Periodic tiling problem in three dimensions

Determine whether every finite translational tile of \(\mathbb Z^3\) that admits a tiling complement also admits a fully periodic tiling complement.

Background

The paper studies when a finite subset of an integer lattice that tiles by translations must possess a fully periodic tiling complement. Although this property holds in dimensions one and two, the authors note that a counterexample exists in sufficiently high dimension, so the general phenomenon is dimension-sensitive.

The authors explicitly identify the three-dimensional case as unresolved. Resolving it would determine whether the rigidity known in the plane extends to Z3\mathbb Z^3, or whether genuinely aperiodic tileability can already occur there.

References

The general periodic tiling problem for $\mathbb Z3$ remains open; see Question~5.1.

— Periodic Tilings of Cardinality Twice a Prime or Nine in Arbitrary Dimension  (2609.26576 - Tan et al., 22 Sep 2026) in Section 1, Introduction, subsection “Background and main result”