Next Sunday, I’m the “visiting artist” at Christopher Danielson’s Math On-a-Stick booth at the Minnesota State Fair. This means I get to plan an arty-mathy activity for kids of all ages and help them do it for four hours. Coloring is always popular, so I have been thinking about coloring quilt blocks, where either the color makes the design visible, or where all the blocks are the same, but the final “quilt” depends on how you arrange those blocks. Then I also started thinking about frieze patterns – these are basically designs that use rigid transformations (such as reflection, translation, rotation, or glide reflection) to create a design that continues infinitely in one dimension, so something that could go around the edge of a bowl, say, or as a wallpaper border (not as actual wallpaper – that’s a different family of symmetries).

My vague criteria for the activity:

- Relatively easy to explain, and doesn’t require constant attention from knowledgeable helpers (I’ll have three or four volunteer helpers, who don’t have to know any math)
- Could have some examples or diagrams available for the more self-directed
- Doesn’t take a wicked long time to accomplish something, but could go on for a while (low floor, high ceiling, that sort of thing)
- Is fun and art-y
- Involves some math, even if it’s not obvious to everyone
- Could be something they take with them, or leave to display in our area, maybe hanging on a fence

Here’s what I’m thinking about so far.

One idea was a double Tumbling Blocks quilt pattern. It’s not that exciting, since it’s basically coloring by numbers – you pick a light, medium, and dark color, and color the chunks numbered 1, 2, and 3 respectively. The pattern would be half of an 8.5 x 11 sheet of paper. See below, plus the “finished” product if we put a bunch of them together.

Another option is the Rail Fence quilt pattern. Again, not very exciting to color, but fun to see how it looks as a “quilt” depending on how you put them together. I’d provide two different half-sheet templates with six to 10 squares on them – one that is arranged in the zigzag, and the other in the pinwheel.

But the more I think about this, the more I’m thinking I might do frieze patterns. I’ve done an activity with them before, where we learned about them, then acted them out physically, and then made them using The Geometer’s Sketchpad software (read about the time we did it with the Drexel Math Contest, and even download the instructions). We could for sure have groups of people acting them out with guidance – we have plenty of room. But I haven’t done the arty-y part with paper and markers before. I wonder how that would work? I sketched out a couple of possible templates – one of triangles, one just squares, both on about a 2″ strip. There are four examples in this picture, then a picture (with a link to a PDF) of all seven types of frieze patterns with their John Conway-given names.

That’s where I’m at right now, and am looking forward to getting some ideas from y’all. I know, especially, that I’ve missed something with the quilt blocks and coloring/symmetry, and am wondering if anyone else has done a frieze pattern coloring activity. Thanks in advance for your contributions!