The (recently solved) inscribed square problem or Toeplitz conjecture posits that every closed, plane continuous (Jordan) curve ${\it \Gamma}$ in $\mathbb{R}^2$ contains all vertices of some square. It appears this theorem was just proven!
Most examples resemble the one on the left, in which the natural continuous parameterization along ${\it \Gamma}$ intersects the square's points in sequence. Let's call such an inscription cyclic.
However, some Jordan curves contain the points of a square in non-sequential order, such as shown at the right. Let's call such an inscription acyclic.
Questions
- Are there Jordan curves that admit only acyclic inscriptions (and not also cyclic inscriptions)?
- Given the inscribed square problem was just answered in the affirmative, can one prove whether acyclic-only curves exist?
- Alternatively, or additionally: can one provide an example of such an acyclic-only curve?
My conjecture is that there are no such acyclic-only Jordan curves. My first approach has been to assume that there is a given acyclic square inscription for a Jordan curve and then prove–invoking continuity assumptions and topological methods–that there must also be a cyclic inscription. Alas, such a proof for even this partial case has been elusive.
Best Answer
The recent work by Jason Cantarella, Elizabeth Denne and John McCleary implies (but does not prove) that the answer to all questions is negative, as I conjectured. Specifically, under certain general conditions one can find a cyclic inscription, and thus precluding the existence of an acyclic-only Jordan curve.