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Can the counter examples found by me qualify as a counter proof for the "inscribed squares problem" (the Toeplitz' conjecture) ?

I ask this here because the problem stands as unsolved for 100 years, and the counterexamples below appear too simple. It is hard to believe that I am the first to find them. So, is there something I missed out?

Here are the curves:

Solution http://automaton.opa.ro/wp-content/uploads/2013/09/solution_4_s30.png Solution http://automaton.opa.ro/wp-content/uploads/2013/09/solution_1_s30.png Solution http://automaton.opa.ro/wp-content/uploads/2013/09/solution_3_s30.png

Mihai Cirneala
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  • Do you have proof for the claim that these curves are counterexamples? – Servaes Oct 26 '13 at 16:14
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    Could it be that you only checked for squares that have corners on your grid? The third curve has a rather obvious square in it, at least. – Karolis Juodelė Oct 26 '13 at 16:14
  • I don't have a formal proof, but I did a numerical verification, which may be or may not be good enough. – Mihai Cirneala Oct 26 '13 at 16:21
  • Yes, the third one has an obvious square. – Mihai Cirneala Oct 26 '13 at 16:27
  • It is not good enough; there is clearly an inscribed square in the first and third curve. In fact, all three of the curves have an inscribed square because they are locally monotone.

    If you intend to find a counterexample, I suggest you first read some of the articles linked on the wikipedia page that prove there exist none for certain kinds of curves.

     – Servaes Oct 26 '13 at 16:29
  • Yes, I did check only the corners, but visually it appeared to me that no square can fit into that. I will try to also check some intermediate points between the corners though. – Mihai Cirneala Oct 26 '13 at 16:47
  • You should note that for any proposed counterexample $C$, you would need to show that for each grid of any given square size, and for any given rotation of $C$, no square can be inscribed in $C$. Since there are uncountably many grid sizes, and uncountably many rotations, your computer, being ever able to check a finite number of possibilities, will never be able to completely say with certainty that $C$ is a counterexample simply by numerical verification. – Casteels Oct 26 '13 at 17:06
  • I realize now that it's not possible to do a numerical verification. Still, it may be possible to use a computer program to find curves that have better chances to be a counter example (if you also have a good grasp of the subject and you know what you look for). And then you have to use analytic methods to prove it. – Mihai Cirneala Oct 26 '13 at 21:55

2 Answers2

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Solution Solution Solution So no, it's not a counterexample

Adam
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The conjecture is known to be true for piecewise $C^1$ curves, which yours certainly are since they are polygons. If you want to solve these types of problems you are going to have to learn hard mathematics, not just write programs. Computers can't do mathematics.

TBrendle
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  • your pictures are not counterexamples. it very easy to find square $\sqrt5\times\sqrt5$ on the 1st picture. – bot Oct 26 '13 at 16:18