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.
Why are trying to time travel … if we can fix someone in time with medicine or love or any strong force that doesnt allow someone to move from a fix point of time.
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