On the Frequency Module of the Hull of a Primitive Substitution Tiling

April Lynne D Say-Awen; Dirk Frettlöh; Ma. Louise Antonette N De Las Peñas; Ma. Louise Antonette De las Penas

On the Frequency Module of the Hull of a Primitive Substitution Tiling

April Lynne D Say-Awen, Dirk Frettlöh, Ma. Louise Antonette N De Las Peñas, Ma. Louise Antonette De las Penas

Research output: Contribution to journal › Article › peer-review

Abstract

Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal {\bb Z}-module, where {\bb Z} is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.

Original language	American English
Pages (from-to)	36-55
Number of pages	20
Journal	Mathematics Faculty Publications
Volume	78
State	Published - Jan 1 2022

ASJC Scopus Subject Areas

Structural Biology
Biochemistry
General Materials Science
Condensed Matter Physics
Physical and Theoretical Chemistry
Inorganic Chemistry

Keywords

frequency module
primitive substitution tilings
aperiodic tilings
quasicrystals

Disciplines

Mathematics

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On the Frequency Module of the Hull of a Primitive Substitution TFinal published version, 4.9 MBLicense: Unspecified

Cite this

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title = "On the Frequency Module of the Hull of a Primitive Substitution Tiling",

abstract = "Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal {\bb Z}-module, where {\bb Z} is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.",

keywords = "frequency module, primitive substitution tilings, aperiodic tilings, quasicrystals",

author = "Say-Awen, {April Lynne D} and Dirk Frettl{\"o}h and {De Las Pe{\~n}as}, {Ma. Louise Antonette N} and {De las Penas}, {Ma. Louise Antonette}",

note = "Say-awen, A. L. D., Frettl{\"o}h, D., & De Las Pe{\~n}as, M. L. A. N. (2022). On the frequency module of the hull of a primitive substitution tiling. Acta Crystallographica Section A: Foundations and Advances, A78(1), 36–55. https://doi.org/10.1107/S2053273321012572",

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language = "American English",

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T1 - On the Frequency Module of the Hull of a Primitive Substitution Tiling

AU - Say-Awen, April Lynne D

AU - Frettlöh, Dirk

AU - De Las Peñas, Ma. Louise Antonette N

AU - De las Penas, Ma. Louise Antonette

N1 - Say-awen, A. L. D., Frettlöh, D., & De Las Peñas, M. L. A. N. (2022). On the frequency module of the hull of a primitive substitution tiling. Acta Crystallographica Section A: Foundations and Advances, A78(1), 36–55. https://doi.org/10.1107/S2053273321012572

PY - 2022/1/1

Y1 - 2022/1/1

N2 - Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal {\bb Z}-module, where {\bb Z} is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.

AB - Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal {\bb Z}-module, where {\bb Z} is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.

KW - frequency module

KW - primitive substitution tilings

KW - aperiodic tilings

KW - quasicrystals

UR - https://archium.ateneo.edu/mathematics-faculty-pubs/185

UR - http://www.scopus.com/inward/record.url?scp=85123036389&partnerID=8YFLogxK

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M3 - Article

C2 - 34967328

VL - 78

SP - 36

EP - 55

JO - Mathematics Faculty Publications

JF - Mathematics Faculty Publications

ER -

On the Frequency Module of the Hull of a Primitive Substitution Tiling

Abstract

ASJC Scopus Subject Areas

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