Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes

TL;DR

This paper proves that determining whether a set of 6 polycubes can tile three-dimensional space through translation is an undecidable problem, extending known results about tiling complexities in higher dimensions.

Contribution

It establishes the undecidability of translational tiling in 3D space with a small set of 6 tiles, advancing understanding of tiling problem complexities.

Findings

Undecidability of 3D translational tiling with 6 polycubes

Extension of known 2D tiling undecidability results to 3D

Supports conjecture of undecidability with fewer tiles in higher dimensions

Abstract

This paper focuses on the undecidability of translational tiling of $n$-dimensional space $\mathbb{Z}^n$ with a set of $k$ tiles. It is known that tiling $\mathbb{Z}^2$ with translated copies with a set of $8$ tiles is undecidable. Greenfeld and Tao gave strong evidence in a series of works that for sufficiently large dimension $n$, the translational tiling problem for $\mathbb{Z}^n$ might be undecidable for just one tile. This paper shows the undecidability of translational tiling of $\mathbb{Z}^3$ with a set of $6$ tiles.