Attempt of Toeplitz Conjecture Proof

In this video I try to prove the Square Peg Problem using basic geometry and topology. This is one of several ways I found to prove it. Not sure if I did it in a correct way, but the video should provide a general vision on the approach. I'm sure there must be simpler and easier way to use my approach, but currently I need to understand if it worth working further on this conjecture.
Criticism is much appreciated.
(btw, there are curves where you can inscribe even number of squares)
Wikipedia article:
en.wikipedia.org/wiki/Inscrib...
I want to thank a lot to all the people making amazing videos visualizing math!! I was inspired by these videos:
discovermaths - Three unsolved problems in geometry
• Three unsolved problem...
3Blue1Brown - Who cares about topology? (Inscribed rectangle problem)
• Who cares about topolo...
Music by Vincent Rubinetti
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...

[@aramarakelyan5712]

Grade video, thank you for that! At 3:35 you claimed that by rotating by 90 degrees we get couple of intersections points. I suppose it is not always the case

[@Asixoid]

Thanks! Yes, it is correct, there are the cases when you get only one intersection point (corner of triangle), but I cover this later in the video. I don't claim there are exactly two, I use term 'couple' which implies 'some positive amount'. Also note that this section is made for demonstration of main principle of curve mapping.

[@aramarakelyan5712]

​@@Asixoid Well, that's very interesting. I have tried to understand, but so far it seems to me that the key statement in given solution the existence of a semi inscribed square (three vertices are on the curve) from any starting point

[@Asixoid]

@@aramarakelyan5712 Yes, that statement is one of the core lemmas.

[@equenos]

Great ideas! I will carefully rewatch the video later to check every step

[@markseagraves5486]

Lovely. May we know how you produce your graphics? What Computational software & NLE? Would you share the work behind the presented proof so we might test it with our own closed curves? More questions than answers, so we’ll done man!

[@Asixoid]

Thanks! Well, I use Rhinoceros with Grasshopper and a bunch of plug-ins. That is my main instrument for work. Not sure about sharing the original files, but I can test your curves and send you results if you want)) I think it is possible to build curve test environment on any popular math platform as Wolfram Mathematica or Jupyter, but I'm not that pro in such software, so maybe someone will build it before me.

[@caspermadlener4191]

The argument works for confex shapes for which for every point, there are two other points which form a 1:1:√2 triangle. This is sadly on always the case. Easy counterexample: tear 💧 There are at most three of those points (easy to prove), so I guess that part is fixable with ugly geometry. Also, the intersection curve can look absolutely horrible when the starting shape is not confex. I am a bit sceptical that you are able to prove that the curve is continuous. Or that there even is a single curve.

[@lillegitimate]

Very nice

[@jakubkonarik578]

Very cool

[@fly7188]

Brilliant

[@AlexanderQ689]

I really liked the pacing, laying out logical next steps without jumping too far I think the computer voice is okay and even has some benefits, but a few phrases seemed a little weird to me, like someone who's first language isn't English. And I don't mean any slight by that, just recommending you / others proofread your scripts more Otherwise, a perfect video and amazing as it is. Subscribing & looking forward to more :)

[@Asixoid]

Thanks! That was one my first videos of the kind, will do my best to improve! Such comments are inspiring tbh!

[@blusham4629]

Great !

[@DiamondSane]

I like math stuff, and these graphs and surfaces popping up and moving around. Seriously this is not my topic, but looks cool.

[@Asixoid]

Actually visual part helps understand the abstraction of math!

[@DiamondSane]

@@Asixoid sometimes it really does

[@pablolecce6931]

Try to use your own voice man. Feel the emotion in your voice will do the videos better!!

[@Asixoid]

There is a reason I don't use my own, but thanks for advice!

[@zyansheep]

@@Asixoid you could use a deepfake 👀. "Math videos by Morgan Freeman" lol

[@lock_ray]

The "inscribed L" construction at 3:30-5:00 is definitely in the right direction, I know that something similar was used to show e.g. the Lipschitz case. I wasn't really able to follow the argument from 5:00 onwards, however.

[@Asixoid]

That would be pretty hard to shine more light in the comment. We can chat using some messenger if you want, drop me an e-mail. I'd be happy to clarify.

[@DS127]

Very good video. Your grammar is off, but your thoughts come through clearly. The script and animations together make what you mean make sense. The mistakes in grammar might hurt the video if your explanations weren't so well done. Your script doesn't go too fast or waste time. Your visuals make things clearer without just being flashy, but still look cool. I see 3blue1Brown's "inscribed rectange problem" video in the sidebar. I feel like he occasionally goes "look! ~*fancy graphics*~" in an explanation without the visuals actually being helpful. Making stylish visuals is fine, but the audience not understanding the way some explanatory graphics "work" might make the ideas behind those graphics seem *more* complicated.

[@NoNameAtAll2]

just wanted you to know by using computer voice, you became my hero

[@NoNameAtAll2]

and it's not even _that_ noticable

[@hahnfelt]

Why…….?

[@Asixoid]

why what?

[@hahnfelt]

@@Asixoid why does this matter? What does it explain? Anybody can filosofy as much as they want about anything, we’re all geeks one way or the other, but do we pay mathematicians to scratch heads on this? If so, for what?

[@stepexgd6628]

@@hahnfelt Because one day, calculations like these might yield practical results. Hell, could you have predicted that integration can be useful in order to calculate motion along a curve?

[@valovanonym]

@@hahnfelt why? Why are you watching if you don't care?

[@harryfan8785]

@@valovanonym To understand why to care.