Rewriting the Toeplitz Conjecture

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In summary, the Toeplitz Conjecture states that all Jordan curves have an inscribed square. It has been stated in the early 1900's and remains an open problem.
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MevsEinstein
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I was thinking of the Toeplitz conjecture in a backward motion. Maybe it makes proving it easier?
The Toeplitz Conjecture (better known as the inscribed square problem) states that all Jordan curves have an inscribed square. It has been stated in the early 1900's and remains an open problem.

I drew a square and then making a ton of curves that touch its four vertices:

inscribed1.PNG

This shows that the square is inscribed in many curves.

Now what if I transformed (dilate, translate, rotate, or reflect) these curves? Well, those curves will also have inscribed squares. From this diagram, I rephrased the Toeplitz conjecture as such: When we make every possible transform on all the curves that have the square above as an inscribed square, they will map on to every other Jordan curve.

So I was thinking, does this make the Toeplitz conjecture easier to prove? I thought that my definition can help with using set notation.
 
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  • #2
It is the other way around: Given any closed Jordan curve, is there always an inscribed square?

330px-Inscribed_square.svg.png


(copyright by Claudio Rocchini, Wikipedia, https://de.wikipedia.org/wiki/Toeplitz-Vermutung)

The black Jordan curve has many inscribed (= contains all 4 vertices) squares.

The already proven special cases (convex, piecewise smooth) make it hard to draw other examples. Plus the fact that it is still open after more than a century.
 
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  • #3
mevseinstein, note that your drawings are of the case of a strongly inscribed square, more restrictive than the ones meant in the conjecture, as explained by fresh42.

may i suggest you try warming up by attempting an easier case?, e.g. try to prove that a jordan curve always contains the vertices of some equilateral triangle. your Idea of using dilation and rotation are sufficient in this case, with some continuity arguments. or perhaps you already know how to do this.
 
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  • #4
MevsEinstein said:
Now what if I transformed (dilate, translate, rotate, or reflect) these curves?
I think that is only possible for "smooth" curves, and not for these "nowhere differentiable" curves (like fractal curves)
 
  • #5
fresh_42 said:
It is the other way around: Given any closed Jordan curve, is there always an inscribed square?

View attachment 304723

(copyright by Claudio Rocchini, Wikipedia, https://de.wikipedia.org/wiki/Toeplitz-Vermutung)

The black Jordan curve has many inscribed (= contains all 4 vertices) squares.

The already proven special cases (convex, piecewise smooth) make it hard to draw other examples. Plus the fact that it is still open after more than a century.
Don't mean to nitpick, just to make sure we're using the same definitions:

A Jordan Curve is , in my understanding, close by definition; homeomorphic to the standard unit circle ( ## x \in \mathbb R^2 : || x||=1 ##)

I guess not every choice of homeomorphism will take inscribed squares to inscribed squares.
 
  • #6
WWGD said:
Don't mean to nitpick, just to make sure we're using the same definitions:

A Jordan Curve is , in my understanding, close by definition; homeomorphic to the standard unit circle ( ## x \in \mathbb R^2 : || x||=1 ##)

I guess not every choice of homeomorphism will take inscribed squares to inscribed squares.
I didn't look it up (and of course don't remember the correct definition anymore). I just read the Wikipedia article on Toeplitz and it said that the cases "piecewise smooth" and "convex" are proven. That led me to the assumption that Jordan is closed and possibly of genus 1.
 
  • #7
fresh_42 said:
I didn't look it up (and of course don't remember the correct definition anymore). I just read the Wikipedia article on Toeplitz and it said that the cases "piecewise smooth" and "convex" are proven. That led me to the assumption that Jordan is closed and possibly of genus 1.
This seems to be leading to Algebraic Geometry. Maybe @mathwonk can clarify here? IIRC, genus 0 curves are those that are parametrizable? Or am I way off? Edit: I thought genus applies to higher dimensional objects, not to curves.
 
  • #8
WWGD said:
This seems to be leading to Algebraic Geometry. Maybe @mathwonk can clarify here? IIRC, genus 0 curves are those that are parametrizable? Or am I way off? Edit: I thought genus applies to higher dimensional objects, not to curves.
I don't care. I just wanted to say "without crossings" a bit more sophisticated. Now I look it up.

Jordan curve = continuous, injective image of [0,1].

This excludes crossings, but requires everything else to be mentioned: closed, rectifiable, 0-homotope, smooth, convex or whatever.
 

Related to Rewriting the Toeplitz Conjecture

1. What is the Toeplitz Conjecture?

The Toeplitz Conjecture is a mathematical problem proposed by Otto Toeplitz in 1911. It states that every positive integer can be written as the sum of at most four squares, where the squares can be repeated.

2. Why is it important to rewrite the Toeplitz Conjecture?

The original formulation of the Toeplitz Conjecture has been proven to be incorrect. Therefore, it is important to rewrite the conjecture in a way that accurately reflects the current understanding of the problem.

3. What progress has been made in rewriting the Toeplitz Conjecture?

There have been several attempts to rewrite the Toeplitz Conjecture, with varying levels of success. Some have proposed alternative formulations that are more accurate, while others have focused on finding counterexamples to the original conjecture.

4. What are the potential implications of rewriting the Toeplitz Conjecture?

If a new formulation of the Toeplitz Conjecture is found and proven to be correct, it could have significant implications for number theory and our understanding of the relationships between numbers.

5. Is rewriting the Toeplitz Conjecture a solved problem?

No, rewriting the Toeplitz Conjecture is an ongoing research topic in mathematics. While progress has been made, there is still much to be explored and understood about this problem.

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