armstrong

Margaret Armstrong was a prolific designer of gold embossed book covers.

Book Cover and Spine Desgin by Margaret Atwood
Book Cover and Spine Design by Margaret Armstrong

Armstrong was actively drawing and designing book covers for fifty years, starting in 1890, which places her directly in the Art Nouveau era. I have to say that once I started looking at her designs, I descended right into her rabbit hole. So much beautiful work to see.

She didn’t just make covers for other authors’ books.

Book Cover by Margaret Armstrong
Book Cover by Margaret Armstrong

Look who the author is! Here’s one of her exquisite drawings from this book.

Indain Paint Brush by Margaret Atwood
Indian Paint Brush by Margaret Armstrong

Here’s a cover that isn’t gold embossed but is every bit as lovely as the others. Or maybe there is some gold embossing here. Not sure.

Bookcover Design by Margaret Armstrong
Book Cover Design by Margaret Armstrong


Looking at these lovely covers I went looking for the books that make up my small collection of gold embossed covers.

Darkness and Daylight in New York
Darkness and Daylight in New York

“My small collection” means two books. I actually enjoy reading the stories of in Darkness and Daylight in New York, though they can be rather brutal to read. Not a good chapter for many people during the era this was written.

Great Composer, Elson
Great Composer, Elson

Haven’t read this one at all. For me, it’s all about the cover.

So many beautiful books. No, I don’t think they are going to ever become obsolete.




Art and Math · Arts in Education · Math with Art Supplies

Thinking Hands


Oh look, that’s me. For once, I’m not the one taking the picture!

Third and last post in this group of postings that are meant to help me clarify and remember where I am at in my thinking about the work that I do with students in schools.

After years doing bookmaking projects to make with children, I realized that many of the art and design skills I use every day align with some of the skills that mathematicians aim to develop. Part of the reason this alignment caught my attention is that I have a great affection for the mathematical thinking that I want to encourage.

Rotating Shapes after school with third graders
Rotating Shapes after school with third graders

In the relatively short time I am in schools with students, I hope to have a positive influence. My experiences and interests have led me to an unusual place where I can use colorful, artful materials to help kids create projects that enrich mathematical thinking. My place isn’t to teach art or teach math, but rather to plant seeds of engagement and excitement.

It seems to me that children intuitively understand concepts that are recognizably both artful and mathful. More and more, my thinking is centered around how to engage and encourage that which is already inside of students.

Spontaneous Scaling of Origami Pocket by First Grader. I see this happen frequently.
Spontaneous Scaling of Origami Pocket by First Grader. I see this happen frequently.

For instance, kids absolutely understand the idea of scale. They realize that their hands are the same, but smaller versions, of dad’s hand. Same with their shoes, their shirts, everything in their world.

Scaling shapes to make decorative border
Scaling shapes to make decorative border

There is no room in the school day for formal study of scale until the intuitive connection to it seems to have long disappeared. Turns out that scale doesn’t only have to do with making large models smaller, but it also is intrinsically connected to relationship thinking, predictive thinking and to the recognition of trends.

Symmetrical frame for a drawing of outer space by a first grade student

Discovering that children are naturally inclined to embrace symmetry has been another exciting area for me to explore with kids. When making books or other structures with students, there is nearly always symmetrical folding going on. I have choices when I teach folding: I can introduce what I do as step-by-step directions, or I can nudge the  students to see the symmetry of what’s going on so that they can predict for themselves what the next fold will be. The latter way gets them to see the project in a more global way, draws them in because they have understanding which includes them, rather than being like a little robot that is being programmed to do this then do that.

Doing some spontaneous symmetry during a few extra minutes at the end of a kindergarten session
Doing some spontaneous symmetry during a few extra minutes at the end of a kindergarten session

Symmetry is deeply embedded in math thinking, so I have been talking to children about connecting symmetry to what they are learning right now in math. Specifically, I talk to them about how when they are looking at an equality, such as 5+3 = 8, that this expression is balanced on both sides. It can also be understood as 5+3= 4+4. If I add 6 to one side of the equation, then I have to add 6 to the other side so that the symmetry of the equation remains true. Talking to students about equations as balanced forms just might help them, later on, when they will have to maintain balance in an equation to solve for x.

Symmetry, Scaling and Fractional Thinking all on display
Symmetry, Scaling and Fractional Thinking all on display

As far as I can tell, the only time symmetry is formally taught in elementary school it’s part of the examination of lines of symmetry in regular geometric objects. I like to be able to at least offer hints that symmetry has richer applications.

Making shapes from other shapes, parts and whole
Making shapes from other shapes, parts and whole

Children seem to have an innate sense of parts that make up the whole, which seems antithetical to the reality that teaching fractions is unfathomably difficult. Is it possible, though, to focus on having students work fractionally from a very early age, way before we introduce the numbers that describe the fractions?

Hexagon, Cube and Thing made from equilateral triangles
Hexagon, Cube and Thing made from equilateral triangles

Playing with blocks was one of my favorite activities as a kid. I certainly noticed halves, fourths, and wholes, but I didn’t make this connection between the blocks and fractions until I was much older. This makes me value not only exposing kids to artful mathematical thinking, but also, sooner rather than later, to help students connect their hands-on activities to the numbers.

Four Fold symmetry Pre-K
Four Fold symmetry Pre-K

There’s more I have to say about all this. but I reminding myself that I have to get to work getting ready for classes.

Am going to end with my list of ideas that I want to keep in mind, not all of which are explained here. Maybe I will get to writing about all these here and there through my teaching season. If not, at least I will have them here to keep me on track.

Art/Math concepts to explore with kids:

  • symmetry
  • Pattern recognition
  • Pattern Building
  • Scale
  • Inverses
  • Continuous magnitudes
  • Trends
  • Relationship thinking
  • Problem solving
  • Parts of a whole
A drawing of mine, that's artful/mathful
A drawing of mine, that’s artful/mathful

The first two posts in this set are here and here.