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- TL;DR Summary
- 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:

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.

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

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.