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…

geometry and paper · geometry and paperfolding · Glue · Math and Book Arts

Sunday Night in the Studio

Building a dodecahedron
In Progress

I’ve been juggling a number of projects lately, working out different problems, nothing yet fully resolved, so this is going to be a post that shows a bit of this and that. As I’m working little things that I want to share keep coming up, so here’s a glimpse at what’s going on here.

Printing paper
Making Copies

One thing that’s going on in the background is printing papers for a workshop that I will be teaching in a couple of weeks. I will need to make a few hundred copies, and I don’t want to leave this until the last minute, nor do I want to be fully out of the room as the copies come out of the machine. Doing these a few at a time

Cutting a Net

The main event of the evening is making a black 12-sided shape, specifically a dodecahedron. I made this template in Illustrator so that it will have certain details that I want, such as a door that allows access to the inside of the completed shape. The template has to be printed on a light colored paper.

I clip my black paper to the yellow with binder’s clips. First step is to press in the score lines, making sure that there is a surface that gives under the pressure of the stylus. For the stylus I use a glitter pen, because I like using glitter pens. Next step is to cut out the shape.

Black Paper Shape
Platonic Solid in progress

Here’s how it looks with cuts and score lines made.

Next I will be gluing colorful pentagons to the black paper. Pentagons were colored with Sharpies, copies were printed at the copy shop, and then I cut them out.

Blackening the Edges
Blackening edges

My wise friend Jane recommended that I go over the edges of the cut out pentagons, using an India Ink based marker. This is a Coptic Brush Multiliner.

Books, Boses, and Portfolios by Franz Zeier, Pages 86,87
Books, Boxes, and Portfolios by Franz Zeier, Pages 86,87

I read this great book by Franz Zeier a few years ago, and, ever since, I’ve been trying to bring my glueing skills up a notch. He is using straight PVA to glue his models together, which is what I am doing as well. Since my grain directions won’t be lining up perfectly I like using the straight, unthinned PVA, as it has so little moisture in it, and dries so quickly.  I have given up on using a brush with this project, as the glue dries quickly on my brushes, too.

My Modest Glue Station
My Modest Glue Station

I have found that it’s neater to use stiff piece of paper to do my gluing. First I pour a bit of glue in the bowl then dip an edge of my paper scrap into the glue.

Putting Glue on a Pentagon
Putting Glue on a Pentagon

I can put a really thin layer of glue down quickly, which is exactly what I want.

I love the look of the paper once the pentagons are glued down. Sort of hate to keep going.

I do keep going though. Such a friendly looking Platonic solid.

Now that’s it for tonight. Glad I don’t have to clean any brushes.





Arts in Education · Math and Book Arts · Math and Paper Folding · Math with Art Supplies

Peek-a-Boo Skip Counting for First-Graders

Peek-a-boo skip counting
What number is under the heart?

For weeks I’ve been burning through piles of papers and ideas trying to work out an engaging skip-counting project to make as part of a math-activities folder for first graders. Having just done a math activities folder with kindergarteners, which went really well, I’ve been wanting to do something similar for first graders. As I’ve also been doing math-with-art-supplies bookmaking projects with second graders, I’ve been keen to design something for the next grade up.

What I’ve  needed to get me going on this is a school to want me to create a project for them. A couple of weeks ago, late in the season, a school called me, asked if I had any time for them, and we struck a deal. We’re doing the project that I’ve been wanting to create.

There will be four hands-on projects in a folder that the students will be making. This post is about just one of the projects, one that supports skip counting, reasoning, and attention to numerical patterns.

Show me what under the butterfly, Oh, it's a 14.
Show me what’s under the butterfly, Oh, it’s a 14. Still not sure what’s under the heart.

Skip counting is a big deal in first grade. Not only does it set the stage to understand multiplication, it also is helps with learning to count money.

My work with second graders has piqued my intereste in skip counting. The projects we’ve been doing, which is making designs with “coins” that add up to $1.00, has been interesting in that I’ve noticed that even though a student can count by fives, you know, 5, 10, 15, 20, 25, 30, 35, 40….,, they have a really hard time doing this same counting by 5’s when you ask them to start at any number other than zero. So, if they have 25 cents plus two nickels they are at a loss as to how to proceed.

Show me what's under the flower. Oh, it a 6, Now can you tell me what's under the heart?
Show me what’s under the flower. Oh, it a 6, Now can you tell me what’s under the heart?


