Tesselation

It is conjectured that a non-periodic tiling of the plane with a set of four prototiles can be realised by a simple decorated monotile and a simple replication-algorithm in a spiral movement. The algorithm uses overlaps and gaps, but there is no need of adding or removing edges. We also show some mutually locally derivable tilings including a set of two asymmetric prototiles never shown before (see Figure 1).

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.