Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes

TL;DR

This paper proves that determining whether a set of four polyhypercubes can tile four-dimensional space through translation is undecidable, advancing understanding of tiling problems in higher dimensions.

Contribution

It introduces a novel technique to lift tiling undecidability from 3D to 4D space, showing that tiling with four tiles in 4D is undecidable.

Findings

Undecidability of 4D tiling with four polyhypercubes.

Extension of 3D tiling undecidability to 4D.

Progress towards general conjecture on tiling undecidability.

Abstract

Recently, Greenfeld and Tao disprove the conjecture that translational tilings of a single tile can always be periodic [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension $n$ is part of the input, the translational tiling for subsets of $\mathbb{Z}^n$ with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable, for some fixed $n$. This paper shows that translational tiling of the $3$-dimensional space with a set of $5$ polycubes is undecidable. By introducing a technique that lifts a set of polycubes and its tiling from $3$-dimensional space to $4$-dimensional space, we manage to show that translational tiling of the $4$-dimensional space with a set of $4$ tiles is undecidable. This is a step towards the attempt to settle the conjecture of the undecidability of translational tiling of the $n$-dimensional space with a monotile, for some fixed $n$.