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A translational tile with no fully periodic tiling in dimension three
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Abstract
We construct a finite translational tile in ℤ3 that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ3, even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Erik D. Demaine, Stefan Langerman, Tiling 3D by Translates of a Single Polycube is Undecidable (arXiv:2610.12392, 8 Oct 2026). Proves that translational tiling of 3D space by a single face-connected polycube is undecidable (co-RE-complete), using "a cyclic encoding adapted from OpenAI's 3D aperiodic tile" with decorated two-prime Sudoku constraints; it cites this preprint as reference [15].
Comments and discussion
- Quanta Magazine "Transformation" (Konstantin Kakaes's column), Cracks appear in OpenAI's wall of proofs (Quanta Magazine, 9 Oct 2026). A short news item on early reactions to the release that links this preprint as its three-dimensional aperiodic tile example and quotes Chaim Goodman-Strauss (Museum of Mathematics) saying it has some novel ideas but is "unnecessarily difficult to read".
Formalization
- OpenAI math release, Lean scope notes for family 155 (lean/docs/155.md, 2026-10-06). Describes the formalized statement: the ℤ³ tile with no complement invariant under a finite-index subgroup, the ℝ³ unit-cube thickening with no fully periodic a.e. tiling under arbitrary real translations, and minimality of dimension three; it links the comparator statement PeriodicTilingThree.lean (
periodic_tiling_counterexample_and_minimality, with asorryplaceholder as comparator challenge files do). - OpenAI, Counterexample.lean (openai/math, lean/OAI/Geometry/PeriodicTiling/). The release's proof file for
OAI.PeriodicTilingThree.periodic_tiling_counterexample_and_minimality; the file itself has nosorry, and its imported lemmas were not inspected. - Erik D. Demaine and Stefan Langerman, translation-tiling-paper: paper and Lean proof (GitHub, accessed 2026-10-21). The follow-up's repository reports a complete Lean formalization of its Wang-to-polycube reduction, adapting and reproving OpenAI's cyclic gadgets and using Meyerovitch–Sanadhya–Solomon for arbitrary-tiling descent. [family-level]
Related prior literature mentioned by others
- Yoonhu Kim, Undecidability of Translational Tiling with 2 Polycubes (arXiv:2508.11725v2, 2026-08-10). Demaine and Langerman's follow-up uses its Theorem 2.5 to make the tile from their adaptation of OpenAI's encoding face-connected while preserving dimension, tile count, and tileability. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Abhishek Khetan, A Periodicity Result for Tilings of ℤ³ by Clusters of Prime-Squared Cardinality (arXiv:2109.14179, v3 16 Jun 2026). Gives weak periodicity of tilings of ℤ³ for tiles of prime-squared cardinality, a weaker notion than the full periodicity the paper rules out.
- Hu Tan, Ying Zhang, Periodic Tilings of Cardinality Twice a Prime or Nine in Arbitrary Dimension (arXiv:2609.26576, 22 Sep 2026). Proves that every finite tile of ℤ^d with cardinality 2q (q an odd prime) or 9 admits a fully periodic tiling, restricting which cardinalities a ℤ³ counterexample such as Theorem 1.1's tile T can have.
- Rachel Greenfeld, Terence Tao, Some variants of the periodic tiling conjecture (arXiv:2505.06757, 10 May 2025). Proves periodicity-type variants of the conjecture, including in ℤ², bearing on the positive results used to argue that three is the least lattice dimension.
- Jaume de Dios Pont, Jan Grebík, Rachel Greenfeld, José Madrid, Translational tilings by axis-parallel polygonal sets (arXiv:2610.15785, 11 Oct 2026). Shows that sets bounded by axis-parallel segments that tile the plane admit quasi-periodic tilings, related to the paper's remark that planar positive results do not settle arbitrary bounded measurable tiles (it does not mention this paper).
- Rachel Greenfeld, Translational tilings: structured or wild? (arXiv:2509.25576, 29 Sep 2025). A survey of translational tilings and the periodic tiling conjecture, describing the state of the field just before this paper's dimension-three claim.
Digests and expositions
- Latent Seminar, Translational tilings without full periodicity (2026-10-07). Explains the Sudoku encoding, stacking, and Euclidean rigidity arguments, distinguishing full periodicity from individual periods; the page states it has not independently verified the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 155
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026,
author = {{OpenAI}},
title = {{A translational tile with no fully periodic tiling in dimension three}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf}{OAI:A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/155.md. The Lean statements have not been checked by people against the paper's claims.
Family 155: A counterexample to periodic tiling in dimension three
Constructs a finite translational tile in ℤ3 that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ3, even with arbitrary real translation vectors.
This is the only paper in this family.