Undecidability of Translational Tiling with 2 Polycubes

Abstract

In this paper, we prove that it is undecidable whether a set of two polycubes can tile $\mathbb{Z}^3$ by translation. The proof involves a new technique that allows us to simulate two disconnected polycubes with two connected polycubes. By expanding this technique to higher dimensions, we also prove that a set of disconnected tiles in $\mathbb{Z}^n$ can be simulated by the same number of connected tiles in $\mathbb{Z}^n$ for $n \geq 3$.