Aperiodic Sets of Prototiles Extracted From the Penrose Rhomb Tiling

Abstract

We present aperiodic sets of prototiles whose shapes are based on the well-known Penrose rhomb tiling. Some decorated prototiles lead to an exact Penrose rhomb tiling without any matching rules. We also give an approximate solution to an aperiodic monotile that tessellates the plane (including five types of gaps) only in a nonperiodic way.