Result 155, Combinatorics

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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Can one shape fill space using only shifted copies, yet never fit into a pattern that repeats in all directions? The manuscript reports such a shape in three dimensions, challenging the link between tiling and repetition.

What changes?

The manuscript constructs a finite set of points in the three-dimensional integer grid. Copies moved only by translation, without rotation, can cover every grid point exactly once. Yet no such covering is fully periodic, meaning invariant under shifts in three independent directions. Replacing each point of the tile with a unit cube gives a tile of ordinary three-dimensional space with the same property. Allowing arbitrary real translation vectors does not restore a fully periodic tiling.

What does that help mathematicians do?

The claimed counterexample places the failure of the periodic tiling conjecture in the smallest possible lattice dimension. It shows why searching only for finite repeating arrangements cannot detect every tile capable of covering the three-dimensional grid: this tile would never appear among the successes. The distinction is precise: the result excludes repetition in three independent directions, not necessarily every possible direction of repetition.

Are there practical applications?

Its immediate value is foundational, clarifying how local pieces can cover space without permitting global periodic order. The passage from grid points to solid unit cubes is especially useful: it carries the obstruction into continuous geometry and shows that allowing translations off the integer grid does not remove it. The supplied material describes no practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A translational tile with no fully periodic tiling in dimension three

September 23, 2026 26 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/155.md.

A counterexample to periodic tiling in dimension three

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The periodic-tiling question asks whether a finite tile that tiles a lattice must admit a periodic tiling. The formalized counterexample is a finite tile in Z3\mathbb Z^3 that tiles but has no complement invariant under a finite-index subgroup. Its unit-cube thickening also tiles R3\mathbb R^3 almost everywhere but admits no fully periodic tiling, even with arbitrary real translation vectors. The result also establishes that dimension three is the least lattice dimension where this failure occurs.

Comparator links

Result Comparator statement
Aperiodic translational tile in dimension three PeriodicTilingThree.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.