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Constructing Penrose-like tilings with 7-fold symmetry

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Abstract

Penrose-like tilings with higher-order rotational symmetry are discussed. Two new examples of recursive rhombic tilings with 7-fold symmetry are derived. The entangled inflation/deflation rules cyclically permute the characteristic 7-fold patches and provide a possibility of expanding the tilings up to infinity without additional ad hoc rearrangements. The derived tilings are self-similar and fully coincide with themselves after inflations and deflations. The main structural motif is presented by repeatedly appearing seven-pointed stars surrounded by similar seven-pointed stars. Characteristic patches demonstrate mutual metamorphoses. The patches cyclically morph each into another while the tilings as a whole remain self-consistent. The results of the study can be used for the development of new artificial materials with heptagonal symmetry.

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Acknowledgments

I thank Jelena R. Kambak for proofreading and writing assistance.

Funding

This work is partially supported by the Ministry of Education and Science of the Russian Federation, Grant 16.8549.2017/8.9.

Author information

Correspondence to A.E. Madison.

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The author declares that he has no conflict of interest.

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Madison, A. Constructing Penrose-like tilings with 7-fold symmetry. Struct Chem 29, 645–655 (2018). https://doi.org/10.1007/s11224-018-1083-7

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Keywords

  • 7-fold symmetry
  • Quasicrystal
  • Self-similarity
  • Substitution tiling
  • Heptagonal tiling