Maybe by now you’ve guess what is under the heart in the photo above. Maybe not. If you need more hints, I can reveal that there is an 8 under the star. This will likely finally be enough for you know know that there’s a 10 under the heart.

We’re not just counting by twos here. I’ve made a paper that slides under the windows that helps with counting by 2’s, 5’s, and 10’s.

I consider this to be an elegant design. One piece of folded paper for the holder, with a one piece of paper for four different number series. The little designs on the peek-a-boo doors are cut with paper punches, which I’ve collected over the years. The rhombus shaped window are made by folding the paper and cutting triangles on the fold.

One of my thoughts with this project is that  it can support students in practicing with going both forward and backwards with their skip counting. For instance, if they see two numbers, say 80 and 85, can they tell me the number that is before the 80 and after the 85? This takes some practice, some thinking, and reasoning, but if they can figure out what number is behind the hidden door, I anticipate the pleasure at solving this puzzle will delight them when the peek-a-boo door reveals the answer.

I do plan to share the template for this after I try it out with some real live first graders. To be continued.

Addendum 6/11/2019

Two classes of first graders made this with me. It went really well! To teach them to use it, I do the demonstration on the board, drawing doors that they could “look” behind for clues.

Here’s a video of what playing with this looked like:

Here’s a template so you can make these yourselves: skip counting first grade

Art and Math · Math and Book Arts

Kindergarten Folder for Making Math

Making Symmetry
Making Symmetry

There are some kindergarten teachers I’ve been working with for years. This year I’ve worked with them to create a math-centered book project for their young students. I launched this with a small class earlier this season then repeated it with about 4 groups, total of about 70 kindergarteners, this past week. It went well.

Actually I’m so delighted with how it went that it’s almost embarrassing.

Making the folder
Making the folder

We made a folder out of a long strip of paper, 35″ x  7.5.”

Folder with four compartments
Folder with four compartments


I put some score lines in to help these 6 years olds get started but they made most of the folds themselves. I make a big deal about how to fold paper.

The folder is basically a four page accordion, with pockets for a different math activity in each of the pockets.

The first pocket has a paper with peek-a-boo flaps to help kids visualize the composition of groups of numbers. This was an  unusual folded structure, but they caught on really quickly, as you can see in the video clip below.

After the folding comes the cutting


Cutting the peek-a-boo flaps
Cutting the peek-a-boo flaps

Then the coloring…

Compositions of the number 4
Compositions of the number 4

…finally they used these images to become more familiar with number compositions. We made these cards for the numbers 2, 3, 4 and 5.

Here’s how it looked watch kids use these to learn their number facts:

Okay, so that was for one pocket.

In another pocket there were squares that the students cut out. I used these to talk about symmetry.

Symmetry with cards
Symmetry with cards

Where one student placed a card on their side of the midline (pencil)  another student mirrored the placement. Seeing symmetry is important in math as students as it is a non-numerical way for them to experience the balance that an equation like 2 +3 = 5 expresses.


I extended this symmetry activity beyond the cards in their pockets. We used items around the classroom to create symmetrical designs, something my twitter community liked and retweeted generously.

We also did a project using beads, reminiscent of an abacus, to make groups of 10.

Separating10 beads into groups
Separating 10 beads into groups

The idea here is to give kids another way to interact with ways to make groupings of 10, contributing to their fluency and grasp of combinations of numbers.

Bead counting book
Bead counting cards


Finally, we did a fortune-teller, aka chatterbox, which many of us made when we were children.

Fortune Teller, Chatterbox
Fortune Teller, Chatterbox


Of course the insides were math themed, using their sight words, too.

Fortune Teller template
Fortune Teller template

Here’s a little clip of the kids playing with these. They absolutely loved this toy.

At first I had a hard time trying to teach this structure to kindergarteners. Once I realized that if I taught it after I worked with them on the symmetry part of this project, the folding would then make more sense to them. It turned out to not be nearly as hard to show them as it originally seemed to be.

The final touch was putting hands on the covers. Literally.

Front and back covers
Front and back covers

Since the kindergarten math curriculum emphasizes using fingers for counting, it seemed highly appropriate to decorate the covers this way.

Whew! What a week!

I was able to meet with each class for a little over an hour three times each.

Looking forward to repeating this project with other groups.

Also, now I want to create something like this for first graders! That’s what I will be working on this week.