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 (github.com/openai/math). The periodic tiling conjecture asserts that every finite subset of a lattice Zd that tiles the lattice by translations also tiles it periodically. S. Bhattacharya proved it for Z2 (arXiv:1602.05738), and R. Greenfeld and T. Tao disproved it for sufficiently large d (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 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. The collection's summary describes this as disproving the conjecture in the smallest possible lattice dimension.
- Source. Manuscript folder in the openai/math repository (commit
1581d4c). - Formalization, as described by the repository. The scope note lean/docs/155.md states that the formalized counterexample is a finite tile in Z3 that tiles but has no complement invariant under a finite-index subgroup; that its unit-cube thickening tiles R3 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
| Date | Work | Relation | Public claim | Public status (checked 15 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | Erik D. Demaine and Stefan Langerman, arXiv:2610.12392, Tiling 3D by Translates of a Single Polycube is Undecidable; repository 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.