Who cares about topology? (Inscribed rectangle problem)

In May 2020, it was proved (arXiv:2005.09193) that if your curve is smooth as well as continuous, then you can not only find a rectangle, but in fact you can find a rectangle of ANY proportions you'd like - that is, given any ratio r, one can find an inscribed rectangle whose side lengths have ratio r.

the hours of work this took.... I can only imagine.

10:38 I laughed at this harder than I should have just because I wasn't expecting it. The specific intonation he used to say "Not helpful!" just captures the feel of the moment.

Watching this now again while learning topology formally for the first time (I'm physicist and decided it's time). It all becomes so much clearer. "oh he means homotopy", "this is an equivalence relation", "hey I know, this is a torus!", etc. It was so much fun!

When you grab a mug and instantly think about a donut, you are either into topology or have a sweet tooth. Ok... maybe both.

I watched this video back in May 2017, just after securing a place for a Maths Master's degree. I loved it at the time, and made me sure that I'd chosen the right degree. 4 years later, with degree in hand, I still love this video, and is all the more impressive to me now. It's fantastic how the reparameterisation of the space of unordered pairs of points is explained and visualised, without using any of the terminology like R/Z that (I now understand) would be so tempting for an experienced mathematician like Grant to use in an offhand way, but would have been lost on my younger self - add too much of this terminology and I would have missed the beauty of the proof in the first place. Bravo Grant, keep being amazing!

as he said "isn't that awesome?", when pairs of unordered points are folded into a Mobius strip, I screamed "Hell yeah!". it's just fascinating man :D

It amazes me how you can make such complex topics so easily understandable to anyone who is willing to think for a few minutes. Definitely one of my favourite videos ever!

I lost a friend recently because of how much he disliked maths. It was his hatred of topology that really torus apart. (In all seriousness, I LOVE topology)

It's totally mindblowing how you can talk about things like continous functions/Homeomorphisms, Product topologies, Quotient topologies and ... - is that a commutative diagramm at 14:03?- and make it understandable and sensefull and usefull and everything even to people who never heard of topology. Thanks for everything!

I’m about to take algebraic topology. Since we’re stuck at home due covid, my topology professor is uploading some of his lectures. I’ve watched some videos already and this videos helped me out a lot to visualise what an homotopy is!

As a mathematian, what really amazes me is the way you show those unbelievably complicated arguements so simply and ellegantly.

There are a lot of awesome videos on YouTube, but this one is probably the video of the year for me. AMAZING, AMAZING! Super interesting problem, excellent explanation, awesome animations. Also happens to be just the right level for me (not too easy, not too hard). So thanks for this amazing work!

14:36 aaand the music starts playing just as the solution becomes beautiful. It's perfect. This video is perfect.

Crazy how a free Youtube video can be more educational than some expensive college lectures.

Oh my, this is my new favorite math video! It's an absolute masterpiece. It's just so awesome that these crazy shapes like the Mobius strip and torus helped us solve a concrete problem. I haven't formally learned any topology yet, but this video has made me excited to learn about it. Thank you so much!

I'm a topologist and I think it's briliant.

I have topology in 2 years and you just hyped the fuck out of me.

10:37 This could be turned into a good meme

Rarely have I felt compelled to comment on a video. I've seen every 3B1B video at least once (including this one) but for whatever reason, YouTube suggested I watch this again and this video stuck out. This is one of the most clear visual and verbal explanations of a complicated topic I have ever seen. Amazing and fascinating. Beautiful video.