Decision problems on geometric tilings

Abstract

We study decision problems on geometric tilings. First, we study a variant of the Domino problem where square tiles are replaced by geometric tiles of arbitrary shape. We show that, under some weak assumptions, this variant is undecidable regardless of the shapes, extending previous results on rhombus tiles. This result holds even when the geometric tiling is forced to belong to a fixed set.Second, we consider the problem of deciding whether a geometric subshift has finite local complexity, which is a common assumption when studying geometric tilings. We show that it is undecidable even in a simple setting (square shapes with small modifications).