ProofAtlas

Tilings · coronas · finite surrounding layers

Heesch's Problem

Collaboration beta

Can plane figures have arbitrarily many complete surrounding layers without tiling the plane—and, more strongly, can every positive finite layer count occur? The source reports a narrow exact lattice computation, while both the standard problem and its stronger exact-value target remain open.

∀n∈ℤ>0,∃S,H(S)=n<∞
Known results and sources
Landscape tiling card with a dark central tile surrounded by two complete rings of congruent copies and an unfinished third ring, labeled as an open question.
A Heesch number measures how many complete coronas of congruent copies can surround a non-tiling seed shape.

Research problem

Exact mathematical statement

For every positive integer nn, does there exist a planar shape with finite Heesch number exactly nn? Equivalently, the source asks whether

∀n∈ℤ>0,∃S,H(S)=n<∞.\forall n\in\mathbb{Z}_{>0},\quad \exists S,\quad H(S)=n<\infty.

Here a Heesch number counts how many complete, nonoverlapping coronas of congruent copies can surround a seed copy before extension stops. The standard external problem asks whether finite Heesch numbers are unbounded; realizing every positive integer exactly is stronger. Revision 10 explicitly says neither the full target nor a transfer from its strict triangular-lattice model to unrestricted motion has been proved.

Problem infographic

Problem at a glance

Landscape three-panel explainer defining a seed tile, complete coronas, finite stopping, and the stronger open exact-value question.
Complete surrounding layers define the Heesch number; stopping after finitely many layers and realizing every exact finite value are separate requirements.

Current mathematical picture

Where work on Heesch's Problem stands

Open problem

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

Useful failureEmpty-ancestry and terminal-edge shortcuts

The complete preflight reports nonzero individual and cross-compatible ancestry for every remaining pair, while the protocol requires equality of the whole canonical state. Complete proof-bearing screens of the thirty remaining pairs and a new distributed boundary state with authentic whole-state recurrence remain viable.

Route status · Narrowed route
Main reductionCurrent reduction

The route fixes an exact triangular-lattice placement model and strict corona topology, enumerates first-corona states, traces paired boundary phases, then attempts exact level-by-level closure or recurrence classification for compact terminal pairs.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeComplete the remaining compact-pair screensTask 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

Heesch's Problem in numbers

4.2kretained lines of mathematical investigation4,195 in the current working snapshot
Argument development
3,257 · 78%
Explored or eliminated routes
78 · 2%
Computational analysis
435 · 10%
Open obligations
332 · 8%
Definitions and setup
93 · 2%
6selected mapped statements1routes investigated6open questions6contribution-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 byHeinrich HeeschProposed1968Status 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

13 selected steps

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

13 selected steps

Scroll horizontally to explore the route

Working route overview for Heesch's ProblemA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can every positive finite number of complete surrounding coronas occur? — Depends on missing premiseCan every positive finitenumber of completesurrounding…Current reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetEvery exact finite height — Depends on missing premiseEvery exact finite heightFour direct 39-contact closures — Depends on missing premiseFour direct 39-contactclosuresStrict lattice model — Depends on missing premiseStrict lattice modelEmpty-ancestry and terminal-edge shortcuts — stoppedEmpty-ancestry andterminal-edge shortcutsComplete the remaining compact-pair screens — OpenComplete the remainingcompact-pair screensProve an authentic repeatable transition — OpenProve an authenticrepeatable transitionTransfer to the intended plane-figure problem — OpenTransfer to the intendedplane-figure problemFull target remains open — OpenFull target remains openThirty compact pairs remain — OpenThirty compact pairs remainWhole-state recurrence gate — OpenWhole-state recurrence gate
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 routeEmpty-ancestry and terminal-edge shortcuts

The complete preflight reports nonzero individual and cross-compatible ancestry for every remaining pair, while the protocol requires equality of the whole canonical state. Complete proof-bearing screens of the thirty remaining pairs and a new distributed boundary state with authentic whole-state recurrence remain viable.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks
01
Complete the remaining compact-pair screensSuggested move: Process the lowest-cost 53/59 phase family with complete ancestry, state fibers, immediate screens, survivor propagation, and exact outgoing-state records.
Ready to work on
02
Prove an authentic repeatable transitionSuggested move: Exhibit equality of the entire canonical boundary state—word, strand pairing, phase, chirality, winding data, and contact graph—under a forced nonterminal transition.
Ready to work on
03
Full target remains open

The source explicitly says the full conjecture is not proved and lists three absent universal gates.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Thirty compact pairs remain

After the reported 39-contact closures, the current work lists thirty remaining compact pairs, fourteen with 44 contacts and sixteen with 45 contacts.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Whole-state recurrence gate

A terminal pair counts as recurrent only when the whole canonical state returns, not merely the terminal edges.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
06
Transfer to the intended plane-figure problemSuggested move: Supply exact-height tuning, finite termination, nontiling, the intended hole convention, and reflection-aware locking from lattice placements to arbitrary congruent motion.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen problem

The standard Euclidean-plane question of whether finite Heesch numbers are unbounded remains open. A finite value of 6 is established, while neither unboundedness nor the current work's stronger every-exact-positive-integer target is established.

