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:

You are likely familiar with the fact that the game of dominoes is played with tiles that are like two squares on top of each other. Much to my delight and surprise, about a year ago, mathematician Justin Lanier enlightened me to the fact that this rectangle, which is twice as long as it is wide, has a specific name and that name is DOMINO.

(pause. paste in Matt Vaudrey’s reaction…

…ok, continue with post)

This may not seems like such a grand reason to celebrate, but, over the past year, by being able to pull this definition out of my back pocket, both my teaching and my structural problem solving have improved and deepened. There is something about calling something by its proper name that has value.

For decades, with thousands of children who have done bookmaking with me, I’ve been trying to figure out how to get students to decorate their books using geometric shapes in a meaningful way. A meaningful way means, to me, getting them to really understand and exploit the characteristics of the shapes. Thousands of children for decades means I can try out one idea after the other, attend to nuance, and maybe even figure out a thing or two.

I’ve known for a long time that starting with squares can help reign in the chaos of making decorations for books. I actually use to cut out and distribute piles of squares to students so that they could work with squares.

Yeah, I would spend hours cutting little squares out brightly colored card stock. Even now it’s painful looking at these squares, remembering those late nights of cutting cutting cutting… and then the students would want to smear glue all over their books and sprinkle the squares like candy on them and, when the glue dried, half of the squares would fall off because the gluing was sub-prime. Ok, I’m sort of exaggerating: Many turned out beautifully, but I still didn’t feel like the kids were getting the most out of the activity: they weren’t making the connection that I was hoping for with the square.

Eventually I realized that I could give students the double-square shape, though at this point I didn’t know that it was called at domino. I show students how to line up two of these double-square perpendicular to each other, from which they cut “on the line” that separates the colors, and snip, snip, they have four squares. If they cut the square again, this time by eye, they have scaled down rectangles, and they can then cut these in half to make baby squares. Yes, baby squares. These are young children and young children like baby squares. I also talk to the about rotating the square, cutting it from point to point to make two triangles, and then cutting the triangles in half to make baby triangles.

Now just this year, during this teaching season, I have a made a point to introduce the double-square shape by name. The students I work with are shape savvy: they know what a rectangle is and they know that the square is a special kind of rectangle, but their eyes light up when I tell them a that a double-square shape is a special shape too, a domino. Does it make a difference for them to call it by name? I answer that with a resounding YES.

It’s been like night and day. Students seem to honor the shape and special qualities of the domino. I know this because, with the hundreds of students that I’ve worked with this year, I saw, for the first time, the overwhelming majority of students being at ease with the idea of working with just the shapes that I talked with them about. There was so much less arbitrary cutting of paper, so much less just slapping down whatever shape that emerged from a hastily cut paper strip.

Across the board, with the introduction to the domino, students approached this part of the project with a sophistication that I had never seen before.

Instead of having to wade through photographs to show the type of work that I think is most valuable to show, I am finding it hard to choose between all the interesting work that these students are making.

I look at these and I can sense that the student is connected to the underlying structure. If you don’t know what I mean by that, just look again at these photos.

It seems to me that I even have metric which tells me that this is not just something I’m imagining. One of the other techniques that I show students is how to make paper spring. This is a challenge for most students, but a worthwhile one, as they love how it gets used in their books. Though not a domino shape, it is made by having an awareness that the two strips of paper they start with are of equal width, and I show them how to weave and rotate the strips down into a square.

After first showing them the techniques of decoration with the domino it was astounding how quickly the students understood how to do the spring.

In fact, in each class, these parts of my presentation went so quickly and smoothly that I kept checking my notes, thinking I had missed some part of the lesson because there was so much extra time.

Needless to say, I have much respect for the domino.

For the last few months I have been casually, now more seriously, studying a folded paper structure in Chinese Folk Art, called the Zhen Xian Boa. My plan is to write a number of posts on this structure. It’s not hard to figure out that part of my fascination with the structure is that, from beginning to end, there’s the domino.

Here’s a project I did with second graders a number of years ago, but, for a specific reason that I will divulge at the end of this post, I chose not write about. Now, having just come across this folder of picture, I liked the images so much that I decided it’s time write about these books.

These second grade student chose to a local bird to research. My job was to design a project that would showcase the results of the research, display some generalized info about the life cycle of the bird, have an “About the Author” section, as well incorporate a diorama that flatten, and which included pop-ups and a paper spring.

I can’t say for sure (though I will dig up my notes and include this info later) but I’d say that this book stand about 10″ high. You can see that it opens from the center to reveal the habitat of the bird.

We were able to do two pop-ups; one in the sky and once on the forest floor. The Blue Jay is attached with a paper spring to give the bird some dimension and movement.

On the backside of the habitat there’s ample room for research and everything else.

Food and Interesting facts go on one of the sides.

Facts about the bird’s appearance and their habitat are written on the far edges of the paper…

….with life cycle info at the center…

…topped off by information about the author.

Now here’s some details to notice. To get the front sections to stay together, the rotated center square is glued on half of its surface, the other half slides under the long strip, which is glued down just at its bottom and top. The details of the decorative elements on the fronts of the books were created with simple, geometric symmetries. I loved the decisions that kids made with the shapes!

Another idea that the students worked with was the idea of using different mediums and methods to make thehabitat. The cloud is foam, there’s cut paper shapes, drawing with markers and crayons, a few shapes created with paper punches (the butterfly and dragonflies) paper springs behind the owls, and both a one-cut and a two-cut pop-up: all with the goal of creating an interesting, texture display.

As you might imagine, these books are made using lots of separate pieces. For this kind of project I generally first have the students make a large origami pocket from a 15″ square paper so that we have container in which to keep everything organized. The classroom teacher, Gail DePace, who I could always count on to enrich my projects with her own personal standards of excellence, had the idea to ask the students to decorate their origami pockets as if they were bird’s nest, complete with appropriately colored eggs.

The students added another dimension to this project by creating their birds in clay and putting them on display along with their books.

At the beginning of this post I said that there was a reason that I hadn’t written about this project. As lovely as the project is, the teacher, who was a spectacular collaborator on this and all projects that we did together, didn’t love this project. She noted that this structure didn’t work well as a book, that it was awkward for the kids to open to the “pages” and read their work when it came time to do their presentation of the final project.

I’d have to agree that this project works much better as a display than as a book. Oh, and it looks great in pictures too. Sometimes, though, the display and the documentation are the priorities, so that’s what I’d keep in mind for this project next time.

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.