Undecidability of Translational Tiling with 2 Polycubes

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 $n\ge 3$.