This video actually started when a student who had gotten interested in doodling Hilbert curves asked if I could animate him one. Fractals are a fun thing to program, and I actually went a little crazy trying out what different space-filling curves look like, hence the supplemental "Fractal charm".
As I was creating it, I thought about how it's very rare for people presenting space-filling curves to mention that the hard part is ensuing these things are well-defined. That is to say, to define what it means for a family of curves to actually have a limit. It also reminded me of a thought I had a while back about how translating visual data to audio data would benefit from using space-filling curve approximations. In short, I realized there was some good fodder for a story there, so the animation for my student turned into a full-blown video.
Music and Measure Theory
There are many facts in math in which I've been able to follow the proof, but have nevertheless been stumped to find an intuitive understanding of what's happening. Such facts have a funny way of becoming topics I want to create a video about. In this case, the fact that you can cover a dense set in the reals with open intervals whose lengths sum to an arbitrarily small constant absolutely blew my mind. After mentally toiling for a while, I finally felt satisfied when I just sat down and thought through what it would feel like to zoom in on an irrational value which was not covered. Thinking about the conditions for being such an irrational reminded me, surprisingly, of when I learned about why 12 is such a natural number of chromatic notes to have in a scale.
After "What does it feel like to invent math", I wanted to do a shorter video, and to experiment with something that lacked narration. Having failed to do due diligence, it was only after I had made heavy progress into the video that I discovered just how unoriginal this method of counting is. Whoops! Nevertheless, I'm glad I made it.
For the first time in one of my videos, I actually filmed something. After using my phone to film my right hand counting to 31, I played around with some image processing techniques. This involved finding the appropriate combination of a Gaussian blur, Canny edge detection, and some more custom operations to thicken and prune the footage.
My goal when I set out to make this video was to explain divergent series with respect to a p-adic metric, keeping in the back of my mind that someday I may want to build up to an explanation of the series 1+2+3+4+...= -1/12 which does not depend on the relatively opaque operation of analytic continuation. The more I worked on it, the more I wanted the explanation to be such that the viewers could imagine that they themselves could have found it, given the appropriate principles. I've come to think this is probably the hallmark of any good explanation. Eventually I shifted the entire theme of the script to center around this idea, leaving the divergent series as a plot device more than anything else. One benefit here was that I got to work in thoughts I had about the relationship between invention and discovery in math, which felt worth sharing.
Euler's Characteristic Formula
This is one of my favorite proofs of all time, for a few reasons:
Here's the solution to the circle division problem. One of my hopes for this video is to illustrate several components to the problem solving process:
The original description of the problem is here:
The original 3blue1brown video project is a three part series on Euler's formula. This includes a video showing how it can be intuitive, a poem, and a visualization for how complex derivatives come into play. For more on how to think about exponentials, see this article. I had wanted to share these perspectives on what exponential functions are really doing, and also how to visual complex functions, as they've given me a great deal of satisfaction with a lot of beautiful math that otherwise felt opaque to me. In hindsight, I cringe at how quickly I talk over certain conceptually heavy points, but nevertheless I hope any viewer willing to pause and ponder finds it worth the effort.