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A simpler approach to Penrose tiling with implications for quasicrystal formation - Nature

  • ️Jeong, Hyeong-Chai
  • ️Thu Aug 01 1996
  • Letter
  • Published: 01 August 1996

Nature volume 382, pages 431–433 (1996)Cite this article

Abstract

QUASICRYSTALS1have a quasiperiodic atomic structure with symmetries (such as fivefold) that are forbidden to ordinary crystals2,3. Why do atoms form this complex pattern rather than a regularly repeating crystal? An influential model of quasicrystal structure has been the Penrose tiling4, in which two types of tile are laid down according to 'matching rules' that force a fivefold-symmetric quasiperiodic pattern. In physical terms, it has been suggested1 that atoms form two or more clusters analogous to the tiles, with interactions that mimic the matching rules. Here we show that this complex picture can be simplified. We present proof of the claim5 that a quasiperiodic tiling can be forced using only a single type of tile, and furthermore we show that matching rules can be discarded. Instead, maximizing the density of a chosen cluster of tiles suffices to produce a quasiperiodic tiling. If one imagines the tile cluster to represent some energetically preferred atomic cluster, then minimizing the free energy would naturally maximize the cluster density6. This provides a simple, physically motivated explanation of why quasicrystals form.

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Authors and Affiliations

  1. Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania, 19104, USA

    Paul J. Steinhardt

  2. Institute for Physical Science and Technology, University of Maryland, College Park, Maryland, 20742, USA

    Hyeong-Chai Jeong

Authors

  1. Paul J. Steinhardt

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  2. Hyeong-Chai Jeong

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Steinhardt, P., Jeong, HC. A simpler approach to Penrose tiling with implications for quasicrystal formation. Nature 382, 431–433 (1996). https://doi.org/10.1038/382431a0

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  • Received: 18 March 1996

  • Accepted: 11 June 1996

  • Issue Date: 01 August 1996

  • DOI: https://doi.org/10.1038/382431a0