Toeplitz’s Square Peg Conjecture (General Jordan Curves)

Establish the validity of Toeplitz’s Square Peg Conjecture in full generality by proving that for every Jordan curve C ⊂ ℝ² there exist four points on C that are the vertices of a non-degenerate square, thereby resolving the open case for arbitrary Jordan curves.

Background

The Inscribed Square Problem, also known as Toeplitz’s Square Peg Problem, asks whether every Jordan curve in the plane contains four points forming a square. The conjecture dates back to 1911 and has been proven in several restricted settings, including convex and piecewise analytic curves, C¹-smooth curves, curves of finite total curvature, and generic C¹ curves.

Despite these advances, the general case for arbitrary Jordan curves remains unresolved. This work uses a visual diffusion model to approximate inscribed squares directly in pixel space, highlighting the multiplicity of possible solutions and the suitability of diffusion models for exploring such geometric configurations. The open problem itself, however, is purely mathematical and concerns the universal existence of an inscribed square for all Jordan curves.

References

Formally, the conjecture states that for every Jordan curve $C \subset \mathbb{R}2$, there exist four points ${p_1,p_2,p_3,p_4} \subset C$ such that $p_1,p_2,p_3,p_4$ are the vertices of a non-degenerate square. More recent work demonstrates validity under additional low-regularity assumptions, yet it remains open for the general Jordan curve case.

— Visual Diffusion Models are Geometric Solvers  (2510.21697 - Goren et al., 24 Oct 2025) in Inscribed Square Problem, Problem Statement (Section)

As of today the question remains open.

— Plane separating continua inscribe rectangles  (2609.25574 - Morales-Fuentes et al., 22 Sep 2026) in Section 1, Introduction

The more general question of wether the same conclusion remains valid for every Jordan curve, is still open. This is known as the rectangular peg problem.

— Plane separating continua inscribe rectangles  (2609.25574 - Morales-Fuentes et al., 22 Sep 2026) in Section 1, Introduction

Does every plane separating continuum inscribe squares?

— Plane separating continua inscribe rectangles  (2609.25574 - Morales-Fuentes et al., 22 Sep 2026) in Section 4, Conclusions and further questions

The problem remains open for arbitrary Jordan curves; see the survey of Matschke .

— Rectangular Pegs on Jordan Curves of Finite $p$-Variation  (2609.04918 - Li et al., 4 Sep 2026) in Section 1, Introduction