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.

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.

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:

I’m in the busy part of my art-in-ed, itinerant artist season. The challenge is to keep what I do relevant to the students, to the curriculum, to the teachers, and to myself. Most of the work that I do in schools is done with teachers I’ve worked with in previous years. Usually I repeat a project each year with the teachers’ new classes, though there are always tweaks that are made. Then, sometimes, it’s time to retire a project that’s been working well for years.

I’ve just finished up quite a few projects in classrooms, many of which were new this year. I’m going to attempt to write a number of posts about these projects before the next set of classes that I teach start up.

There’s been a shift in my approach to what I offer to the schools. Whereas I used to think of my work as a way to motivate and celebrate literacy, now I am more focused on using our bookmaking projects in a way that supporting the teachers’ math goals. I’ve been realizing that the math part of the curriculum is where many teachers most appreciate support. There is so much in the paper and book arts that can support the math that students need to learn that making this shift has been thoroughly enjoyable to me.

Symmetry is a theme that kept emerging in the projects that I presented these last few weeks. This is partly to do with the nature of making books, but I also deliberately focussed on it more than it other years. I’ve realized, just recently, that the symmetry of shapes is the visual equivalent of mathematical expressions. I probably won’t express this well, but here goes. Think about doing any sort of math problem that has an equal sign in it. 5+3 = 8. It’s balanced. If you add a 3 to one side you have to add a 3 to the other side to keep the expression true. Math calculations are all about symmetry and balance. It is, therefore, completely appropriate and desirable, to help kids develop their natural affinity to symmetry.

One of the projects that I did with kindergarten students had to do with these piles of square cards. Students worked in teams. The first student puts down a card, then the partner puts down a card that is a symmetrical reflection of color and shape. They take turns putting down a card and then reflecting it.

It took a bit of doing for these 6 year olds to get the hang of what we were doing, but, still, quickly, patterns emerged.

These cards, by the way, are an element within a larger math activity book that we made.

Since these pieces are made from paper, I suggested to the teacher, as this becomes easy for the kids, to cut out one of the smaller squares from some of the cards so that mirroring the shape transformations becomes a bit more challenging.

Another symmetry project I tried out for the first time was with the Pre-K crowd. My friend Joan, who has worked with this age group, showed me this activity that she had developed with kids she had worked with. I’ve been excited to try it out.

What I did here was define a line of reflection. Then these five-year olds did the same kind of reflection symmetry that I described above, each taking turns putting down a stick, then the partner reflects it with one of their sticks.

Again, it was a struggle to get these students started, but it didn’t take long for them to catch on.

After a short while I combined groups so, instead of working in pairs, there were four people in a group, which led to different kinds of designs. The pattern above was made near the end of the activity. From start (first handing out the sticks) to finish, this activity took a mere twenty-two minutes, which was how long it took them to began to lose interest. At this point I suggested that they just used the sticks to make whatever arrangements that they wanted to make. Surprisingly, many started trying to use them to spell out their names. I heard their teacher remark something about the fact that they struggle to write their names but they seem to be able to construct them just fine. Which gave me an idea, which I will show in my next post. Now, though I want to jump back to the photo at the top of the post, which is the books made by seventh graders.

I’ve been doing this project with the seventh grade for many years. I give them a large piece of paper (23″ x 35″), which they fold and tear to make a pamphlet.

I don’t explicitly talk about the symmetry of the folding we do, but I will talk about it in the future. The fact that the sequence of fold and tears results in a scaled down version of the original sheet is something I want them to be aware of.

In fact, every aspect of making this book is symmetrical, even the pattern of the thread that sews the pages together is totally symmetrical.
When building just about anything, even a book, symmetry rules.

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.

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.

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

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.

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.

Students seemed to enjoy creating these images.

After they created the outlines they added color.

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 …

…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.

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.

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.

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

I 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.

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

Some drawings were big and bold.

Some drawings were delicate and detailed.

I 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.

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.

News Flash: A Codified Language Exists to Describe Patterns. I’ve been so excited to discover the way to speak about patterns.

I’ve been teaching decorative techniques for a long time now. I’ve started trying to use more precise terminology in my teaching, and I suspected there was more to know. I started out looking at artistic and graphic design sites, really I did. I looked on lynda.com, I looked on youtube, and poked around the internet in general. Then Maria Droujkova pointed me in the direction of something called Wallpaper Groups, and guess what, I landed on sites that described pattern making with precision, using the language of mathematics.

