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Discrete geometry · convex bodies · congruent packing density

Ulam’s Packing Conjecture

Collaboration beta

Is the Euclidean ball the three-dimensional convex body whose densest congruent packing has the lowest possible density?

δ(K)≥δ(B)=π18
Known results and sources
A sphere and several centrally symmetric convex solids occupy comparable packing cells with visibly different void spaces.
Ulam’s conjecture asks whether the ball has the lowest optimal congruent packing density among all convex bodies in three dimensions.

Research problem

Exact mathematical statement

For every three-dimensional convex body KK, let δ(K)\delta(K) be the supremal density of a packing of Euclidean space by congruent copies of KK, allowing arbitrary rotations and reflections. If BB is a Euclidean ball, Ulam’s packing conjecture is

δ(K)≥δ(B)=π18\delta(K)\ge \delta(B)=\frac{\pi}{\sqrt{18}}

for every convex body K⊂ℝ3K\subset\mathbb R^3. The governing source explicitly states that the full conjecture is not solved in the project.

Problem infographic

Problem at a glance

Three packing panels compare congruent spheres and other convex bodies, emphasizing occupied volume, voids, and unrestricted orientations without claiming a solution.
Each body may be translated, rotated, or reflected; the open question compares the best achievable density for every convex body with the sphere-packing density.

Current mathematical picture

Where work on Ulam’s Packing Conjecture stands

Open conjecture

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

Useful failureUniversalization of scoped local or lattice arguments

The source identifies curvature-saturated high-frequency escape for the local program and unbounded period/orientation complexity for the full problem; it also records exact counterexamples or ceilings for several lower-level motifs. Joint HCP linear-plus-quadratic control, the signed cubic arithmetic envelope, and a separately proved unrestricted-packing compactness or reduction theorem remain source-proposed avenues.

Route status · Narrowed route
Main reductionCurrent reduction

The source splits the problem into a local near-ball program and a global centrally symmetric lattice program. Locally it combines linear HCP contact gain with quadratic volume loss; globally it derives a critical-stress representation and seeks a sharp signed cubic arithmetic-envelope inequality. Neither program currently reaches arbitrary convex bodies and unrestricted congruent packings.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeClose the frequency-uniform odd near-ball gate.Task status · Ready to work on
Research-record correctionResearch-record correction

We 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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Ulam’s Packing Conjecture in numbers

2kretained lines of mathematical investigation2,024 in the current working snapshot
Argument development
1,709 · 84%
Explored or eliminated routes
46 · 2%
Computational analysis
75 · 4%
Open obligations
91 · 4%
Definitions and setup
103 · 5%
6selected mapped statements1routes 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.

Proposed byStanisław Ulam (attributed)Proposed1972Status checkedAug 15, 2026
Source recordedAug 15, 2026

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

10 selected steps

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

10 selected steps

Scroll horizontally to explore the route

Working route overview for Ulam’s Packing ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.The ball should be the least densely packable convex body in three dimensions — Depends on missing premiseThe ball should be the leastdensely packable convex bodyin…Current reduction — Depends on missing premiseCurrent reductionExact packing-density target — Depends on missing premiseExact packing-density targetAll-odd-degree HCP multiplier — Depends on missing premiseAll-odd-degree HCPmultiplierClosing target — Depends on missing premiseClosing targetFixed-direction HCP theorem — Depends on missing premiseFixed-direction HCP theoremUniversalization of scoped local or lattice arguments — stoppedUniversalization of scopedlocal or lattice argumentsClose the frequency-uniform odd near-ball gate. — OpenClose the frequency-uniformodd near-ball gate.Prove the signed cubic arithmetic-envelope inequality and preserve common-lattice signs. — OpenProve the signed cubicarithmetic-envelopeinequality…Build the unrestricted-packing bridge. — OpenBuild theunrestricted-packing bridge.
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 routeUniversalization of scoped local or lattice arguments

The source identifies curvature-saturated high-frequency escape for the local program and unbounded period/orientation complexity for the full problem; it also records exact counterexamples or ceilings for several lower-level motifs. Joint HCP linear-plus-quadratic control, the signed cubic arithmetic envelope, and a separately proved unrestricted-packing compactness or reduction theorem remain source-proposed avenues.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Close the frequency-uniform odd near-ball gate.Suggested move: Analyze curvature-saturated blow-up sequences using the joint HCP linear response and quadratic detector without assuming a bounded harmonic degree.
Ready to work on
02
Prove the signed cubic arithmetic-envelope inequality and preserve common-lattice signs.Suggested move: Work within each critical-lattice fiber and test affine inequalities sharp on rotational mixtures of FCC stresses before addressing nonsmooth limits.
Ready to work on
03
Build the unrestricted-packing bridge.Suggested move: Seek a structural reduction or compactness theorem controlling period and orientation complexity without silently replacing congruent packings by lattice packings.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

The global three-dimensional claim that the ball minimizes optimal congruent packing density among all convex bodies remains open. A published theorem establishes a strict local result for sufficiently close origin-symmetric bodies, not the unrestricted global comparison.

