Are the “juggling roots” related to Aztec Diamond tilings?

I was working over at MIT today and brought the print I made overnight so that I could sand it after it cured:

Overnight print of the roots of x^5 + tx +1 as t varies over the circle of radius 2 centered at 0 in the complex plane. #math #mathchat pic.twitter.com/Dsi99yTBKF

— Mike Lawler (@mikeandallie) May 9, 2017

Occasionally there’s a grad student that I chat with and he walked by today and asked what the print was. I showed him the “juggling roots” from the John Baez tweet:

Watch animations of "juggling roots". As you change the coefficients of a polynomial, its roots move in fun ways: https://t.co/iGXCtAcrhO pic.twitter.com/hXYXN3ePDy

— John Carlos Baez (@johncarlosbaez) May 8, 2017

Here are the two recent projects that we’ve done after seeing that tweet.

Sharing John Baez’s “juggling roots” tweet with kids

Sharing John Baez’s “juggling roots” post with kids part 2

Seeing the rotating roots, he said “Oh, that’s related to the Aztec diamond tilings.” Unfortunately he had to run to a meeting so I didn’t get to learn what the relationship was.

But . . . here’s a picture of the Aztec diamond:

Here are a few of the projects that we’ve done on the Aztec diamond tilings:

The Arctic Circle Theorem

TA second example from tiling the Aztec diamond

It is funny the relationships you see when you know what you are looking at. I don’t see the connection, but I’m excited to learn what it is!

Sharing John Baez’s “juggling roots” post with kids part 2

Yesterday I saw this incredible tweet from John Baez:

Watch animations of "juggling roots". As you change the coefficients of a polynomial, its roots move in fun ways: https://t.co/iGXCtAcrhO pic.twitter.com/hXYXN3ePDy

— John Carlos Baez (@johncarlosbaez) May 8, 2017

We did one project with some of the shapes this morning:

Sharing John Baez’s “juggling roots” tweet with kids

The tweet links to a couple of blog posts which I’ll link to directly here for ease:

John Baez’s “Juggling Roots” Google+ post

Curiosa Mathematica’s ‘Animation by Two Cubes” post on Tumblr

The Original set of animations by twocubes on Tumblr

Reading a bit in the comment on Baez’s google+ post I saw a reference to the 3d shapes you could make by considering the frames in the various animations to be slices of a 3d shape. I thought it would be fun to show some of those shapes to the boys tonight and see if they could identify which animated gif generated the 3d shape.

This was an incredibly fun project – it is amazing to hear what kids have to say about these complicated (and beautiful) shapes. It is also very fun to hear them reason their way to figuring out which 3d shape corresponds to each gif.

Here are the conversations:

(1)

(2)

(3)

(4)

(5)

(6) As a lucky bonus, the 3d print finished up just as we finished the last video. I thought it would be fun for them to see and talk about that print even though (i) it broke a little bit while it was printing, and (ii) it was fresh out of the printer and still dripping plastic 🙂

The conversations that we’ve had around Baez’s post has been some of the most enjoyable conversations that I’ve had sharing really advanced math – math that is interesting to research mathematicians – with kids. o