translation design pattern


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, 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.

Example of Pattern Rotation


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.

Translation repeat pattern


 A Translation takes a motif and repeats it exactly.

Example of Vertical Reflection

Vertical Reflection

Vertical Refection mirrors 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.

Example of Glide Pattern


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.

Example of Horizontal Reflection

Horizontal Reflection

Horizontal Reflection mirrors the design across an imaginary horizontal line.

Example of Translation Pattern


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.

A bounty of books

Eight Humongous Rubber Band Books, showing off their colorful spines

The sixth grade English teacher in this school likes the idea of each of her students making a book that they can use as (her words) a memory catcher. Writing, pictures, and ephemera will go into these books. The design challenge is that I can’t count on having more than 40 minutes to work with the students. I want them to end up with something large, sturdy, and I want them to enjoy making it.

Setting out the Papers

Setting out the Papers

On my day with these sixth graders, they walked in the library, saw the colorful papers and were immediately delighted. “Do we get to do this today?!” They were all so happy! My papers here are tabloid size, 11″ x 17″ 67lb papers (which, by the way, are getting more expensive and harder to source every time I look).

Choosing papers

Choosing papers

Each student chooses eight papers. We have plenty of space to work. It’s interesting to notice how each student chooses to arrange their stash.

setting out papers

Getting ready to work

Some students choose to work alone and spread their papers out all out in front of them

Getting ready to work

Getting ready to work

Other students work two, three or four to a table and have to stack their papers.

Next step is to fold the papers then nest them together in groups of two.


I’ve worked with these students many times before, and they are all have expert paper-folding skills.


The trick to accurate paper-folding is to hold the paper with one hand, then slide the other hand towards the curl.

foldingThese students have been using my bone folders just about every year they’ve been in school. If I forget to hand them out they will ask for them. In schools I refer to them “folding tools” to avoid  vegetarian discussions. If the fact that they are made of bone comes up, I advise vegetarians not to eat them.

These papers will  get nested together in groups of two.

These papers will get nested together in groups of two.

The students end up with four groups of two folded papers. This grouping is completely non-intuitive: students want to nest them all together, one inside of the other, and wrap one rubber band around the spine and be done. In fact, the book would work just fine that way, but I’m here to show them something different, and, arguably, better. By asking them to make four groups of paper they will end up with a thicker, and much cooler looking book spine, one which shows off some of the colors in the book.

Snips off the top and bottom of the spine edge of the paper

Snips off the top and bottom of the spine edge of the paper

Once the pages are grouped together, there’s one more step before the assembly starts. The corners of the tops and bottoms of the folds are snipped off. These snips create valleys that the rubber bands will settle into.

Attaching nested groups of paper

Attaching nested groups of paper

Two groupings of papers are set next to each other side-by-side, opened in the middle. The rubber band slides over the four adjacent pages, binding the page groupings together.  I use  Quill Brand Rubber Bands, 7Lx1/8″W which are humongous in just the right way. Smaller rubber bands will actually work for this, but the tighter the rubber band stretches, the sooner it will rot and break. I want these books to stay together for a good long time.

Attaching signatures together

Binding together two groupings of papers, with the rubber band

On goes the rubber band! This is done until all four sections are linked, in sequence, one group right next to each other. This book can be made to be just about any number of pages long.

Book made with humongous rubber bands

Book made with humongous rubber bands

It’s a good idea to decorate the cover of this book right away, as the flexible nature of the spine can make it tricky to figure out which page is the front once it’s been opened and looked through. Students make pockets to go on the front and back covers, to store items that will be eventually attached into the books. I’ve been making these books with this school’s sixth graders for a number of years, but I don’t get to see them finished. Students, however, will joyfully tell me about them, and they will also tell me, oh I remember when my brother made these! From what I understand, they hold a plethora of memories.

Pieces of Paper


This story begins in a teachers’ lunchroom, a couple of years ago, in Upstate NY. I was sitting with some teachers when another member of the staff started talking to a first grade teacher, Mrs. K, about a new math mandate. It was something about using manipulatives to create a variety of shapes. I’m a bit foggy on this part but it seems to me that they were required to use rhombuses (or rhombi, both are correct) for their shape building.

A Rhombus

A Rhombus

Upon being told that she would have to incorporate these manipulatives into her math unit Mrs. K asked if there was any money in the budget for manipulatives. The answer was no.

