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Tilings and Coverings

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Crystallography of Quasicrystals

Part of the book series: Springer Series in Materials Science ((SSMATERIALS,volume 126))

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Abstract

Tilings fill space without gaps and overlaps, they can be periodic, quasiperiodic or nonperiodic. If decorated with atoms or larger atomic arrangements, tilings can serve as models for quasiperiodic structures. One-, two-, and three-dimensional examples will be discussed in detail. Beside substitutional sequences such as the Fibonacci and Octonacci sequences, also sequences with almost continuous and singular continuous spectra will be discussed. The tilings underlying really existing quasicrystals with 5-, 8-, 10-, 12-, and 14-fold symmetry or their approximants are treated in detail. Finally, the three-dimensional Penrose tiling is dealt with as example for the quasilattice of icosahedral quasicrystals. Furthermore, coverings will be discussed, which are important for understanding the geometry of cluster structures. Contrary to packings and tilings, coverings fill the space without gaps but with partial overlaps. There is always a one-to-one correspondence between coverings and tilings.

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Authors and Affiliations

  1. ETH Zürich, Department of Materials, Laboratory of Crystallography, Wolfgang-Pauli-Str. 10, 8093, Zürich, Switzerland

    Professor Dr. Walter Steurer & Dr. Sofia Deloudi

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  1. Professor Dr. Walter Steurer
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  2. Dr. Sofia Deloudi
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Correspondence to Walter Steurer .

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Steurer, W., Deloudi, S. (2009). Tilings and Coverings. In: Crystallography of Quasicrystals. Springer Series in Materials Science, vol 126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01899-2_1

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  • DOI: https://doi.org/10.1007/978-3-642-01899-2_1

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