When I asked the boys what they wanted to talk about this morning, they said that they wanted to talk more about Dan’s shapes. Instead of the whiteboard again, today we used Mathematica and had a really fun follow up project.

I stared out by showing them some simple code for the project. That code uses the Sin() and Cos() functions in Mathematica. I did not explain why I used these functions in any detail, but just jumped in to talking about the function Dan was studying:

After talking about a few of the simple cases in the first part, we moved on to talk about some of the more complicated cases.

Dan actually made a gif of how the map of the circle changes as you increase the number of terms in the series:

It was fun to hear the boys’ thoughts about the shapes in this part – including a couple of “whoa”‘s!

Next we explored another of Dan’s ideas – what about the images of circles having a radius other than 1. We explored a few smaller circles and a few larger circles. Lots of “whoa”‘s here. Seems like the ideas here are a great way to get kids to talk about geometry.

Finally, I thought it would be fun to look at a few contour plots of the map. The ideas here are a little more advanced and I’m not sure that the boys fully understood what they were looking at – which is fine. I just wanted to show a few alternate ways to view maps of complex functions:

So, a fun morning studying yesterday’s project in a little more depth. It sure was nice to year that they wanted to learn more about Dan’s shapes đź™‚

After we finished, my younger son asked if he could play around a little more with the shapes on the computer – an hour later he just asked me again if me can play more. Awesome!