When I asked my younger son what he thought was the most beautiful math he’d seen, he replied “fractals” and specifically mentioned the Mandelbrot set. We haven’t done a project about the Mandelbrot set, so it seemed like a good idea to talk about it today.

As I say in the introduction below – it isn’t just a pretty picture, there is some really cool math.

So, I started the project by talking (or probably more accurately stumbling) through an explanation of the map that defines the Mandelbrot set. After that, we worked through a few examples:

Having looked at 0, -1, and 1, we now moved on to looking at some complex numbers. The next numbers we tried were i and -i. It turns out that both of these numbers are part of the Mandelbrot set, but the calculations are slightly (really, just slightly) more complicated.

At the middle of this video we produced a crude map of the Mandelbrot set with snap cubes, and then at the end we discussed a little bit about how a computer program to plot the Mandelbrot set would work.

Now we moved to the computer to study the Mandelbrot set in Mathematica. Luckily Mathematica has a function – MandelbrotSetPlot[] – that makes this part of the project pretty easy for us. In this part we talked a little bit about what happens when you vary the number of iterations, and also what happens when you zoom in.

Determining the coordinates for zooming in was also a nice little mathematical discussion with the boys.

The coordinates to zoom in on turned out to be a more interesting topic than I was expecting. With the camera off we had a long discussion about the coordinates of the location that they wanted to see more carefully. I’m sorry I turned the camera off, actually, but oh well.

I love the discussion and general thoughts from the kids about the shapes we were seeing here. My younger son is right – this really is beautiful math for kids to see!