Who cares about topology? (Old version)

3Blue1Brown explanation of eigenvectors and eigenvalues is very insightful and intuitive [1]. One of my favorite videos from his channel is "Who cares about topology? (Inscribed rectangle problem)" [2]. They way a Torus and Möbius Strip come up as solution to the problem is so elegant. [1]: [LINK] [2]: [VIDEO]

I really like 1blue3brown, but this one didn't quite resonate with me. To me, it felt like a long winded way to make it intuitive that the point (1, 1 ... 1) has distance sqrt(n) from the origin, where n is the number of dimensions. I think there'd be move value in explaining things like 'most volume is near the edge' and 'most points are far apart'. Granted, that is less about building intuition, and more about specifically dispelling heuristics from 2 and 3 dimensions. I'd still wholeheartedly recommend his other videos though. Especially [1] where he gives a very nice topological result regarding inscribed rectangles in closed loops. In the same vein is [2] proving the borsuk-ulam theorem. [1] [VIDEO] [2] [LINK]

That's really interesting. I found that I loved, similar to the occasional video numberphile does, watching Grant explain how one might realistically solve an interesting problem. A particular favourite is the one on the Inscribed Square Problem, which, although introducing some elements of topology, engages me more from the get go. He poses a problem and my curiosity keeps me engaged until the very end, it feels a bit like a detective story, I get the same satisfaction as some of the better Sherlock Holmes stories where in the course of reading I think "Damn, I could have come to that conclusion if I'd paid a bit more attention!". [VIDEO]

Here's a nice 3blue1brown video about the easier version of the inscribed rectangle problem, that some inscribed rectangle exists: [VIDEO] And I remember thinking this was a nice overview of some aspects of the "square peg problem" (the Toeplitz conjecture): [LINK]