View source for The periodic tiling conjecture in dimension three

'''The periodic tiling conjecture in dimension three''' is the subject of a manuscript dated 23 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]). The periodic tiling conjecture asserts that every finite subset of a lattice '''Z'''<sup>''d''</sup> that tiles the lattice by translations also tiles it periodically. S. Bhattacharya proved it for '''Z'''<sup>2</sup> ([https://arxiv.org/abs/1602.05738 arXiv:1602.05738]), and R. Greenfeld and T. Tao disproved it for sufficiently large ''d'' ([https://arxiv.org/abs/2211.15847 arXiv:2211.15847]). This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== The original result ==
* '''Claim.''' There is a finite translational tile in '''Z'''<sup>3</sup> that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in '''R'''<sup>3</sup>, even when arbitrary real translations are allowed. The collection's summary describes this as disproving the conjecture in the smallest possible lattice dimension.
* '''Source.''' [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026 Manuscript folder] in the openai/math repository (commit <code>1581d4c</code>).
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/155.md lean/docs/155.md] states that the formalized counterexample is a finite tile in '''Z'''<sup>3</sup> that tiles but has no complement invariant under a finite-index subgroup; that its unit-cube thickening tiles '''R'''<sup>3</sup> almost everywhere but admits no fully periodic tiling, even with arbitrary real translation vectors; and that three is the least lattice dimension where this failure occurs.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 15 October 2026)
|-
| 2026-10-08 || Erik D. Demaine and Stefan Langerman, [https://arxiv.org/abs/2610.12392 arXiv:2610.12392], ''Tiling 3D by Translates of a Single Polycube is Undecidable''; repository [https://github.com/edemaine/translation-tiling-paper edemaine/translation-tiling-paper] || Use || Deciding whether a single connected polycube tiles three-dimensional space by translations is co-RE-complete, hence undecidable; dimension three is optimal. The reduction is from Wang tiling, using the decorated two-prime Sudoku construction of Greenfeld and Tao and, in the abstract's words, "a cyclic encoding adapted from OpenAI's 3D aperiodic tile", with a reduction of Kim to make the prototile face-connected. || arXiv preprint, version 1. The repository README describes a complete Lean formalization of the reduction, the main results and the mathematical inputs, and a separate LLM-written technical overview.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]