Undecidability of Translational Tiling with 2 Polycubes

TL;DR

This paper proves that determining whether two polycubes can tile three-dimensional space by translation is undecidable, introducing a novel simulation technique that extends to higher dimensions and connected tiles.

Contribution

It establishes the undecidability of translational tiling with two polycubes and introduces a new technique for simulating disconnected tiles with connected ones in higher dimensions.

Findings

Proves undecidability for tiling with two polycubes in 3D.

Develops a technique to simulate disconnected tiles with connected tiles.

Extends simulation method to higher dimensions for connected tiles.

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 $n \geq 3$.