Three unsolved problems in geometry

The Toeplitz conjecture and perfect cuboid problem are among easy-to-understand geometry problems that remain unsolved.

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[GlorifiedTruth]

I have successfully inscribed squares in 140 million different Jordan curves. I figure that's about half way to infinity, so I'm well on my way to proving the conjecture.

[nzuckman]

I like the 3rd problem a lot. I like the idea of disproving the perfect cuboid by deriving various restrictions and getting two which contradict each other

[gedlangosz]

I've not heard of any of these before! Thanks - another excellent video.

[LucenProject]

Ahh! I've been working on the perfect cuboid problem as something I randomly thought up! I didn't know it had a name! Thank you for this! Now I can look into it properly!

[manypeople]

I love the video, but there seems to be a small mistake. Should be 1+2^(n-2) instead of 1+2^(n-1).

[rylaczero]

Interesting. Would love to know more of the unsolved problems in other mathematical areas.

[dickyarya]

I'm not even a math student yet I'm here lol

[joshuasparber]

It seems to me that the Jordan curve problem might need a proof similar to that of the Poincare Conjecture. I would believe that whether a square can be constructed from a curve, any curve, would depend on whether the curvature on distinct points was such that the tangents or the limit the tangent is going to would be perpendicular to each other. This could be put into mathematical language. For instance, look at the instances in which this would work, the hypocycloid. The limits of the tangents at the cusps are such that a square can be constructed from the points. But flatten one of the points a little and this would no longer be the case. This is a problem in which there are so many cases it might be better off to work it out intuitively rather than with a computer. This would also be the same as saying that there are no two lines in which a Cartesian product domain can be constructed. One would think that there would be cases in which this could not be done. So possibly, not every curve that can be constructed would have 2 lines within which such a Cartesian product could be constructed.

[kasuha]

I spent some time on the Toeplitz conjecture some time ago (inspired by a 3blue1brown video) and I'd swear I have a proof for any curves that aren't fractal surfaces without defined tangent at uncountable number of points. And it should hold even for these too, I just did not find how to prove that. But I'm no mathematician and given how long the problem is known most likely someone else already tried my approach and found some shortcomings I couldn't see. Also, if the conjecture holds for Jordan curves, it holds for any curves containing a loop since such curves always contain a Jordan curve.

[francogonz]

I think I actually have a counterexample to Toeplitz Conjecture. Im working with a professor from my university on that

[kashnigahbaruda]

The perfect cuboid was very surprising not to have been solved or found!

[arekkrolak]

The perfect cuboid problem I saw recently on the list of 5 open number theory problems :)

[moer]

If someone were to come up with a solution, where would they go to show their work?

[DumAndSmart]

I’m working on the third problem I figured out what the last two digits are for each number when you’ve squared it with just a calculator well I haven’t figured them out but figured three possibilities
sides 64, 36 diagonal 61, 89
Sides 44, 56 diagonal 81, 69
Sides 04, 96 diagonal 21, 29
Both one side and the spatial diagonal must end in 25 and one of the diagonals( the 44, 56/04, 96 diagonal) must end in two zeros

[robert-skibelo]

Great material, nice graphics too. It's so refreshing to hear a calm, clear presentation, as opposed to all those frenzied, shouty Americans elsewhere on Youtube. And to hear Z pronounced properly. ;-) You've misspelt Toeplitz in at least one place and the vowel of the first syllable should be more like that of "berth", not "go".

[IntegralKing]

I love how these problems are so easy to describe, but immensely difficult in their execution

[mouryapaladugu]

The second one is so simple. I wonder why it's so difficult to prove.

[artsmith]

1:40 Typo on the screen. "1 + 2^(n-2)" is spoken, but "1 + 2^(n-1)" is displayed.
The wiki page says the conjecture is for "1 + 2^(n-2)".
A table of the values would have been interesting to see:
n 1 + 2^(n - 2)

4 5
5 9
6 17

[RJSRdg]

By my reckoning, the third problem may have solutions which satisfy h^6 - h^5 - h^4-h^2 - h + 1 = 0 where h is the square of the cube root of the shortest side.

Now I just need to find a way of solving that equation (or not!).

[CatPerson]

Now I'm wondering is it possible to have an unsolvable equation in mathematics