Undecidability of the tiling problem for fewer than three polygons

Determine whether the decision problem of whether a finite set of polygons tiles the Euclidean plane is computationally undecidable for sets of size two and for a single polygon, thereby establishing the minimal cardinality of a polygonal tile set that yields undecidability.

Background

The paper discusses the classical tiling decision problem and notes that it is known to be undecidable for at least three polygons, citing recent advances. The author emphasizes that the status for smaller sets remains unresolved, making the exact minimal number of polygonal prototiles required for undecidability a key open question.

Resolving undecidability for two or one polygon would clarify the boundary of algorithmic intractability in plane tilings and potentially connect to broader complexity-theoretic questions in combinatorial geometry.