Undecidability of Tiling the Plane with a Set of 5 Polyominoes

TL;DR

This paper proves that it is undecidable to determine whether five polyominoes can tile the plane by translation, introducing a novel edge-labeling method involving a dedicated polyomino.

Contribution

It presents a new undecidability proof for tiling the plane with five polyominoes using an innovative edge-labeling technique.

Findings

Proves undecidability for tiling with five polyominoes

Introduces a new edge-labeling method for polyominoes

Uses a dedicated polyomino for labeling process

Abstract

In this paper, we give a proof that it is undecidable whether a set of five polyominoes can tile the plane by translation. The proof involves a new method of labeling the edges of polyominoes, making it possible to assign whether two edges can match for any set of two edges chosen. This is achieved by dedicating 1 polyomino to the labeling process.