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Symmetry 2017, 9(2), 19; doi:10.3390/sym9020019

Cyclotomic Aperiodic Substitution Tilings

Visualien der Breitbandkatze, Am Mitterweg 1, 85309 Pörnbach, Germany
Academic Editor: Hans Grimmer
Received: 28 September 2016 / Revised: 18 December 2016 / Accepted: 4 January 2017 / Published: 25 January 2017

Abstract

The class of Cyclotomic Aperiodic Substitution Tilings (CASTs) is introduced. Its vertices are supported on the 2 n -th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmetry D n or D 2 n , i.e., the tiling contains an infinite number of patches of any size with dihedral symmetry D n or D 2 n only by iteration of substitution rules on a single tile. View Full-Text
Keywords: cyclotomic aperiodic substitution tiling; discrete metric geometry; quasiperiodic; nonperiodic; minimal inflation multiplier; minimal scaling factor; substitution matrix; Girih tiling cyclotomic aperiodic substitution tiling; discrete metric geometry; quasiperiodic; nonperiodic; minimal inflation multiplier; minimal scaling factor; substitution matrix; Girih tiling
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Pautze, S. Cyclotomic Aperiodic Substitution Tilings. Symmetry 2017, 9, 19.

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