Exploring the Language of Patterns

Translation

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.

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

 A Translation takes a motif and repeats it exactly.

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.

Glide

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

Horizontal Reflection mirrors the design across an imaginary horizontal line.

Translation

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?

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.

Graphic Design Elements for First Graders

Book Covers Designed by First graders

Book cover design has its challenges for all ages.  When the books that I make with children have a title page,  rather than repeat the title page info up front,  I like students to design a visually stunning cover. I used to give first graders lots of bits of cut paper, with the directions that they should cut and glue a nice design on to their covers. Those of you who work with first graders know exactly how that goes…..I’ve tried many approaches to the cover graphics, and recently I tried something out that is an elementary version of something I have done with adults. I was happy with the results, so here it is.

2'" x 8" strips of paper, plus assorted bits

I started out by letting students pick just one strip of 2″ x 8″ paper. They then took just two small pieces of paper from an assortment of bits of paper that were in a box. The technique that I encouraged was specific:

• make just three or four cuts in the larger paper, to create small cut-outs
• make just one cut in just one of the smaller pieces
• arrange the papers on the cover so that the cut-outs from the larger paper are near to the places they were cut from
• arrange the other piece as desired and glue down well.

I think I also mentioned something about “less is more” to these budding graphic designers. Most of the students were able to resist the urge to cut and paste any which way, and they came up with a variety of really fine book covers.

Modern Art Design Covers by First Graders

An added bonus to this technique, besides looking good it took only minutes for these designs to be completed.

Now, here’s some cut paper designs, using a similar method of working, that I do with adults.

Cut Paper Designs by a Grown-Up

 Most first graders aren’t up to creatively cutting long edges of paper, but that will change in a few years. For now, a few snips here and there, and thoughtful placement works just fine.

Hexagon-Flexagon: Post #4, Student Work

I’m ending this series of posts on hexagon-flexagon with images of work done by students in Michele Gannon’s art classes in the Adirondacks.  I taught the students how to do the paper-folding. Michelle’s considerable artistic influence supported the students’ creative output.  Each of images that I am showcasing here is followed by the same image in different configuration on the same structure. Notice how the patterns change.

This lovely design was created with Prismacolor pencils. Some, but not all, of the Prismacolors show up well on black paper.

This next set shows a really simple design that works really well, as the look of it changes dramatically when the image is flexed.

Ta-Dah!

Since each hexagon-flexagon has three distinct surfaces to design and decorate I  brought in a variety of novel tools to help motivate the students to keep working. In the photos below the students used my Chinese Dragon paper punch. Paper punches are a great motivational tool, and the dragon is the most popular of all that I have.

By the way, the students that made these images were about eleven years old.

I like the way the relationship between the dragons changes when the flexagon flexes.

Another tool that makes a big hit with students is Crayon Gel Markers. These markers take about 15 seconds to fully show up on the paper, thus adding a bit a magic to the process of decorating.

I also use some of the Crayon colored pencils. Only some of them work well on black paper. I used to be able to find Crayona FX (?) pencils that worked on black paper, but I haven’t seen them around for awhile.

I have to say that that only reason that I felt comfortable teaching this structure to these students is that I knew I would be working with small groups. I don’t think that I had more than 10 or 12 students in each class.  The fact is, the folding  for this structure is so specific, that I doubt I could successfully teach it to large classes. But I think it would be a great project to teach to home schoolers, or in an after school art class.

The hexagon below has a completely different look when it’s flexed.

The middle circle totally changes, and that white line becomes a whole different thing, too.

These were so much fun to teach.  It’s a real pleasure to watch the expressions on the students’ faces when they see their own design be transformed when the hexagon flexes. That’ s my favorite part of the process, too.

Related posts:

Intro to hexagon-flexagons

Fractions Flexagaon

How to Make a Hexagon-Flexagon