The more I learn the more I understand that what math does is enhance the way that people can describe what’s in the world. It appears that hundreds of years ago mathematicians figured out how to understand and talk about patterns.

This summer I’ll be teaching a week’s worth of classes to young children at our community center. I enjoy showing students decorative techniques, so my immediate interest has been to develop a modest curriculum that focuses on making books that are embellished with style. Even though many of the students will be at an age where they are still struggling with concepts such as “next to” and “underneath” I hope to introduce them to ways of thinking about concepts of transformation.

Strip Symmetry is where I landed when I was surfing for a way to find words to describe the kind decorations I’ve been thinking about. In other words, the patterns I am looking to teach will have a linear quality in the way that they occupy a space, as opposed to being like a central starburst, or an all-over wallpaper pattern. It turns out that there are only a handful of words that are used to describe every single repeating linear pattern ever made.

A Translation takes a motif and repeats it exactly.

Vertical Refectionmirrors a motif across an imaginary vertical line. The name of this particular transformation confused me at first, as the design itself extends in a horizontal direction, but once I prioritized the idea of the vertical mirror, it made more sense.

Glide Reflection can be described as sliding then flipping the motif,, but that description sounds confusing to me. Instead, understand glide reflection by looking at the pattern we make with our feet when we walk; Our feet are mirror images of each other, and they land in an alternating pattern on the ground. Imagine footsteps on top of each of the paper turtles you might better be able to isolate the glide refection symmetry.

Horizontal Reflection mirrors the design across an imaginary horizontal line.

Here’s a translation that shifts horizontally, but there’s no such thing as a strip symmetry that translates top to bottom. Instead, convention dictates that the viewer turns the pattern so that it moves from left to right.

Rotation rotates a design around an equator. The pattern above, as well as the first image of this post, I had considered these both to be rotatation( ( I imagined the equator drawn across the middle of the page), especially if it’s 7 year-olds that I am talking to, but close inspection reveals more. To highlight that I am presenting these concepts with broad strokes, here is what Professor Darrah Chavey wrote about the image above (the one with the leaves) when I asked for his input:

“As to this particular pattern, there’s a slight problem in viewing these leaves as a strip pattern. The leaves you show are made from a common template, but that template isn’t quite symmetric, and the way the leaves are repeated across the top isn’t quite regular. For example, the stem of the maple leaf in the top row, #1, leans a little to the left, and has a bigger bulge on the left. If we view this as a significant variation, then the maple leaves on the top row go: Left, Right, Left, Left, Right, Right, which isn’t a regular pattern, i.e. it doesn’t have a translation. On the other hand, if we view those differences as being too small to worry about, then the leaves themselves have a vertical reflection, successive pairs of leaves have vertical reflections between them, and the strip pattern on the top is of type pm11. The bottom strip is a rotation of the top strip, but if we view those differences as significant, then it still isn’t a strip pattern (it would be a central symmetry of type D1), and if we view those differences as insignificant, then it would be a pattern of type pmm2, since it would have both vertical reflections, and rotations (and consequently also have horizontal reflections).”

I was excited to get this response to the leaves image, as it reminded me that my newly acquired understanding of symmetries, though useful, is simply just emerging.

So that’s it:

Translation

Reflection (horizontal or vertical)

Rotation

Glide Reflection

Darrah Chavey, who is a professor at Beloit college, turned out to be the hero in this journey of mine, for having made and posted videos on youtube. Here’s a link to one of his many lectures on patterns: Ethnomathematics Lecture 3: Strip Symmetries

Now here’s some nuts-&-bolts of what I’ve learned from making the samples that I’ve posted here:

the book I made was too small (only 5.5″ high) because the cut papers then had to be too small to handle easily. I’m thinking that any book I make with students needs each page to be at least 8.5″ tall.

It was easier to create harmonious looking patterns when I started out with domino rectangles (rectangles that have a 2:1 height to width ratio), then cut them in half and half again to make squares, tilted squares,triangles and rectangles.

I like the look of alternating plain paper and cubed paper. Folding paper that has cubes printed on just one side accomplishes this.

I am going to enjoy teaching these college level concepts to young elementary children.