I remember being blown away when I first read Mandelbrot's "The Fractal Geometry of Nature" about how utterly reasonable the definition of fractal dimension is. Moreover, as the title of the video suggests, I think that was the first time I saw that the original motivation for defining fractals did not have self-similarity as the central focus, but instead centered around capturing the idea of roughness that persists at all levels.
If I could revisit the video, one thing I might change is how I discuss the definition of a fractal. I wanted something concrete that a viewer could come away with, a way to fill in the blank as they say "fractals are not self-similar, instead they are _______". But I fear that focusing on non-integer dimension might not do appropriate justice to the nuances in trying to define roughness. I opted not to say that fractals are shapes whose fractal dimension exceeds their topological dimension because I didn't want to get into topological dimension. But even that definition misses a few rough objects. A rather nice analogy that one commenter referenced is that defining fractals for mathematicians is a bit like defining life for biologists: We all know what it is that we're trying to encapsulate, but every time you try to nail it down with a concrete definition it seems to neglect some of what you want, or to admit some of what you don't want.
This one is definitely a very different video for me. When a friend of mine was getting a math tattoo, I decided to record it with my phone just in case it might make a fun video. After pondering it, and deciding that there could be a nice narrative about the man-made permanence of conventions along with a mini-lesson on how to interpret trig quantities, I decided to just go for it.
While working on (well, procrastinating from) an "Essence of Calculus" video, I was playing around with visualizing various complex functions. When I put in the zeta function, it struck me as so pretty that I just had to make it the next project. This was especially true when I realized it could give a nice way to motivate the idea of analytic continuation, which is one of the more opaque aspects of this function that stalls people trying to read more into it.
Hopefully, this can help motivate people to learn about complex analysis more generally.
Binary, Hanoi and Sierpinski
This pair of videos includes several really surprising relationships, between binary/ternary counting, the Towers of Hanoi puzzle, and Sierpinski's triangle. I was also pretty excited to be able to do a project with Keith Schwarz, a phenomenal CS lecturer at Stanford. Anyone curious should look at some of his project at http://www.keithschwarz.com/
This is also surprisingly relevant to the space filling curves video I did some time ago. In those, I included a brief animation of a curve that fills Sierpinski's triangle, but I'll admit that it was rather unintuitive to me. This here was the first time that I felt a deep intuitive reason for why such a curve should exist, and can naturally be defined.
Who cares about topology?
This is, quite possibly, my favorite pieces of math of all time. As I say in the video, I think it was the first time I realized why mathematicians care about the properties that certain shapes have which don't change as you squish and morph around that shape. Sure, the mobius strip, torus and Klein bottle are mildly interesting to look at, but until you see one used to solve a problem, contemplating these shapes feels more like intellectual indulgence than math.
I never actually define topology in the video, doing so only really makes sense if you have some history with more abstract math, but I do think the viewer who understands the cleverness here has an appreciation for the spirit of topology itself.
Triangle of power
Who doesn't love a notation rant? I came across this idea on math stack exchange (credit to Alex Jordan), and thought the symmetry here made it well worth sharing. Also, this has a bit more of the "popular appeal" factor than some of my other videos.
I have huge respect for Steven Strogatz, and the work he's put into math outreach. So when I saw that he had tweeted some nice things about my early videos, I reached out to see if he wanted to work on something together. Doing the Brachistochrone, and communicating Mark Levi's clever insight, was all his idea, and I'm really glad he suggested it.
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.