Cyclotomic Aperiodic Substitution Tilings

Pautze, Stefan

doi:10.3390/sym9020019

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cyclotomic aperiodic substitution tiling

discrete metric geometry

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minimal inflation multiplier

minimal scaling factor

substitution matrix

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

Cyclotomic Aperiodic Substitution Tilings

Stefan Pautze

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

Full-Text | PDF [2984 KB, uploaded 25 January 2017] |

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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}

D_{2 n}

, i.e., the tiling contains an infinite number of patches of any size with dihedral symmetry

D_{n}

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

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Pautze S. Cyclotomic Aperiodic Substitution Tilings. Symmetry. 2017; 9(2):19.

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Pautze, Stefan. 2017. "Cyclotomic Aperiodic Substitution Tilings." Symmetry 9, no. 2: 19.

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