Dan Anderson’s complex mappings part 2

Yesterday we did a fun project about a mapping in the complex plane that I saw from Dan Anderson on Twitter:

That project is here:

Dan Anderson Project part 1

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 🙂

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!

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A neat complex number program from Dan Anderson

I drove back to Boston from NYC today and was pretty tired when I got home. BUT, upon arrival I saw this incredible tweet from Dan Anderson:

I had to ask Dan (and click through a few of the related tweets) to figure out what was going on, but the final shape in the Gif is the image of the unit circle under the map:

z -> z + z^4 / 4 + z^9 / 9 + \ldots + z^{n^2} / n^2 + \ldots

Amazing!

It was clear that the boys would find Dan’s tweet interesting, so I thought up a short project over dinner. Before diving in to that project, though, I just showed them the gif that Dan made and asked for their thoughts. The pings that you hear from my phone during the four videos are a barrage of more neat gifs from Dan!

 

After hearing what the kids thought about the shape, we moved to the white board to talk a little bit about complex numbers. The kids have heard a bit about complex numbers in the past – just not recently. Once we finished a quick review of some of the basics of complex numbers, we talked about what the image of the unit circle looks like when you take powers.

The talk here was obviously not intended to be a comprehensive talk about complex numbers. The ideas here are just what came to mind when I saw Dan’s picture plus my attempt to explain those ideas to kids on the fly.

 

Now I explained the picture Dan was making. When you first write down the map (as above) it looks really complicated. We tried to simplify as much as possible by walking through the images of some easy numbers. I was happy that we were actually able to make some good progress here. (Also happy that my older son thinks every infinite series adds up to -1/12. ha ha – thanks Numberphile!)

 

After we finished this part, Dan actually published a gif showing the first 20 steps – that picture helps you see the images that we talked about in this video:

Not having Dan’s latest gif handy during our project, we went back to his original picture to see if we had understood how 1, -1, i, and -i behave. It is really neat that you can explain the behavior of these special points to young kids 🙂

We end this video by looking at another new gif that Dan made while we were working out our project – it shows the images of circles with radius ranging from 0 to 1 under the map. Great work by Dan!

 

So, a really fun project for kids thanks to Dan’s incredible programming work. Such a great way to introduce kids to the cool behavior of complex numbers.