Today I went into to MIT and totally by accident learned about a really cool idea in math that I’d never heard of before – the “Arctic Circle Theorem.” Even more fun, it turns out that one of the mathematicians who proved the theorem is Jim Propp whose amazing blog Mathenchant has already inspired a couple of our projects.
Way, way, waaaaaaay oversimplifying, the Arctic Circle theorem says that if you randomly tile a shape called an “Aztec Diamond” with dominoes you’ll almost certainly end up with a really simple pattern near the corners of the diamond.
A picture is worth way more than words here, so here’s a picture of one of the shapes we looked at tonight:
and here’s a description of the idea that is
probably a bit better than I could give:
A discussion of the Aztec Diamond / Arctic Circle theorem on Wikipedia
and here’s some software I found that helps you play with tilings of Aztec diamonds:
Dan Romik has software for playing with the Aztec Diamond
So, after hearing about the problem I spent the day thinking that kids would probably really enjoy hearing about / looking at this problem. After finding the software above I couldn’t resist sharing the idea with my kids tonight!
First I had my older son look at the program as we looked at larger and larger tilings. I didn’t explain much of anything about what was going on, I just wanted to hear what he had to say:
Next we went through it again, but I told him a little bit more about what was going on to see
Now I repeated the same process with my younger son – here are his thoughts when he was seeing the tiling patterns for the first time:
and here are his new thoughts after I explained just a little bit more about what he was seeing:
I really think there’s a great project for kids hiding in here somewhere! Maybe it is a computer project, or maybe it is a simple project with snap cubes. It is really amazing to see how order comes from randomness here. Can’t wait to think a little more about how to share the ideas here with kids 🙂