Undecidability of Translational Tiling of the Plane with Four Tiles

TL;DR

This paper proves that determining whether four disconnected polyominoes can tile the plane through translation alone is an undecidable problem, extending previous results that required more tiles.

Contribution

It establishes the undecidability of translational tiling of the plane with only four disconnected polyominoes, reducing the known minimum number from five to four.

Findings

Undecidability established for four polyominoes

Extends previous undecidability results to fewer tiles

Uses novel reduction techniques in discrete geometry

Abstract

The translational tiling problem, dated back to Wang's domino problem in the 1960s, is one of the most representative undecidable problems in the field of discrete geometry and combinatorics. Ollinger initiated the study of the undecidability of translational tiling with a fixed number of tiles in 2009, and proved that translational tiling of the plane with a set of $11$ polyominoes is undecidable. The number of polyominoes needed to obtain undecidability was reduced from $11$ to $7$ by Yang and Zhang, and then to $5$ by Kim. We show that translational tiling of the plane with a set of $4$ (disconnected) polyominoes is undecidable in this paper.