Tiling 3D by Translates of a Single Polycube is Undecidable

Abstract: We prove co-RE-completeness, and thus undecidability, of the following problem: given a single (connected) polycube, decide whether it tiles 3D Euclidean space by translations. We reduce from Wang tiling using the decorated two-prime Sudoku construction of Greenfeld and Tao and a cyclic encoding adapted from OpenAI's 3D aperiodic tile, and apply a reduction of Kim to make the prototile connected (via faces). Dimension three is optimal: translational monotiling is known to be decidable in $\mathbb{Z}<sup>2$ and for a single (possibly disconnected) polyomino in $\mathbb{R}<sup>2$.