Hilbert's Curve

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.

Binary Counting

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.

Inventing Math

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:

• The culmination of relevant facts into a proof comes as quite a surprise (or at least it did for me when I first heard it).  Such surprise constitutes the heart of any "slick proof".  
• It touches on the notion of duality, which is an incredibly far-reaching concept.  Duality is undeniably beautiful in the symmetries it produces, and generally unknown outside the higher echelons of mathematics education.  
• It provides a nice different perspective on a very classic theorem.

Circle Division

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:  

• Ask simpler questions to warm up.
• Clarify what you do know, even if you don't know if it will be useful.
• See if you can relate your problem to deeper and more general truths in math.
• Don't settle for a concrete formula unless you feel like you really understand it, in the sense that you can explain why it has the properties that it does.  

The original description of the problem is here:

Euler's Formula

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.