Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Geometry P-inscribed-square

Inscribed square problem

Inscribed square problem Does every Jordan curve admit an inscribed square?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

Start working on it Submit a claim Follow
Cite 
@misc{cairn-inscribed-square,
  title        = {Inscribed square problem},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/inscribed-square}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

inscribed_square_problem. Inscribed square problem Does every Jordan curve admit an inscribed square?

inscribed_rectangle_problem. Inscribed rectangle problem Does every Jordan curve admit inscribed rectangles of any given aspect ratio?

The inscribed square problem or Toeplitz conjecture asks whether every Jordan curve (i.e. simple close curve in ℝ²) admits an inscribed square, i.e. a square whose vertices all lie on the curve. There are several open and solved variants of this conjecture.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.InscribedSquare (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

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
theorem inscribed_rectangle_problem :
    answer(sorry) ↔ ∀ (γ : Circle → ℝ²) (hγ : IsEmbedding γ) (r : ℝ) (hr : r > 0),
      ∃ t₁ t₂ t₃ t₄, IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) r

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.