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. 🙂

summer art/math

Making Game Cards with Kindergarteners

5 year-olds in the summertime could use a bit of number play. Get them invested, make it a game, get them coming back for more.

Numbers PDF   

Kent Haines wrote about a card game that he played with his daughter using a deck of cards. No point in spending two bucks on a deck of cards when there’s twenty 5 year-olds looking for something to do, right? We started out making our own deck by coloring in numbers 1 – 10. Of course I couldn’t find the perfect typeface so I made one I liked.

We did the coloring as sort of side project to other activities, so a took a few sessions to color enough for forty cards, which was my goal. Full disclosure: I probably did about 10 of them myself.

Next, kids put the “right” number of plant items with the numerals. I photographed them, then put them through my graphic program.

I printed and cut out the cards at home, then we played!

My rules for the game were a bit different than Kent’s, though we both start our games with laying out ten cards (two rows of five each), and the ultimate goals of our games are the same, which, as Kent points out, is to give children practice with counting, cardinality and comparing numbers.

In my version of the game I make sure that all ten numbers are in front of the child, but hidden, and in scrambled order. A random card gets turned over then it’s up to the child to determine where it goes in the number line up. The card that’s now bumped out of its own starting place gets turned over and the child decides where it needs to go, and so on.  If it turns out that two cards just switch places so that there is now no new card looking for a new place, the child can turn over a random card. This sounds confusing until you play, then it makes perfect sense.

There is no winning or losing, just finishing. Sometimes the kids played in pairs, some were slow and thoughtful, some were super fast, but they all loved the game. YAY! And they recognized the cards that they had in hand in making, and loved this connection. Too much fun.

Since we are moving the cards around I’m calling this version of Kent’s game “Recycle.” So PC.

Here’s a couple of video clips:


Addendum April 10, 2020


Here’s the a PDF of deck of recycle cards 2 types that we made, that you can print out to use or to use to make your own.

geometry and paper · holiday project · Ornament · Paper Ornament · Uncategorized

Round-up of Holiday Season Projects

A Bevy of Paper globes
A Bevy of Paper globes Directions at

Making things out of paper seems to be something we do during the holiday season.

Here’s some projects that I’ve written about with links to the full posts that explain them, that seem appropriate for the season.

This first one, the Spiraling Ornaments was surprisingly well-liked. All year, when I visited different places, I’d see ones that people I know had made.

Video tutorial (not mine) for Kaleidocycle



Here’s a template for a kaleidocycle. Not particularly holiday-ish, but fun and colorful, folds into something like the image below. More about this at



Next, directions for a six-sided snowflake. My big tip is to use paper napkins, as they already are the right shape: no extra prep needed! Also, paper napkins cut quite easily. They are perfect for snowflakes.

how to make a paper snowflake


If you want to understand how the cuts of your snowflakes affect the final design, see below:



Festive Jumping Jacks are quite fun. I’ve made these with kids just a few times, as all the knot tying makes this an intense project for anything more than a small group, but so worth the effort!

Jumping Jumping Jack

To work out how to make these you might have to look at a few posts, which are all listed at

The stars below are tricky to make, until you get the hang of them. I still have the ones I made on display from last year.

The original post contains a good bit of discussion about the geometry embedded in these shapes.

Finally, making little books with stories or messages is always worth doing.

Origami books made from a multiple folded papers, to create a Star Book and a Cascading Book, aka Origami Caterpillar Book
Origami books made from a multiple folded papers, to create a Star Book and a Cascading Book, aka Origami Caterpillar Book

Here’s a post that can get you started on some simple books to make with kids

For more an overwhelming amount of other book ideas, check out what I’ve tagged as making books with children

There you have it. Enough to do to keep you out of trouble at least until January.

