# P13: Square peg problem

- ID: `P13`
- Reference: `square-peg-problem`
- Page: https://theoremdb.org/statements/P13
- Record maturity: Reviewed problem with recorded work

## Problem

For every simple closed curve \(C\subset\mathbb{R}^2\), there are four distinct points of \(C\) that are the vertices of a square.

### Context

The difficulty lies in passing from regular curves to arbitrary continuous embeddings while preventing inscribed squares from degenerating.

### Problem setup

- **Definition (A simple closed plane curve).** A simple closed plane curve is a continuous injective image of a circle in the plane, also called a Jordan curve.
- **Definition (The square's four vertices must lie on the curve; its interior may cross the curve).** The square's four vertices must lie on the curve; its interior may cross the curve.
- **Remark.** The difficulty lies in passing from regular curves to arbitrary continuous embeddings while preventing inscribed squares from degenerating.

### What counts as a solution

- Prove that every Jordan curve contains four distinct vertices of a square, or construct a Jordan curve and prove that it contains no such square.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: Chambers proves the result for curves sufficiently close to a C2 Jordan curve. Greene and Lobb obtain squares for a further area-versus-diameter class. The general continuous case remains open. Exact unresolved remainder: Prove every Jordan curve inscribes a nondegenerate square, or certify a Jordan curve with none. [1](#reference-1) [2](#reference-2) [3](#reference-3)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: Chambers proves the result for curves sufficiently close to a C2 Jordan curve. Greene and Lobb obtain squares for a further area-versus-diameter class. The general continuous case remains open. Exact unresolved remainder: Prove every Jordan curve inscribes a nondegenerate square, or certify a Jordan curve with none.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: Chambers proves the result for curves sufficiently close to a C2 Jordan curve. Greene and Lobb obtain squares for a further area-versus-diameter class. The general continuous case remains open.

Exact unresolved remainder: Prove every Jordan curve inscribes a nondegenerate square, or certify a Jordan curve with none.

### Background and intake notes

- Original intake status: The cited scholarly paper states that the problem is open for general embedded plane curves. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited paper on 2026-07-22.
- The problem is proved for many smoother or otherwise restricted classes of curves. A proposed proof must cover arbitrary Jordan curves.

- Recorded example: A circle contains infinitely many inscribed squares.

### Open directions

- **Route 1** (reported): Prove that every Jordan curve contains four distinct vertices of a square, or construct a Jordan curve and prove that it contains no such square. [1](#reference-1)

### Computational notes

- Polygonal approximation can locate squares on particular curves but does not by itself control a limiting square on every Jordan curve.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `square-peg-problem`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Benjamin Matschke, “On the Square Peg Problem and Its Relatives”. arXiv:1001.0186 (2009). Benjamin Matschke, arXiv:1001.0186, abstract and introduction https://arxiv.org/abs/1001.0186
   - Also cited at Abstract and introduction
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:1001.0186, checked 2026-07-31; checked 2026-07-31
   - Source use: original_summary
   - The cited scholarly paper states that the problem is open for general embedded plane curves. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - Packet-linked survey and general status.
   - Source named by the research packet.
2. <a id="reference-2"></a>Gregory R. Chambers, “On the Square Peg Problem”. Discrete & Computational Geometry 73(4) (2025), 1144-1153. DOI 10.1007/s00454-025-00720-x. Abstract and main theorem https://doi.org/10.1007/s00454-025-00720-x
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Positive result for curves close to a C2 curve.
3. <a id="reference-3"></a>Joshua Evan Greene and Andrew Lobb, Floer Homology and Square Pegs, current author manuscript. maths.dur.ac.uk checked 2026-08-01. Abstract and introduction https://maths.dur.ac.uk/users/andrew.lobb/JF.pdf
   - preprint; primary source; Current author manuscript checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Rectifiable-curve rectangle interval and conditional square result.