[3][4]
External progress

What the literature has established

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

  1. Peer reviewedA later peer-reviewed tilings article still described 6 as the largest known finite Heesch number.[4]
  2. Peer reviewedBašić constructed a plane figure with Heesch number 6 and stated the basic unboundedness question as open.[3]
  3. Peer reviewedMann surveyed Heesch's tiling problem and the then-known finite Heesch-number-5 milestone.[2]
  4. Historical sourceHeesch's monograph established the historical setting for finite surrounding layers around a non-tiling plane figure.[1]
4 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusHeesch's Problem
Stronger or generalized formexact realization of every positive finite Heesch number

Realizing every positive integer exactly would imply unbounded finite Heesch numbers, but unboundedness alone would not establish every exact value.

[3]
Solved special caseexistence of a plane figure with Heesch number 6

A figure with finite Heesch number 6 is known; this single value does not settle unboundedness or exact-value coverage.

[3][4]

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 must fix the class of plane figures, congruent-copy convention, corona definition, hole convention, and finite-versus-infinite status.
  • Formalization targetA lattice-restricted upper bound needs a separate theorem before it can control arbitrary congruent Euclidean motions.
  • Formalization targetAn every-exact-value result requires a uniform family, strict lower bounds, exact termination, and proof that the final objects do not tile the plane.

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

Corrected the research recordCorrection note

Correction details

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 statements6 open questions1 narrowed routes
Statements by mathematical role6 selected mapped statements
  • theorem candidate1 of 61
  • reduction1 of 61
  • lemma3 of 63
  • computational claim1 of 61
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementCan every positive finite number of complete surrounding coronas occur?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementEvery exact finite heightintermediate
  • retained route statementStrict lattice modelintermediate
  • retained route statementFour direct 39-contact closuresintermediate
  • 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 failureEmpty-ancestry and terminal-edge shortcutsreported failure
  • Research targetComplete the remaining compact-pair screensopen
  • Research targetProve an authentic repeatable transitionopen
  • Research targetTransfer to the intended plane-figure problemopen
  • Research targetFull target remains openopen
  • Research targetThirty compact pairs remainopen
  • Research targetWhole-state recurrence gateopen
  • ComputationThe source reports exhaustive direct state/bridge screens for indices 20, 26, 42, and 43, with exact per-index partitions and an independent set-level transfer comparison.Those source-reported screens close four narrow fourth-level transition candidates in the strict lattice model; they do not prove recurrence, arbitrary-motion locking, unboundedness, or exact realization of every finite height. · reported unreproduced
  • Narrowed routeEmpty-ancestry and terminal-edge shortcutsThe complete preflight reports nonzero individual and cross-compatible ancestry for every remaining pair, while the protocol requires equality of the whole canonical state. Complete proof-bearing screens of the thirty remaining pairs and a new distributed boundary state with authentic whole-state recurrence remain viable.
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 bridgeComplete the remaining compact-pair screens

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

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Prepared starting pointComplete the remaining compact-pair screens

Heesch's Problem · ready to start

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Research contextPrepared context for any AI agent

Can plane figures have arbitrarily many complete surrounding layers without tiling the plane—and, more strongly, can every positive finite layer count occur? The source reports a narrow exact lattice computation, while both the standard problem and its stronger exact-value target remain open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 15, 2026

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

  1. 1
    Reguläres Parkettierungsproblemoriginal source · Heinrich Heesch · Springer · 1968 · DOI 10.1007/978-3-322-98547-7 · accessed Aug 14, 2026
  2. 2
    Heesch's Tiling Problempeer reviewed result · Casey Mann · The American Mathematical Monthly 111(6), 509–517 · 2004 · DOI 10.1080/00029890.2004.11920105 · accessed Aug 14, 2026
  3. 3
    A Figure with Heesch Number 6: Pushing a Two-Decade-Old Boundarypeer reviewed result · Bojan Bašić · The Mathematical Intelligencer 43(3), 50–53 · 2021 · DOI 10.1007/s00283-020-10034-w · accessed Aug 14, 2026
  4. 4
    Solutions to Seven and a Half Problems on Tilingspeer reviewed result · Bojan Bašić, Aleksa Džuklevski, Anna Slivková · The Electronic Journal of Combinatorics 30(2), P2.50 · 2023 · accessed Aug 14, 2026

Important qualifications

  • This bounded primary-source pass is not an exhaustive literature, priority, or mathematical review.
  • The private packet's submitted URLs were not fetched, and no packet attachment was executed, compiled, or rendered.
  • The standard external problem asks whether finite Heesch numbers are unbounded; the current work's every-exact-positive-integer target is stronger and remains separately stated.
  • The current work's strict triangular-lattice finite screens remain source-reported and were not independently reproduced by this metadata pass.

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