Math Conjectures

Toeplitz Square-Peg Conjecture

OPENLandmarkConjectureProposed 1911 · Full conjecture

Canonical statement

For every continuous injective map γ:S1→R2\gamma:S^1\to\mathbb R^2, the Jordan curve γ(S1)\gamma(S^1) contains four distinct points that are the vertices of a nondegenerate Euclidean square.
View source LaTeX
For every continuous injective map
\(\gamma:S^1\to\mathbb R^2\), the Jordan curve \(\gamma(S^1)\) contains
four distinct points that are the vertices of a nondegenerate Euclidean
square.
DG-016:[Toeplitz1911SquarePeg][Matschke2014SquarePeg][Chambers2025SquarePeg]
Discrete geometryGeometry and topology

Notes

The conjecture is known under smoothness, convexity, and other regularity hypotheses. The case of an arbitrary Jordan curve, with no regularity beyond continuity and injectivity, remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.