Over the last few days I’ve been preparing a project with the boys based a great fractal geometry example I found in this wonderful book:

We finally got going with that project this morning. The starting point was watching this Vi Hart video which gives a “proof” that

After we watched the video we sat down to talk about the strange result and what they thought was going on. They seemed to gravitate to the idea that the jagged edges were causing problems, but the fact that the zigs and zags were getting smaller and smaller – and would eventually have a height of 0 – was still a bit confusing:

After talking about the Vi Hart video I introduced the kids to the Gosper curve by showing them the figures in the book that inspired this example. We also made use of an amazing program that Dave Radcliffe shared when I asked for a little help on Twitter:

Playing around with this program really helped the boys see the first couple of shapes in the sequence that eventually leads to the Gosper curve. I definitely owe Dave a big favor!

The next part of the project was to build the first couple of shapes that lead to the Gosper curve out of our Zometool set. The initial hexagon was easy, obviously, but the shape at step 2 gave them a little difficulty. In the video below they talk about building the shapes and then explore a connection between the hexagon from step 1 and the shape from step 2. The fun part here is that the boys saw some of the important connections that lead you to the Gosper curve.

Next we built a level 3 shape. It was lucky that we had the program from Dave Radcliffe since that allowed the boys to a little more confident that they had the right shape. It is interesting to see the 6 new level 2 shapes surrounding the original level 2 shape. Too bad our living room isn’t big enough to make a level 4 shape!

One interesting comment from my younger son is that he thinks that as we increase the size we’ll get closer and closer to a shape that looks like the original hexagon.

For the final part of the project we used our 3d printer to make 7 of the (approximate) Gosper curves. Here’s the shape we used from Thingiverse (our shape is the 2nd of the three shapes, but I can’t get that one to link properly):

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

The punch line for the project is the same punch line that caught my attention in the book – when you increase the linear size of the Gosper curve by 3, the area inside the curve increases by a factor of 7 rather than a by a factor of 9. Everything that the boys have learned about scaling up to this point is that area scales as the square of the linear factors, but fractals have a different property. Pretty amazing!

Also, sorry for not explaining the analogy between the two boundaries right. Felt as if I was wrong as I was explaining it, but didn’t see what I got wrong until just now.

As a fun end to the project, I showed them Dan Anderson’s modification of the Gosper Island shape – sort of a combination of the Sierpenski Triangle / Menger Sponge shapes and the Gosper Island:

So, a fun project giving the kids an introduction to fractal dimension – a concept that I never would have guessed could be made accessible to kids. Really happy to have had the luck of running into this fun idea last week.

## Comments

I wonder what your sons would think of how this relates to the blob pythagorean theorem in Barry Mazur’s Numberphile video? it seems that Professor Mazur had misled us, the property division only works if the plots have dimension 2. Is 2 even an upper bound on the dimension of the plots? Does the answer depend on whether the kingdom is in Iowa vs Nepal?

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