A super fun fractal project for tomorrow

It all started with a Steven Strogatz tweet from a few weeks ago:

It wasn’t the Lorenz attractor that caught my eye, though, it was the purple book behind it with the curious title: Fractals Chaos Power Laws.

With a little bit of free time on my hands yesterday I finally decided to track down that book. It wasn’t too surprising to find out that the MIT library had it (though it was sort of a surprise to see that it was cataloged in the quantum field theory section!). So . . . off to Cambridge.

I spent the afternoon reading through the book and was absolutely blown away. Among the things that caught my attention was this fractal example in the first chapter (you’ll have to scroll up a bit to see the full example):

A cool example of how fractal dimensions work

It looked like a really fun example to talk through with the boys, too. The more I thought about it the more I liked it, and it also seemed like being able to hold the objects in your hand would make the project even better. One slight problem on that front, though, was that since we are in the middle of a move I don’t have access to the computer I normally use to make the 3D printing files . . . .

A little shout out to twitter produced some helpful suggestions and basically a miracle save from Dave Radcliffe:

The code he shared is going to be a fantastic resource for the project tomorrow, but the bit of super amazing luck was that the shape actually had a name – the Gosper Island.

A quick check on Thingiverse revealed that there was already a couple of different choices for prints. I chose this one:

The Gosper Curve on Thingiverse (I printed the middle one)

Here’s the picture of the finished set of 7 Gosper curves (each one took about an hour to print):

The plan that I’ve sketched in my mind for the project tomorrow goes something like this:

(1) Start with the “standard” proof that \pi = 4 to show that strange things can happen when you have infinite jagged edges. Something along the lines of this old Vi Hart video:

(2) Next jump to the hexagons. We can make the first few iterations of the Gosper curve with our Zometool set. Something like this, but maybe the next iteration, too:


(3) After making a few iterations ourselves, we’ll jump to the program that Dave Radcliffe shared to see how the shape evolves with a few more iterations.

(4) Finally we’ll play with our set of 7 3d printed (almost) Gosper curves and find the surprising scaling result – multiplying the linear size of the figure by 3 scales the area by a factor of 7 rather than the expected scale factor of 9. Just like the \pi = 4 “proof” the infinite jagged-ness is causing something unexpected to happen.

I’m super excited to try this all out tomorrow!