A second example from tiling the Aztec diamond

Yesterday I learned about the Arctic Circle theorem and used it for a fun talk with the kids:

The Arctic Circle Theorem

This morning we had a fun little coincidence as one of the problems that my son was working on was the proof that 1 + 2 + 3 + \ldots + n = (n)(n+1) / 2.  The coincidence is that the number of different tilings of the nth Aztec Diamond is 2^{(n)(n+1)/2}, so for a quick project this morning we looked at the sum and then tried to find the 8 different tilings of the level 2 Aztec Diamond:

Part 1 is a short discussion of the sum:

 

Part 2 is looking at the tilings of the Aztec Diamond – counting the number of tilings of the level 2 diamond is a pretty good challenge for kids.

 

So, a lucky second project with the Aztec Diamond. I definitely want to think more about how to share the ideas in the Arctic circle theorem with kids. I think the ideas here are something that kids will really love.

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The Arctic Circle Theorem

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:

 

Screen Shot 2016-03-02 at 7.07.20 PM

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 🙂