Open problemChecked July 24, 2026

Inscribed Square Problem

A century-old question about finding a square on every simple closed plane curve.

Geometry Topology Jordan curves

Mathematical statement

Every Jordan curve $\gamma:S^1\to\mathbb{R}^2$ contains four points that are the vertices of a nondegenerate square.

Statement source: Therefore editorial statement

Statement terms: Source-specific

Editorial wording by Therefore. No statement reuse license is granted.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

inscribed_square_problem

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem inscribed_square_problem :    answer(sorry) ↔      ∀ (γ : Circle → ℝ²) (hγ : IsEmbedding γ),        ∃ t₁ t₂ t₃ t₄,          IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) 1 := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

AMS survey on the square peg problemsurvey Inscribed square problem overviewbackground