After school I sought out Mrs. K and showed her some paper-folding and shape transformations that referenced rhombuses. This teacher seemed delighted with what I was showing her. I volunteered to send her something that I thought she might find useful, then went home and created these images for her, which are equilateral triangles that become a rhombus.

screen shot rhombi

I never asked Mrs.K if she used what I sent her. I recognize that what I sent was, unfortunately, not a project. Instead, it was just the bones, the beginning of a project that needed to be developed.  Every so often I’ve revisited these images, wondering what I could do with them. Then a few nights ago Malke, from Indiana, asked me about projects for a family night.

It was late, and we decided to resume the conversation the next day. The next morning, before Malke and I reconnected, I saw this post from Simon Gregg, in France:

I had an Aha! moment. It suddenly came together. I sent off this note to Simon:

Malke, who I included in the conversation, responded with a reference to a beautiful manipulative that I wasn’t familiar with, but which also showed that she immediately recognized what I was getting at with my DIY paper version of manipulatives.

Since Malke seemed to know exactly what I was thinking about I got to work creating the pieces for this activity. I’m pretty happy with how this has developed. It requires triangle paper, and matching paper shapes that can be printed on colorful papers. My thought is that simple, bold shapes can be created in sort of a free form way…

Big Shapes

Big Shapes

…or more challenging shapes can be drawn on to the paper…

Drawn Shapes

Drawn Shapes

…and filled in, while trying to make as few cuts as possible and being mindful about cutting along the lines defined by the triangles.

Filling in drawn shapes

Filling in drawn shapes

So, where can you get these papers to do a do-it-yourself shape building set? Right here. I’ve created a couple of PDF’s to get you started:

Triangle paper and Rhombuses

Make beautiful shapes. Send photos. Thank you.

Addendum: Take a look at Malke’s post on hands-on math: she collected and organized many interesting perspectives. It’s a fabulous piece of writing.

Addendum #2 (April 2016)  Malke liked working with smaller rhombuses so I made her this  PDF rhombi with spaces So far she is planning on using them without the triangle grid paper. Here’s a link to some images she’s created as samples for an upcoming project rhombusphotos

A Pre-K Project

A Pre-K Project

Tonight I’m finishing up gathering supplies for the first day of what is always my most challenging, and most satisfying, school visit of the my teaching-artist season. I have been visiting schools in the Adirondacks for many years, but I have spent the most time in this one particular school. I get to work with nine grade levels, pre-K through 7th grade.  I need to create nine completely different projects, which will go from beginning to completion over six days, spread out through the month of March.

In the interest of finishing up the details, and getting to bed (last night, daylight savings time kicked in, so getting up tomorrow morning will be a challenge) I am going to list the nine projects for the nine grade levels, then I’m going to try to write about them over the course of the month.

Here goes.

PreK: the teachers asked that we do a project with the students’ names. We’ll thread beads and cover weight papers on to shoelace-tipped yarn, write a letter on one side of the card, and a picture which starts with that letter on the back. See photo above,

Kindergarten: Accordion Book with pockets, a variation of structure in the picture below.

An Accordion Book with Pockets for Kindergarten Sight Words

An Accordion Book with Pockets for Kindergarten Sight Words

First Grade:  A  folding triptych about Alaska and an  animal that lives in Alaska. Will include a pop-up, a pocket to hold research papers, and a poetry page. We’ll color the sky with Northern Lights.

Second Grade: A book that folds up like a valise, that has pockets within for a “passport,” a folding map, postcards, a boarding ticket, and little books with information about a country that the student is studying.

Third Grade: We’ll make a journal for the students to use however they want.

Fourth Grade: This is the class that will be making a Zero to One Fractions  book that I’ve been writing about

Fifth Grade: I still have some planning to do on this project, but it will likely be a social studies based project made from units of an Origami Base, which opens and closes in a dynamic way.

Sixth Grade: This group will use tabloid size papers, folded in half, and bound, in four separate sections, with large rubber bands. The students will use these with their English teacher, between now and the end of the year, as a memory catcher.

Seventh Grade: We’ll fold down and trim a large, 35″ x 23 ” paper into an 8.75″ x 5.75″ pamphlet, which students will sew, glue in to a hinge piece, add soft covers, and decorate. The book will go with them to their English class, for content to be added between now and the end of the school year.

I keep everything organized ( I hope) in a notebook that I can make in about five minutes, that looks like this.

How to Bind Loose Leaf Papers by Paula Beardell Krieg

Hopefully I will be posting all of these projects. But now it’s time to wrap things up for the night.

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