Geometric optimization and rigid motions

Ambidextrous moving sofa

Collaboration beta

How large can a connected planar body be if it must negotiate both left and right unit-width corridor turns? A sharp abstract support-profile bound is available in the source, while its comparison with the body's ordinary area remains open.

\sup_{S \in S_{amb}} | S | = β + \cot β - 2, 4 \tan^{3} β + 3 \tan β = 1

Known results and sources

Illustration: One rigid connected shape must negotiate both turns; the disk is only a feasible example.

Research problem

Exact mathematical statement

Let $\mathcal S_{\rm amb}$ be the compact connected planar sets of finite Lebesgue area that admit continuous planar rigid motions through both handed right-angle turns in unit-width corridors, with a complete straight-corridor placement between the turns. Motions allow wall contact and zero-width connectors; they do not allow reflections, deformation, or leaving the plane. Prove

\mu_{\rm amb}:=\sup_{S\in\mathcal S_{\rm amb}} |S| =A_R=\beta+\cot\beta-2,\qquad 4\tan^3\beta+3\tan\beta=1,\quad 0<\beta<\frac{\pi}{2},\quad 0.298<\tan\beta<0.299.

The target is equality of areas. Smoothness, symmetry, strict convexity, finite contact structure, and uniqueness of the physical set are not assumed.

Problem infographic

Problem at a glance

Illustration: Right-turn, straight-placement and left-turn corridor diagrams show the same disk, followed by the open candidate area equation. — Illustration: Maximize area over compact connected bodies that pass both turns, including a straight placement. The displayed disk is an example, not an optimizer.

Current mathematical picture

Where work on Ambidextrous moving sofa stands

Open problem

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

Useful failureUnqualified functional enclosure from local geometric regularity

The lower core travels to the right; the source's corrected exact positive defect survives the terminal decimal repair. A comparison using genuine exposed envelopes, or a theorem restricted to attained global maximizers with justified geometric variations, remains open.

Route status · Narrowed route

Main reductionCurrent reduction

For an attained global maximizer, prove AMS-ENC-F: |S| ≤ M(h_conv S,δ). Combined with the source-reported partial functional inequality and the feasible candidate, this would force δ=0 and the target area equality. R-FIBER offers the separate sufficient bound ∫(U_K−L_K) dx ≤ A_R on the exact admissible nonempty-fiber domain.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeDerive an exact ordinary-fiber-area-minus-F expression accounting for exposure, endpoint terms, and multiplicity.Task status · Ready to work on

Work mapped so far

Ambidextrous moving sofa in numbers

1.2kretained lines of mathematical investigation1,200 in the current working snapshot

Argument development: 791 · 66%
Explored or eliminated routes: 59 · 5%
Computational analysis: 87 · 7%
Open obligations: 116 · 10%
Definitions and setup: 147 · 12%

9selected mapped statements2routes investigated3open questions3contribution-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.

Status checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Derive an exact ordinary-fiber-area-minus-F expression accounting for exposure, endpoint terms, and multiplicity.

Suggested move: Suggested next step: Write the proposed identity in terms of genuine niche envelopes and evaluate it on the candidate, single disk, two-disk positive defect, shave, and tip-cut controls.

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

14 selected steps

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

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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

2 recorded

Narrowed routeUnqualified functional enclosure from local geometric regularity

Route status · Narrowed route

Narrowed routeCompact tail realization from H1 outer data alone

A compact body's support profile is Lipschitz, so no admissible tail in that abstract example has a compact realization. The abstract H1 optimization and inequality remain valid; compact realization remains in the current research map for Lipschitz outer data, including actual compact hull supports.

Route status · Narrowed route

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Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Derive an exact ordinary-fiber-area-minus-F expression accounting for exposure, endpoint terms, and multiplicity.Suggested move: Suggested next step: Write the proposed identity in terms of genuine niche envelopes and evaluate it on the candidate, single disk, two-disk positive defect, shave, and tip-cut controls.

Ready to work on

Exclude reverse-oriented core traversal for attained global maximizers, or pay for its area contribution from the available corrected deficit.Suggested move: Suggested next step: Specify a permissible shape modification and prove that it preserves connectedness and both continuous turning motions before invoking maximality.

