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Describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components

Polyhedral tilings are often used to represent structures such as atoms in materials, grains in crystals, foams, galaxies in the universe, etc. In the previous paper, we have developed a theory to convert a way of how polyhedra are arranged to form a polyhedral tiling into a codeword (series of numb...

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Autores principales: Nishio, Kengo, Miyazaki, Takehide
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Nature Publishing Group 2017
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5240342/
https://www.ncbi.nlm.nih.gov/pubmed/28094254
http://dx.doi.org/10.1038/srep40269
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author Nishio, Kengo
Miyazaki, Takehide
author_facet Nishio, Kengo
Miyazaki, Takehide
author_sort Nishio, Kengo
collection PubMed
description Polyhedral tilings are often used to represent structures such as atoms in materials, grains in crystals, foams, galaxies in the universe, etc. In the previous paper, we have developed a theory to convert a way of how polyhedra are arranged to form a polyhedral tiling into a codeword (series of numbers) from which the original structure can be recovered. The previous theory is based on the idea of forming a polyhedral tiling by gluing together polyhedra face to face. In this paper, we show that the codeword contains redundant digits not needed for recovering the original structure, and develop a theory to reduce the redundancy. For this purpose, instead of polyhedra, we regard two-dimensional regions shared by faces of adjacent polyhedra as building blocks of a polyhedral tiling. Using the present method, the same information is represented by a shorter codeword whose length is reduced by up to the half of the original one. Shorter codewords are easier to handle for both humans and computers, and thus more useful to describe polyhedral tilings. By generalizing the idea of assembling two-dimensional components to higher dimensional polytopes, we develop a unified theory to represent polyhedral tilings and polytopes of different dimensions in the same light.
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spelling pubmed-52403422017-01-23 Describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components Nishio, Kengo Miyazaki, Takehide Sci Rep Article Polyhedral tilings are often used to represent structures such as atoms in materials, grains in crystals, foams, galaxies in the universe, etc. In the previous paper, we have developed a theory to convert a way of how polyhedra are arranged to form a polyhedral tiling into a codeword (series of numbers) from which the original structure can be recovered. The previous theory is based on the idea of forming a polyhedral tiling by gluing together polyhedra face to face. In this paper, we show that the codeword contains redundant digits not needed for recovering the original structure, and develop a theory to reduce the redundancy. For this purpose, instead of polyhedra, we regard two-dimensional regions shared by faces of adjacent polyhedra as building blocks of a polyhedral tiling. Using the present method, the same information is represented by a shorter codeword whose length is reduced by up to the half of the original one. Shorter codewords are easier to handle for both humans and computers, and thus more useful to describe polyhedral tilings. By generalizing the idea of assembling two-dimensional components to higher dimensional polytopes, we develop a unified theory to represent polyhedral tilings and polytopes of different dimensions in the same light. Nature Publishing Group 2017-01-17 /pmc/articles/PMC5240342/ /pubmed/28094254 http://dx.doi.org/10.1038/srep40269 Text en Copyright © 2017, The Author(s) http://creativecommons.org/licenses/by/4.0/ This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
spellingShingle Article
Nishio, Kengo
Miyazaki, Takehide
Describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components
title Describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components
title_full Describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components
title_fullStr Describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components
title_full_unstemmed Describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components
title_short Describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components
title_sort describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5240342/
https://www.ncbi.nlm.nih.gov/pubmed/28094254
http://dx.doi.org/10.1038/srep40269
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