The inscribed square problem (Toeplitz's square peg problem)

Does every closed curve in the plane that does not cross itself pass through the four corners of some square? Toeplitz asked this in 1911. The answer is yes for every reasonably regular curve, including all smooth curves, all polygons and all convex curves, but curves that are rough at every scale remain out of reach.

Problem

A Jordan curve is a continuous injective map $\gamma$ from the circle into $\mathbb{R}^2$, that is, a simple closed curve. Four points of the plane are the vertices of a square if they are the corners of a non-degenerate square, in some order.

Question. Does every Jordan curve contain the four vertices of some square?

A natural strengthening asks for inscribed rectangles of every prescribed aspect ratio.

Background

Otto Toeplitz posed the problem in 1911. Most proofs are topological, of a homological nature, and need some regularity of the curve. For a general continuous curve they break down because inscribed squares might shrink to a point when one approximates the curve by regular ones, and no limiting argument is known to prevent this. Matschke's survey describes the many partial results and variants.

What is known

• Every Jordan curve contains the vertices of some rectangle, but without control over its aspect ratio.
• Squares exist for piecewise analytic curves, including all polygons (Emch, 1916), for smooth curves (Shnirelman, 1944), and for locally monotone curves, which include all polygons and all convex curves (Stromquist, 1989).
• Tao (2017) proved the result for curves that are the union of two graphs of Lipschitz functions with Lipschitz constants less than 1; Greene and Lobb (2024) extended this to constants less than $1 + \sqrt{2}$.
• Greene and Lobb (2021) proved that every smooth Jordan curve contains rectangles of every aspect ratio, using symplectic topology.
• The formal-conjectures file records the inscribed square for $C^2$ curves as known, and states both the square and rectangle questions for general Jordan curves as open.

What would count as progress

• A proof for new classes of rough curves, for example all Lipschitz curves, all rectifiable curves, or graphs with larger Lipschitz constants.
• A proof that inscribed squares cannot shrink to a point under approximation for some broad class of curves, which would let the topological arguments pass to the limit.
• Progress on the simpler variants proposed by Tao (2017), including a periodic version that is open even for regular curves.
• A formal proof of the Lean statement inscribed_square_problem, or of one of the known special cases stated in the same file, Wikipedia/InscribedSquare.lean.

References

1. B. Matschke, A survey on the square peg problem, Notices Amer. Math. Soc. 61 (2014), 346.
2. W. Stromquist, Inscribed squares and square-like quadrilaterals in closed curves, Mathematika 36 (1989), 187–197.
3. T. Tao, An integration approach to the Toeplitz square peg problem, Forum Math. Sigma 5 (2017), e30.
4. J. E. Greene and A. Lobb, The rectangular peg problem, Ann. of Math. 194 (2021), 509–517.
5. J. E. Greene and A. Lobb, Square pegs between two graphs, arXiv:2407.07798 (2024).

Source

Imported from Formal Conjectures Wikipedia/InscribedSquare, curated by Google DeepMind and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.