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. 🙂

Decoration · Making Books with children · Making books with elementary students

The Biography Project, with colorful borders and letters

book cover of Biography Project
The Biography Project Book cover, A bookmaking Project with Second Graders

During the last couple of weeks one of the in-school projects that I’ve been involved in is the Biography Project with Second Graders.   This project has been a real treat to work on, for many reasons, not the least of which has been the fine collaborative opportunities I’ve been able to be part of with the teachers.  At the time that I proposed the structure I wasn’t sure what to do with the cover page. Between watching what most motivated the students, as well as integrating ideas from the teachers, the covers were this blend of typography, student made decoration, and a silhouette.

I like framing the title of the book color. I had noticed that students were incredibly enthusiastic when I had given them outlines of letters to fill in, which was the cover of one of the mini-books that is housed inside the pages of the big book.

Here’s one of the mini-books, which is a book which will contain clues about the person who the student is studying.

Here’s one of the Clues books, which shows the questions at the edge of the pages.  The  readers are given clues which will help them guess whose biography is written here. On the next page (which is not shown in this post) there is a writing and drawings which are relevent to the subject..

I’m always searching for  ways to get the job done (create a compelling book structure for students to use) while adding details that make the  process more individualized as well as being more enjoyable for students. Giving the students these outlines to fill in, both on the frame of the book and the outlines of the letters, was a fun addition to this project.

Here’s some more samples:


Right now I’m deep into the busiest part of the season that I visit schools. What I am going to try to do is post as I go along, with the intent of focussing on details that have captured my interest when working with the students.  What captured my interest this week was this way of providing blank spaces which students could enliven with color.


WOW designs

I have been working with many young children lately. My goal is create a project that pleases the children, the teacher, the school and me. With young children this means that the projects often start out highly structured. Even though I know that this is a good thing there is a part of me that just wants to let the children run wild with color. This, sigh, generally does not turn out to be a good thing.

Page Decoration by first graders

I struggle to find a place where I can let students loose with color and design, but still reign in their designs in a way that shows them a new and unexpected way of approaching their work.

For years I have been trying out different ways of introducing decoration. I feel like there’s been a good bit of success in working with letters of the alphabet. But I have found that it’s good to even simplify this approach. Whereas I used to demonstrate ways of using the letters Z,O,V,W,I,C,X, and S as decoration I recently decided to just try out using WOW and I….by the time I met with the third class of the day I was down to using just WOW.

Children love to watch an adult draw.I am always happy to stand at the board and draw for them. To demonstrate this technique I draw some W’s and O’s, emphasizing the decorative possibilities of these letter shapes, and then I draw WWWWWWWWW and say that sound….the insightful teachers know just how long to let the students imitate the sound before they ask their class to quiet down. But I like that they spend a long moment with the WWWWWWWWWWWWW and OOOOOOOOOOOooooOOOO’s because it seems that vibration they make gets deep inside of them, priming them for good work.

And, of course, the coolest thing is that, even with the structure that I impose…

… their designs turn out so differently from each other’s.


decorating the page, by a first grader

Books Made from one sheet of folded paper · Decoration · Making Books with children · Making books with elementary students · origami pamphlet

Individualizing with Color

Making Books with First Graders

Origami Pamphlet Books, fully opened

This past week I worked with three classes of first graders. My goal was to help the students create books which honor their writing. I want the books to be good-looking, dynamic and individualized. I have three 75 minute sessions to accomplish this.

Origami Pamphlet Made by Chase, First Grade
Origami Pamphlet Book by Chase

All students begin by making an Origami Pamphlet using the same color paper. No choice there. But I am able to give them choice in the decorative details.

One of my favorite decorative techniques is to ask the students to create designs with geometric shapes. Just the mention of color rivets students’ attention. I try to find a place to lay out their color choices attractively. I’ve figured out that making colors available to students in a carte blanche kind of way results in designs that descend into chaos. Now I am more orderly in the distribution of color. Perhaps I am delusional, but I try to convey the concept that there are advantages in practicing restraint.

For decorative accents, students choose four colors from my palette of Brite cover weight papers . These strips of paper are 5 inches x 1.25  inches (if I were in metric-land I would cut these stips to be 3cm x 12cm). Then everyone cuts SQUARES ONLY. They do this by creating an “L” with the strips, then cutting on the line that defines where the strips overlap.

Books by First graders

Students can use their squares as squares, turn them to become diamonds, cut them in half to create triangles, or cut them into lines. I do not allow them to explore any other options. This makes me feel mean, but I explain to these budding artists they can try out all sorts of decorative options on all their works for the rest of their lives, but, for right now I want them to do it my way so that they will learn a new technique. I promise them that although they are all getting the same instructions, that their books will each have their own look.

 Book page by First Grader

They work a bit on each page, then go back and add more after each page has been treated.

We use glue sticks to adhere the shapes to the paper. I bring in the 1.41oz (40g) size of UHU Glue sticks. Then I threaten students that the shapes will fall off the page unless they apply enough glue and pressure to the papers.

The result: Same But Different.

There are other decorative technques that the students use. But that’s another post.
To be continued.