## Exploring the Language of Patterns

### June 2, 2015

*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 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.

* 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:

*both*vertical reflections, and rotations (and consequently also have horizontal reflections).”

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.

June 3, 2015 at 2:40 pm

Hi Paula,Thank you for your interesting post and lovely design ideas. All of the geometry terms you’ve mentioned are utilized in our grade 5 maths curriculum in Saskatchewan. Children always feel so powerful using “math language”. I also like to make small books with designs in them when doing this unit, but I think that yours are inspiring and rather spectacular. Tammy

Date: Wed, 3 Jun 2015 02:54:02 +0000 To: jyd_2002@hotmail.com

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June 3, 2015 at 7:00 pm

HI Tammy! It’s good to hear from you! Thanks for your comments about the designs, but I’ve been thinking that maybe I got a bit too carried away. I am thinking of following up with much simpler examples of these symmetries. Thanks for reenforcing for me how much students really do like using real math language. One experience stays with me concerning this: I was working with pre-schoolers,- 4-olds!- who had a teacher that was way into dinosaurs. Well those little children could rattle off names and facts about every big-named dino that would leave the most educated adult tongue-tied.It’s great to have the opportunity to gift these words to children…Glad to hear you’re still making books!

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June 4, 2015 at 3:31 pm

Some adults and even children feel totally, hopelessly stuck in art. It breaks my heart to see it. Here is a nice surprise. This language of mathematics can get people unstuck in art! I’ve seen it again and again. People are so happy and thankful when that happens. I understand the psychology involved. But still it feels like magic!

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June 4, 2015 at 3:44 pm

From what I remember from geometry textbooks, what you are calling glide should be called glide reflection.

Nevertheless, this is a very pretty and well done post.

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June 5, 2015 at 6:02 am

Yes! Glide Reflection! I’ve gone in and done this edit. I’m glad you caught that as this post has gotten more views right out of the gate than any of my other posts and I want it to be right. Thank you.

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June 11, 2015 at 10:11 pm

[…] exploring the language of symmetry, which I wrote about in my previous post, I started trying to figure out a straightforward way to teach this symmetry to the children that I […]

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