Undecidability of Tiling the Plane with a Set of 5 Polyominoes

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.