Let T be a tile in Z^n. A tiling by translated copies of T is periodic if it is invariant under a nonzero translation of Z^n. Periodic tiling conjecture. If T tiles Z^n with translated copies, then it can tile Z^n periodically with translated copies. This conjecture would imply decidability of translational tiling by a single tile in each fixed dimension. It was disproved by Greenfeld and Tao, who constructed a tile admitting a translational tiling but no periodic translational tiling.

Periodic tiling conjecture for translational monotile tilings

The conjecture asks whether every translational tiling of $\mathbb{Z}^n$ by one finite tile has a tiling invariant under a nonzero translation. Greenfeld and Tao constructed a counterexample in sufficiently high dimension, disproving the conjecture there.

Known results

Dimension $1$ : the conjecture was already known to hold.
Dimension $2$ : Bhattacharya proved it in $2020$ ; Greenfeld and Tao gave an alternative proof.
Sufficiently large dimensions: Greenfeld and Tao constructed aperiodic translational monotiles.

September 2022 counterexample

Greenfeld and Tao announced an explicit, computable finite tile in $\mathbb{Z}^d$ for sufficiently large $d$ that tiles translationally but admits no periodic translational tiling. Later accounts describe the result as published, but the smallest counterexample dimension is not identified and the claim is treated here as unverified.

Current status (as of September 2026): The conjecture is false in sufficiently large dimensions and true in dimensions $1$ and $2$ ; the three-dimensional and intermediate-dimensional cases, and the minimal counterexample dimension, remain unresolved.

Periodic tiling conjecture for translational monotile tilings

Primary source

Progress summary

Solutions 0