[2][3]
External progress

What the literature has established

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

  1. Peer reviewedKallus placed the result in the broader theory of local pessimal packing shapes and recorded the attribution caveat surrounding the name Ulam's conjecture.[3]
  2. Peer reviewedKallus proved that every sufficiently spherical origin-symmetric convex solid can be packed more densely than the ball.[2]
  3. Peer reviewedHales proved the Kepler conjecture, establishing the optimal three-dimensional sphere packing density π/√18 used as Ulam's benchmark.[1]
3 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusUlam's packing conjecture
Dependency or reductionKepler sphere-packing theorem

The Kepler theorem supplies the exact optimal density of congruent balls but does not compare that value with every other convex body.

[1]
Solved special caseOrigin-symmetric near-ball local pessimum theorem

The ball is a strict local packing pessimum among sufficiently close origin-symmetric convex bodies; this is local and symmetry-restricted rather than the global conjecture.

[2]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA statement-aligned formalization of congruent packing density for arbitrary rotating and reflected convex bodies was not identified.
  • Formalization targetThe global comparison over all three-dimensional convex bodies has no known proof artifact.

Research-record corrections

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

Corrected the research recordCorrection note

Cited passages corrected
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task. 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. 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.

4 standing statements2 proposed statements3 open questions1 narrowed routes
Statements by mathematical role6 selected mapped statements
  • theorem candidate1 of 61
  • reduction1 of 61
  • lemma3 of 63
  • equivalence1 of 61
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included
  • retained route statementThe ball should be the least densely packable convex body in three dimensions
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact packing-density targetintermediate
  • retained route statementFixed-direction HCP theoremintermediate
  • retained route statementAll-odd-degree HCP multiplierintermediate
  • 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
  • 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 failureUniversalization of scoped local or lattice argumentsreported failure
  • Research targetClose the frequency-uniform odd near-ball gate.open
  • Research targetProve the signed cubic arithmetic-envelope inequality and preserve common-lattice signs.open
  • Research targetBuild the unrestricted-packing bridge.open
  • Research targetFrequency-uniform odd gatesuperseded
  • Research targetSigned cubic arithmetic envelopesuperseded
  • Research targetUnrestricted-packing bridgesuperseded
  • Narrowed routeUniversalization of scoped local or lattice argumentsThe source identifies curvature-saturated high-frequency escape for the local program and unbounded period/orientation complexity for the full problem; it also records exact counterexamples or ceilings for several lower-level motifs. Joint HCP linear-plus-quadratic control, the signed cubic arithmetic envelope, and a separately proved unrestricted-packing compactness or reduction theorem remain source-proposed avenues.
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 bridgeClose the frequency-uniform odd near-ball gate.

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.

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Prepared starting pointClose the frequency-uniform odd near-ball gate.

Ulam’s Packing Conjecture · ready to start

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Research contextPrepared context for any AI agent

Is the Euclidean ball the three-dimensional convex body whose densest congruent packing has the lowest possible density?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    A proof of the Kepler conjecturepeer reviewed result · Thomas C. Hales · Annals of Mathematics · 2005 · DOI 10.4007/annals.2005.162.1065 · accessed Aug 15, 2026
  2. 2
    The 3-ball is a local pessimum for packingpeer reviewed result · Yoav Kallus · Advances in Mathematics · 2014 · ARXIV 1212.2551 · DOI 10.1016/j.aim.2014.07.015 · accessed Aug 15, 2026
  3. 3
    Pessimal packing shapespeer reviewed result · Yoav Kallus · Geometry & Topology · 2015 · ARXIV 1305.0289 · DOI 10.2140/gt.2015.19.343 · accessed Aug 15, 2026

Important qualifications

  • Scoped to primary papers for the sphere-density benchmark, modern attribution, and the published origin-symmetric local-neighborhood theorem; the packing literature was not exhaustively surveyed.
  • The historical attribution to Ulam is qualified because the modern primary discussion traces it to a Gardner-attributed remark rather than a surviving formal statement by Ulam.
  • No maintained formalization or universal computation was identified in the scoped search.

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