Patch frequencies in Penrose rhombic tilings
Jan Maz\'a\v{c}
TL;DR
This paper introduces an algorithm for precisely calculating patch frequencies in Penrose rhombic tilings, extending existing methods and applying them to specific large patches, with similar approaches for Ammann-Beenker tilings.
Contribution
It presents a novel algorithm for exact patch frequency calculation in Penrose tilings, based on dualisation and vertex configuration extension.
Findings
Successfully computed frequencies of large patches in Penrose tilings
Extended the method to Ammann-Beenker tilings
Provided a practical algorithm for exact patch frequency determination
Abstract
This short exposition presents an algorithm for an exact calculation of patch frequencies for the rhombic Penrose tiling. We recall a construction of Penrose tilings via dualisation, and by extending the known method for obtaining vertex configurations, we obtain the desired algorithm. It is then used to determine the frequencies of several particular large patches which appear in the literature. The analogous approach is also explained for the Ammann-Beenker tiling.
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Taxonomy
TopicsQuasicrystal Structures and Properties · graph theory and CDMA systems · Mathematics and Applications