Actually, I posted this video a couple weeks ago, but between my podcast and professional stuff, I haven’t gotten around to reposting it here. I’m trying to catch up on actual blog posts I have planned, so I thought I should share it. Check it out.

This video is a tribute to Mathologer’s video about multiplication tables and the Mandelbrot set a few years ago. You can see it here.

This is an animation of modular arithmetic, which is basically the kind of arithmetic you do on a clock, except instead of 12 hours or 60 minutes, the circle adds up to an arbitrary number. In this case, there are 360 points on the circle.

The lines show multiplication by the number in the top left corner. Take multiplication by 2, for example. Go around the circle, and draw a line connecting 0 to 0 (which is just a point), 1 to 2, 2 to 4, 3 to 6, and so on. When you get to 180, you have to start over, so you connect it to 0, 181 to 2, and so on. You can see this at 00:06 in the video. The resulting heart shape formed by the lines is a cardioid, which is the largest bulb in the Mandelbrot set. Each number up to 360 produces a different pattern, many of which are interesting.

This isn’t the only video of animated modular multiplication tables on YouTube. Mathologer’s original video spawned many copycats, a few of which are larger and more complex. But I did do something that no other video I saw does. My version pauses at many of the interesting numbers so that you can see the patterns more clearly. Without that, it’s hard to tell what’s going on, much less appreciate the complexity of the structures.

I animated this whole thing using Python. Maybe I’ll post the code later if I decide on an appropriate venue. The music comes from Tchaikovsky’s “Pathétique” Symphony, specifically the second movement, which is a waltz in 5/4 time instead of the usual 3/4. I thought the odd time signature befitted the strange spinning of the patterns in the video. Public domain recording courtesy of Musopen.

I have another, more complex math animation that I’m working on. If you want a sneak peak…well, I haven’t rendered anything yet. But you can click here for the inspiration.