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Patch frequencies in rhombic Penrose tilings.

Jan Mazáč1

  • 1Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, Bielefeld 33501, Germany.

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|July 24, 2023
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Summary

This study introduces an efficient algorithm for calculating patch frequencies in rhombic Penrose tilings. The method extends existing techniques to accurately determine frequencies of large patches and applies to Ammann-Beenker tilings.

Keywords:
dualization methodpatch frequencytiling

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Area of Science:

  • Mathematics
  • Crystallography
  • Quasicrystal research

Background:

  • Penrose tilings are aperiodic structures with unique geometric properties.
  • Calculating the frequency of specific local arrangements (patches) in these tilings is computationally challenging.
  • Existing methods for vertex configurations do not directly address patch frequency calculations.

Purpose of the Study:

  • To develop an efficient algorithm for the exact calculation of patch frequencies in rhombic Penrose tilings.
  • To extend this methodology to other related aperiodic tiling systems, such as the Ammann-Beenker tiling.
  • To determine frequencies for significant patches found in existing literature.

Main Methods:

  • A construction of Penrose tilings using dualization is reviewed.
  • The known method for obtaining vertex configurations is extended.
  • An efficient algorithm for exact patch frequency calculation is derived from the extended method.
  • The algorithm is applied to specific large patches and the Ammann-Beenker tiling.

Main Results:

  • An efficient algorithm for exact patch frequency calculation in rhombic Penrose tilings is presented.
  • The algorithm successfully determines frequencies for several large, known patches.
  • An analogous approach is demonstrated to be effective for the Ammann-Beenker tiling.

Conclusions:

  • The developed algorithm provides an efficient and exact method for analyzing patch frequencies in Penrose and related aperiodic tilings.
  • This work facilitates a deeper understanding of the statistical properties of quasicrystalline structures.
  • The generalized approach offers a valuable tool for researchers in mathematics and materials science.