8 1/2" x 11" Book Making · Accordion Books · Art and Math · geometry and paperfolding · Making Books with children · Making books with elementary students

Five Days of Summer Workshops with 3rd and 4th Graders

Pop Ups with more Pop ups
Pop Ups with more Pop ups

In the summertime, when school is not in session, I’m on my own in terms of deciding on what kinds of projects that I want to teach in workshops. Last week I taught for five days  at the local community center.  My sessions with the kids were 40 minutes long, and although I prepared for 30 rising third and fourth graders, there was no telling how many students would attend each day. I had originally thought I would make a plan for the week, but quickly realized that it was more satisfying to create projects each day based on what I found interesting in the children’s work from the day before.

Making Pop-ups in an Accordion Structure
Making Pop-ups in an Accordion Structure

My own goal for the week was to do explorations with shapes and symmetry.  On Day 1 we made a four-page accordion book and did some cut-&-fold to make pop-ups. The students were amazing paper engineers;  With impressive ease, they created inventive structures.

Pop-up Worksop
Pop-up Worksop

There were plenty of counselors in the room, and from this very first project, these counselors joined right in with creating their own projects.

Overlapping Rotated Squares
Overlapping Rotated Squares

I was so impressed with the students’ folding skills that the next day I helped them create an origami pamphlet that contained more pop-ups, as well as some interesting other cut-outs. What turned out to be the most interesting work on Day 2 was how much the kids liked the little bit of rotational symmetry that I encouraged them to do: I gave them each a square of paper, asked them to trace it on to the cover of their book, then rotate it and trace again.

summer squares 3

These students like the shapes created by shapes, so the next day I brought in a collections of shapes and asked them to arrange tracings of these shapes on a piece of heavy weight paper, which was folded in half.

Tracing Shapes to Create  More Shapes
Tracing Shapes to Create
More Shapes

Students seemed to enjoy creating these images.

summer shapes 2

After they created the outlines they added color.

Colored Shapes
Coloured Shapes

When the coloring was done we folded the paper, and attached some pagesto the fold so that the students had a nice book to take home. The kids seemed to like this project and made some lovely books, but I ended up  feeling like there wasn’t anything particularly interesting going on with this project in terms of explorations of building with shapes. So …

Building Stars and Hexagons with Regular Rhombuses
Building Stars and Hexagons with Regular Rhombuses

…the next day I brought in colored papers that were printed with rhombuses, as well as some white paper printed with a hexagon shape. Each student filled in their own hexagon with 12 rhombuses.

Making a Hexagon with a Star in the Middle
Making a Hexagon with a Star in the Middle

My plan for this project was to have each student make their own individual hexagon then put them all together on a wall so that it would be reminiscent of a quilt.

Paper Hexagon Quilt
Paper Hexagon Quilt

Here’s our paper quilt made from 22 hexagons!

The next day, Day 5, was my last day at this program. I liked the engagement with and results of how the students worked with shapes when they were given structure. There’s a balance that I try to honor of providing structure while allowing individual choices. For my last day, then, I decided to give the students a page that I created that is based on the geometry that uses intersecting circles and lines to create patterns.

A work in progress by one of the couselors
A work in progress by one of the counselors

If you look closely at the photo above you’ll see many different lines and curves overlapping and crisscrossing.

summer geometryI asked students to look for shapes that they liked, to use the lines that they wanted to use, and to ignore the lines that they did not want. It was interesting to watch how the students worked; I was particularly interested in seeing how some children chose to start looking at designs starting in the center, while other children gravitated to the outside edges first.

summer geometry 8

Some students filled areas with color, while others were happy to make colorful outlines of shapes.

summer geometry 2Some drawings were big and bold.

summer geometry 5

Some drawings were delicate and detailed.

summer geometry 4I think that every one of the teenage counselors sat and made their own designs, right alongside of the students. Actually, I think that my favorite unexpected outcome of the week was how involved the teenagers got with the projects.

summer geometry3

This last project of the week was my own personal favorite (though the quilt project runs a really close second). I had never done anything quite like this before with students, and was really surprised to see how much they enjoyed this work, and how differently they each interacted with the lines and curves. This kind of surprise is what’s so great about summertime projects.