Aperiodic Sets of Prototiles Extracted From the Penrose Rhomb Tiling

TL;DR

This paper introduces new aperiodic prototile sets derived from Penrose tilings, including an exact tiling set without matching rules and an approximate monotile that enforces nonperiodic tessellations.

Contribution

It presents novel aperiodic prototile sets based on Penrose rhomb tilings, including an exact set without matching rules and an approximate monotile for nonperiodic tiling.

Findings

Decorated prototiles produce exact Penrose tilings without matching rules

An approximate monotile tessellates the plane nonperiodically

The monotile includes five types of gaps in tessellations

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.