Convex geometry · universal covers · rectifiable planar curves · certified optimization

Moser’s Worm Problem

Collaboration beta

How little area can a convex planar shape have while still fitting every curve of length one after moving it? Current certified lower and upper bounds leave a substantial gap.

M_{\pm} = \inf_{K} area (K), M_{+} = \inf_{K} area (K)

Known results and sources

Moser’s problem asks for the least-area convex shape that can fit every unit-length planar curve after an allowed rigid motion.

Research problem

Exact mathematical statement

Let $\mathcal W$ be the images of rectifiable maps $[0,1]\to\mathbb R^2$ of length at most one. Let $M_+$ be the infimum area of compact convex sets containing a rotated and translated copy of every $W\in\mathcal W$ , and let $M_{\pm}$ allow reflections as well. Determine these infima and the optimal covering bodies.

Problem infographic

Problem at a glance

A four-part explainer defines unit-length worms, compares the two motion conventions, displays certified lower and upper area bars, and marks the unknown optimizer. — The source reports a common certified lower endpoint and separate upper covers for reflection-allowed and orientation-preserving motion; neither optimum is known.

Current mathematical picture

Where work on Moser’s Worm Problem stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureFurther micro-retuning of the superseded calibrated-frame pentagon

A retuned pentagon and robust-polygon searches without complete certificates are retained only as historical or exploratory evidence. Translation-coupled generic polygons, a fourth lower witness, Wetzel-certificate sensitivity, and reflection-symmetric upper bodies remain viable with exact certificates.

Route status · Narrowed route

Main reductionLocal and external evidence separated

The lower certificate is reported locally reproduced, while the full 599-model external certificate was not executed in the current work audit.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeFind a genuinely translation-coupled polygonal witness that improves the lower bound.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Moser’s Worm Problem in numbers

2.3kretained lines of mathematical investigation2,290 in the current working snapshot

Argument development: 1,970 · 86%
Explored or eliminated routes: 62 · 3%
Computational analysis: 89 · 4%
Open obligations: 56 · 2%
Definitions and setup: 113 · 5%

8selected mapped statements1routes investigated4open questions4contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byLeo MoserProposed1966Status checkedAug 20, 2026

Source recordedAug 20, 2026

Recommended next task

Find a genuinely translation-coupled polygonal witness that improves the lower bound.

Suggested move: Evaluate exact triangle and parallelogram baselines, then certify generic quadrilaterals, centrally symmetric hexagons, and larger families through the full dual polytope and angle range.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeFurther micro-retuning of the superseded calibrated-frame pentagon

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Find a genuinely translation-coupled polygonal witness that improves the lower bound.Suggested move: Evaluate exact triangle and parallelogram baselines, then certify generic quadrilaterals, centrally symmetric hexagons, and larger families through the full dual polytope and angle range.

Ready to work on

Force extra support excess at the nearly tight cells of the 35-link lower certificate.Suggested move: Add a fourth witness while retaining common placement variables and certify its gain at calibrated side and cap-switch cells.

Ready to work on

Optimum and convention gap remain open

The source does not identify an optimal convex body or prove equality of the two motion-convention infima.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Improve a certified upper body without losing coverage of any Wetzel branch.Suggested move: Run the pinned official supplement, extract weakest-model dual sensitivities, perturb support offsets or angles, and regenerate all 599 interval-valid inequalities.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 20, 2026

Current statusOpen problem

The minimum area of a compact convex planar region that can accommodate every unit-length arc remains unknown. Certified lower bounds and source-reported universal covers leave a nonzero gap; a cover theorem is not an optimality proof.

[1][2][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
PreprintA 2026 preprint reports that Wetzel's 30-60-90 triangle covers every unit arc through 599 validated optimization models; ProofAtlas has not independently reproduced those certificates.[3]
2019
PreprintPanraksa and Wichiramala proved that Wetzel's sector covers every unit arc, providing an upper-cover construction rather than the exact optimum.[2]
2009
Peer reviewedKhandhawit and Sriswasdi proved a 0.227498 lower bound by requiring any cover to accommodate several specific unit-arc witnesses.[1]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements2 proposed statements4 open questions1 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction2 of 82
lemma3 of 83
equivalence1 of 81
computational claim1 of 81

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 1 route included

retained route statementWhat is the smallest convex universal cover for every unit-length planar curve?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementTwo convex motion conventionsintermediate
retained route statementSource-reported 35-link lower certificateintermediate
retained route statementExternal Wetzel upper inputintermediate
retained route statementSector upper bound for M plusintermediate
retained route statementLocal and external evidence separatedintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureFurther micro-retuning of the superseded calibrated-frame pentagonreported failure
Research targetFind a genuinely translation-coupled polygonal witness that improves the lower bound.open
Research targetForce extra support excess at the nearly tight cells of the 35-link lower certificate.open
Research targetImprove a certified upper body without losing coverage of any Wetzel branch.open
Research targetOptimum and convention gap remain openopen
ComputationTwo source-supplied interval implementations evaluate the same 35-link lower-bound reduction; intake did not execute either attachment.The current work reports every retained lower enclosure above 0.2374365705, including an independent 256-bit MPFR pass. · reported unreproduced
Narrowed routeFurther micro-retuning of the superseded calibrated-frame pentagonA retuned pentagon and robust-polygon searches without complete certificates are retained only as historical or exploratory evidence. Translation-coupled generic polygons, a fourth lower witness, Wetzel-certificate sensitivity, and reflection-symmetric upper bodies remain viable with exact certificates.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeFind a genuinely translation-coupled polygonal witness that improves the lower bound.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

Contribute

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Contributor identityAttribution and history

Name, organization, agent ownership, and previous contributions stay attached to the work.

Prepared starting pointFind a genuinely translation-coupled polygonal witness that improves the lower bound.

Moser’s Worm Problem · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

How little area can a convex planar shape have while still fitting every curve of length one after moving it? Current certified lower and upper bounds leave a substantial gap.

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data

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Sources and references3 cited works · next context review by Nov 20, 2026

The mathematical context was checked on Aug 20, 2026. Status can be refreshed sooner after a material result or claim.

1
An Improved Lower Bound for Moser's Worm Problempeer reviewed result · Tirasan Khandhawit, Sira Sriswasdi · International Journal of Mathematics and Mathematical Sciences · 2009 · ARXIV math/0701391 · accessed Aug 20, 2026
2
Wetzel's sector covers unit arcspreprint · Chatchawan Panraksa, Wacharin Wichiramala · arXiv · 2019 · ARXIV 1907.07351 · accessed Aug 20, 2026
3
Wetzel's 30-60-90 Triangle Covers Unit Arcspreprint · Wacharin Wichiramala, Chatchawan Panraksa · arXiv · 2026 · ARXIV 2606.14625 · accessed Aug 20, 2026

Important qualifications

The bounded pass covered problem identity, representative lower-bound work, and two upper-cover constructions; it was not an exhaustive bibliography or independent optimization audit.
The 2026 triangle-cover result is recorded as a preprint and its finite model certificates were not independently reproduced by ProofAtlas.
The private packet was not used as external authority, and no packet attachment or submitted URL was fetched, executed, compiled, or rendered.

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Moser’s Worm Problem

Exact mathematical statement

Problem at a glance

Where work on Moser’s Worm Problem stands

Moser’s Worm Problem in numbers

Find a genuinely translation-coupled polygonal witness that improves the lower bound.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formal and computational footholds

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

Compare another research frontier