[#P2892] Plane tilings by every five-cell lattice animal

A mathematical schematic of Plane tilings by every five-cell lattice animal. — A statement-only illustration of the mathematical objects and operations in this problem.

Problem. Let \(A\subset\mathbb Z^2\) have exactly five elements, with no connectivity assumption. Must \(\mathbb Z^2\) admit a partition into sets of the form \(g(A)+t\), where \(t\in\mathbb Z^2\) and \(g\) is a rotation or reflection preserving the square lattice?

1Context

Disconnected animals have unbounded relative offsets, so the central issue is finding a finite reduction or an invariant. Periodic tiling certificates and failed offset families are directly reusable.

2Definitions

Definition 1. The set A represents the union of the five closed unit cells whose lower-left corners are its elements; such a possibly disconnected set is called a five-cell lattice animal.

Definition 2. A tiling is an exact partition of lattice cells: every element of Z^2 belongs to exactly one transformed copy, so overlaps and uncovered cells are forbidden.

3What counts as a solution

Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists.
A positive computational classification must state a finite reduction for arbitrarily separated cells and provide periodic fundamental-domain certificates for every reduced class; bounded-box enumeration alone is insufficient.

1Status

Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow and Mathematics Stack Exchange pages have no solution. The source author reported in 2021 that the author of the four-cell theorem knew of no later progress, and no subsequent exact classification was found. Exact unresolved remainder: Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists. A positive computational classification must state a finite reduction for arbitrarily separated cells and provide periodic fundamental-domain certificates for every reduced class; bounded-box enumeration alone is insufficient.^[1][2][3]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow and Mathematics Stack Exchange pages have no solution. The source author reported in 2021 that the author of the four-cell theorem knew of no later progress, and no subsequent exact classification was found. Exact unresolved remainder: Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists. A positive computational classification must state a finite reduction for arbitrarily separated cells and provide periodic fundamental-domain certificates for every reduced class; bounded-box enumeration alone is insufficient.^[1][2][3] Relevance to this problem For Plane tilings by every five-cell lattice animal, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2, Reference 3 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The MathOverflow and Mathematics Stack Exchange pages have no solution. The source author reported in 2021 that the author of the four-cell theorem knew of no later progress, and no subsequent exact classification was found. Exact unresolved remainder: Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists. A positive computational classification must state a finite reduction for arbitrarily separated cells and provide periodic fundamental-domain certificates for every reduced class; bounded-box enumeration alone is insufficient. Open full record →
	Trace	Reported
attempt · Route 1 Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists. A positive computational classification must state a finite reduction for arbitrarily separated cells and provide periodic fundamental-domain certificates for every reduced class; bounded-box enumeration alone is insufficient.^[1] Relevance to this problem For Plane tilings by every five-cell lattice animal, record five-cell-lattice-animal-plane-tiling-next-route-20260801 (“Complete the stated acceptance conditions”) documents a concrete method, search boundary, or failed route. The record states: Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work against the displayed statement and preserve every hypothesis and quantifier. Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists. A positive computational classification must state a finite reduction for arbitrarily separated cells and provide periodic fundamental-domain certificates for every reduced class; bounded-box enumeration alone is insufficient. Any computation must retain a replayable witness and a matching exclusion or completeness certificate. Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-27. The MathOverflow and Mathematics Stack Exchange pages have no solution. The source author reported in 2021 that the author of the four-cell theorem knew of no later progress, and no subsequent exact classification was found.

On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for MathOverflow question 380490; comments settle only special shapes such as a 2 by 2 block plus one cell.
Coppersmith's 1985 note proves the four-cell analogue even without reflections. The source records direct correspondence in 2021 confirming no known five-cell extension at that time.
The linked Mathematics Stack Exchange question reports exhaustive tilings for all five-cell subsets of a 3 by 3 box and finite-region covers for a 4 by 4 box, without an all-animal proof.
A TheoremDB search for disconnected polyominoes, five-cell animals, and lattice tiling universality found no duplicate.
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. Every five-cell subset of a 3 by 3 lattice box was reported to tile using translations and rotations, but that finite family does not cover animals with widely separated cells.

Computational notes

The source reports that every five-cell subset of a 4 by 4 box covers a 10 by 10 square; a finite cover is weaker than a plane tiling and should be stored separately.

2See also

Conway’s thrackle conjecturediscrete geometry
Borsuk’s conjecture in four dimensionsdiscrete geometry
Completing a line arrangement to triangular bounded cellsdiscrete geometry

How to cite

TheoremDB contributors, “Plane tilings by every five-cell lattice animal,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/five-cell-lattice-animal-plane-tiling

This page as plain text: five-cell-lattice-animal-plane-tiling.md

This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net^[1], current as of August 1, 2026.

1References

Packet source. MathOverflow question 380490, “Plane tilings by every five-cell lattice animal,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Question 380490 and all visible comments, checked through the Stack Exchange API on 2026-07-27.Also cited at Full question, answers, and visible comments concerning Plane tilings by every five-cell lattice animal; checked 2026-08-01.Also cited at Editorial research route recorded 2026-08-01.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Plane tilings by every five-cell lattice animal: This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.Source named by the research packet.
Mathematics Stack Exchange question 3955595, related discussion for “Plane tilings by every five-cell lattice animal,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at question statement and update requesting progress on the five-cell case after Coppersmith's four-cell theorem.Source used to assess the problem's recorded status.For Plane tilings by every five-cell lattice animal, this source is an exact duplicate and discovery discussion for the five-cell target, with no accepted resolution.
Don Coppersmith, “Each Four-Celled Animal Tiles the Plane,” Journal of Combinatorial Theory, Series A 40(2) (1985), 444-449. DOI 10.1016/0097-3165(85)90105-0. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at main theorem that every four-celled animal tiles the plane.Source used to assess the problem's recorded status.For Plane tilings by every five-cell lattice animal, this source establishes the sharp known lower-size boundary and does not handle five-celled animals.

This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.

[#P2892] Plane tilings by every five-cell lattice animal

1Context

2Definitions

3What counts as a solution

1Status

1Records

Computational notes

2See also

1Lean verification

1References

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Definitions

What counts as a solution

What's known

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1Context

2Definitions

3What counts as a solution

1Status

1Records

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance