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.
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
Oh, with respect to getting ones head around that that embedding of a mobius strip such that the edge lies on a plane but the strip doesn't intersect itself:
The edge of your twisted paper strip describes kind of a folded figure 8 in space, where the crossing doesn't intersect itself (by the width of the strip). Grab one of the two the two tips of that figure 8 - which lie next to one another because the 8 is folded - and pull it up and over to unfold the 8.
The unfolded 8 has 4 side points, and two of them are next to the bit under the crossing point. Grab one of them and pull down and across to turn the 8 into an 0.