Tiling 3D by Translates of a Single Polycube is Undecidable

2026-10-08

AI Summary

This study investigates whether a single connected polycube can tile three-dimensional Euclidean space via translations. By synthesizing Wang tiling reductions, the Greenfeld–Tao decorated twin prime Sudoku construction, OpenAI cycle encoding, and Kim’s connectivity reduction, this work provides the first rigorous proof that the problem is undecidable in three dimensions and establishes it as co-RE-complete. This research transcends the known decidability limitations in two dimensions by demonstrating that three dimensions constitute the minimal dimension in which the translational tiling problem for a single polycube becomes undecidable. Consequently, it determines the optimal critical dimensional boundary for the algorithmic decidability of this problem.

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 $\mathbb{Z}^2$ and for a single (possibly disconnected) polyomino in $\mathbb{R}^2$.

Problem

• Polycube
• Tiling
• 3D Euclidean space
• Undecidability
• Translational monotiling

Innovation

• Undecidability
• Polycube Tiling
• Wang Tiling
• co-RE-completeness
• Aperiodic Tile