A specific example regarding the inscribed square problem

Question

Toeplitz' conjecture (also called inscribed square problem) says that:

For every Jordan curve $\mathscr C$, there exists four distincts points $A$, $B$, $C$ and $D$ belonging to $\mathscr C$ such that $ABCD$ is a square.

A Jordan curve is a non self-intersecting continuous loop.

Here is a drawing to illustrate the situation, and a link to the Wikipedia page if you want to find out more about this conjecture.

The conjecture has already been proven in several cases, including when $\mathscr C$ is piecewise analytic.

So we know that for these two figures, there exists an inscribed square.

The question is how do I find those squares?

@Aretino Not exactly... I was trying to convince myself that Toeplitz' conjecture is indeed true, so I tried it on a few example, but I did not manage to find a square in those particular ones. — E. Joseph, Feb 06 '17 at 21:02
Are you looking for a general algorithm to find a square in any regular curve? — Intelligenti pauca, Feb 08 '17 at 18:38
@Aretino I don't dare to hope that such algorithm would exist (does it?), I was just wondering about that particular example. — E. Joseph, Feb 08 '17 at 19:07

score 7 · Accepted Answer · edited Dec 14 '19 at 07:15

7

There is. If we draw an orthogonal line with respect to the one that is aligned with the curve, then you can build the square.

edited Dec 14 '19 at 07:15

emonHR

answered Dec 14 '19 at 05:45

1 Answers1