This open problem taught me what topology is
If you’ve ever wondered what topology is, this problem is one of the best examples I know of to give an authentic sense of what it’s all about.
Books have a notion of a second edition, and while YouTube videos usually don’t, I wanted to make a new edition of one of the earliest videos on the channel, as this is one of my favorite pieces of math.
Aside from animating this beautiful piece of math better, I had an itch to address new research that’s happened since, and to pull in numerous other mind-bending connections.
Grant,
I would think intuitively that a proof for the rectangle and the box would start like this:
Take a fractal pattern like the Koch Snowflake but do not put in all the infinite lower levels.
I would believe that a series of boxes and rectangles would form, possibly depending on symmetry; say, having an even number of thorns. Then add more and more lower layers towards a geometric limit of infinite sublayers.
For such a symmetric fractal the boxes and rectangles might expand out towards a circle or converge into a circle as the thorns go infinitely down. This seems a lot like your circular geometric proof for the Basel theorem on the complex polar plane. And here symmetry might have a role to play (the usual symmetry/entropy interaction).
Further thoughts—I can see two limiting cases using the Koch Snowflake:
1) large boxes around the inside with the asymptote a circle
2) small box that diminish to dots with the asymptote a circle
I always struggled with topology. Now I have a much better introduction to it. Thank you, and thanks for all the wonderful videos—they are the stuff of legend. Congratulations! Keep doing what you're doing, and I'm definitely restacking this with a note of my own! I