Open problem, Geometry, Topology, posed 1911

Inscribed square problem (Toeplitz)

Open

Every Jordan curve $\gamma \subset \mathbb{R}^2$ (a continuous, simple closed curve in the plane) contains four distinct points that form the vertices of a nondegenerate square.

Research frontier as of 2026

As of 2026, Toeplitz's inscribed square problem remains open for general continuous (for instance, fractal or nowhere-differentiable) Jordan curves $\gamma \subset \mathbb{R}^2$ . For smooth Jordan curves, Joshua Evan Greene and Andrew Lobb (`arXiv:2005.09193`, Annals of Mathematics 2021) proved the stronger rectangular peg conjecture for all aspect ratios and later extended it to all cyclic quadrilaterals. In 2024 (`arXiv:2407.07798`), Greene and Lobb introduced spectral invariants from Jordan Floer homology to push Tao's Lipschitz-graph inscription theorem from Lipschitz constant $1$ up to $1 + \sqrt{2}$ . The central obstacle for general continuous curves is that when a smooth curve is approximated by smooth curves $\gamma_k \to \gamma$ , the side lengths of the inscribed squares on $\gamma_k$ could shrink to $0$ in the limit.

Best known results

Every smooth ( $C^\infty$ ) Jordan curve inscribes a rectangle of every aspect ratio and every cyclic quadrilateral (Greene and Lobb, 2020–2021).
Every locally monotone Jordan curve (without sharp cusps) and every curve bounded by two Lipschitz graphs of Lipschitz constant $< 1 + \sqrt{2}$ inscribes a square (Stromquist 1989; Tao 2017; Greene–Lobb 2024).

Tools and where they stop

Tool	Achieved	Where it stops
Symplectic geometry and Lagrangian intersection theory	Encodes pairs of chords with equal midpoint and length as intersections of Lagrangian submanifolds in $\mathbb{C}^2$ , resolving the rectangular and cyclic-quadrilateral peg problems for smooth curves (Greene–Lobb, 2020).	Requires smooth tangent vectors (or mild Lipschitz regularity via Floer homology); for rough continuous curves, infinitesimal squares along the curve cannot be separated from true macroscopic squares.
Topological parity and configuration-space invariants	Proves that generic smooth or piecewise-analytic curves have an odd number of inscribed squares, and that every continuous Jordan curve inscribes a rectangle (Vaughan, 1977).	In the limit of rough curves with corners or fractal oscillations, a sequence of nondegenerate squares on smooth approximations can collapse to a single vertex.

Open questions

Does every continuous Jordan curve in $\mathbb{R}^2$ (without any smoothness or Lipschitz hypothesis) contain the four vertices of a square?
Does every continuous Jordan curve inscribe a rectangle of every aspect ratio, or can fractal curves avoid specific aspect ratios?

References

Joshua Evan Greene, Andrew Lobb (2021). The rectangular peg problem · DOI:10.4007/annals.2021.194.2.4 · arXiv:2005.09193
Joshua Evan Greene, Andrew Lobb (2024). Square pegs between two graphs · arXiv:2407.07798 [preprint, not peer-reviewed]
Benjamin Matschke (2014). A survey on the square peg problem · DOI:10.1090/noti1100