Tiling with arbitrary tiles

TL;DR

The paper proves that any finite tile in integer lattice space can tile some higher-dimensional integer lattice, resolving a conjecture and advancing understanding of tiling problems in discrete geometry.

Contribution

It establishes that every finite tile in al^n can tile al^d for some d, confirming Chalcraft's conjecture.

Findings

Any finite tile in al^n can tile al^d for some d

Resolution of Chalcraft's conjecture in tiling theory

Advances understanding of tiling possibilities in higher dimensions

Abstract

Let $T$ be a tile in $\mathbb{Z}^n$, meaning a finite subset of $\mathbb{Z}^n$. It may or may not tile $\mathbb{Z}^n$, in the sense of $\mathbb{Z}^n$ having a partition into copies of $T$. However, we prove that $T$ does tile $\mathbb{Z}^d$ for some $d$. This resolves a conjecture of Chalcraft.