Ready to work on

Construct a sharp dual inequality for the genuine nonempty-fiber envelope objective across the full normalized admissible domain.Suggested move: Suggested next step: Develop an exposed-piece or local ribbon-exchange charge and check it on asymmetric, flat-faced, and free-height data without assuming set-extension uniqueness.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen problem

The maximum area of a shape that can negotiate both left and right turns in a unit-width corridor remains open in the inspected sources. Romik's explicit shape is a candidate, not a proved optimum.

[1][2]

External progress

What the literature has established

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

2018
Peer reviewedRomik derived an explicit ambidextrous sofa candidate of area approximately 1.64495 from differential equations; its global maximality was not established.[1]

Mathematical neighborhood

Claims, milestones, and routes in the current map

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

7 standing statements2 proposed statements3 open questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction2 of 92
lemma3 of 93
computational claim1 of 91
counterexample2 of 92

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included

retained route statementDetermine the maximum area that can negotiate both handed unit-corridor turns.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementRetained unconditional bracketintermediate
retained route statementSharp abstract tail optimizationintermediate
retained route statementRough H1 realization obstructionintermediate
retained route statementCompetitive core regularityintermediate
retained route statementConnected two-disk negative controlintermediate
retained route statementGenuine fiber-area routeintermediate
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 counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · 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 counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · 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 failureUnqualified functional enclosure from local geometric regularityreported failure
Useful failureCompact tail realization from H1 outer data alonereported failure
Research targetDerive an exact ordinary-fiber-area-minus-F expression accounting for exposure, endpoint terms, and multiplicity.open
Research targetExclude reverse-oriented core traversal for attained global maximizers, or pay for its area contribution from the available corrected deficit.open
Research targetConstruct a sharp dual inequality for the genuine nonempty-fiber envelope objective across the full normalized admissible domain.open
Narrowed routeUnqualified functional enclosure from local geometric regularityThe lower core travels to the right; the source's corrected exact positive defect survives the terminal decimal repair. A comparison using genuine exposed envelopes, or a theorem restricted to attained global maximizers with justified geometric variations, remains open.
Narrowed routeCompact tail realization from H1 outer data aloneA compact body's support profile is Lipschitz, so no admissible tail in that abstract example has a compact realization. The abstract H1 optimization and inequality remain valid; compact realization remains in the current research map for Lipschitz outer data, including actual compact hull supports.

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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 bridgeDerive an exact ordinary-fiber-area-minus-F expression accounting for exposure, endpoint terms, and multiplicity.

2 approaches have 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.

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Prepared starting pointDerive an exact ordinary-fiber-area-minus-F expression accounting for exposure, endpoint terms, and multiplicity.

Ambidextrous moving sofa · ready to start

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Sources and references3 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Differential Equations and Exact Solutions in the Moving Sofa Problempeer reviewed result · Dan Romik · Experimental Mathematics 27, 316–330 · 2018 · ARXIV 1606.08111 · DOI 10.1080/10586458.2016.1270858 · accessed Sep 7, 2026
2
The moving sofa problemauthoritative webpage · Dan Romik · accessed Sep 7, 2026
3
Moving sofa companion materialssoftware or dataset · Dan Romik · accessed Sep 7, 2026

Important qualifications

This is a scoped administrative literature search, not an exhaustive survey or an independent mathematical review.
Submitted packets are not external evidence. Their novel claims, corrections and proof obligations retain their separate source-reported posture.
No external code, proof development, data file or certificate was executed or independently reproduced during collection. Public links confer no acceptance or redistribution authority.
An empty formalization or recognition list means no suitable item was established in this search; it does not establish nonexistence. Unknown origin dates and original attributions are left unresolved.

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Ambidextrous moving sofa

Exact mathematical statement

Problem at a glance

Where work on Ambidextrous moving sofa stands

Ambidextrous moving sofa in numbers

Derive an exact ordinary-fiber-area-minus-F expression accounting for exposure, endpoint terms, and multiplicity.

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

Claims, milestones, and routes in the current map

Conditions that would advance the current route

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