CONSTRUCTION OF CONTINUUM FROM A DISCRETE SURFACE BY ITS ITERATED SUBDIVISIONS

Motoko Kotani; Hisashi Naito; Chen Tao

doi:10.2748/tmj.20201225

CONSTRUCTION OF CONTINUUM FROM A DISCRETE SURFACE BY ITS ITERATED SUBDIVISIONS

Motoko Kotani, Hisashi Naito, Chen Tao

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Abstract

Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivisionmethod by applying the Goldberg-Coxeter subdivision and discuss the convergence of a sequence of discrete surfaces defined inductively by the subdivision. We also study the limit set as the continuum geometric object associated with the given discrete surface.

Original language	English
Pages (from-to)	229-252
Number of pages	24
Journal	Tohoku Mathematical Journal
Volume	74
Issue number	2
DOIs	https://doi.org/10.2748/tmj.20201225
Publication status	Published - 2022

Keywords

convergence theory
discrete curvature
Discrete geometry

ASJC Scopus subject areas

Mathematics(all)

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10.2748/tmj.20201225

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title = "CONSTRUCTION OF CONTINUUM FROM A DISCRETE SURFACE BY ITS ITERATED SUBDIVISIONS",

abstract = "Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivisionmethod by applying the Goldberg-Coxeter subdivision and discuss the convergence of a sequence of discrete surfaces defined inductively by the subdivision. We also study the limit set as the continuum geometric object associated with the given discrete surface.",

keywords = "convergence theory, discrete curvature, Discrete geometry",

author = "Motoko Kotani and Hisashi Naito and Chen Tao",

note = "Funding Information: 2010 Mathematics Subject Classification. Primary 52C99, Secondary 53A05, 53C23, 65D17. Key words and phrases. Discrete geometry, discrete curvature, convergence theory. Authors (KM and HN) were partially supported by JSPS KAKENHI Grants Numbers JP17H06465, JP17H06466, and JP19K03488. KM was partially supported by JST, CREST Grant Number JPMJCR17J4, Japan. Publisher Copyright: {\textcopyright} 2022 Tohoku University, Mathematical Institute. All rights reserved.",

year = "2022",

doi = "10.2748/tmj.20201225",

language = "English",

volume = "74",

pages = "229--252",

journal = "Tohoku Mathematical Journal",

issn = "0040-8735",

publisher = "Tohoku University, Mathematical Institute",

number = "2",

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TY - JOUR

T1 - CONSTRUCTION OF CONTINUUM FROM A DISCRETE SURFACE BY ITS ITERATED SUBDIVISIONS

AU - Kotani, Motoko

AU - Naito, Hisashi

AU - Tao, Chen

N1 - Funding Information: 2010 Mathematics Subject Classification. Primary 52C99, Secondary 53A05, 53C23, 65D17. Key words and phrases. Discrete geometry, discrete curvature, convergence theory. Authors (KM and HN) were partially supported by JSPS KAKENHI Grants Numbers JP17H06465, JP17H06466, and JP19K03488. KM was partially supported by JST, CREST Grant Number JPMJCR17J4, Japan. Publisher Copyright: © 2022 Tohoku University, Mathematical Institute. All rights reserved.

PY - 2022

Y1 - 2022

N2 - Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivisionmethod by applying the Goldberg-Coxeter subdivision and discuss the convergence of a sequence of discrete surfaces defined inductively by the subdivision. We also study the limit set as the continuum geometric object associated with the given discrete surface.

AB - Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivisionmethod by applying the Goldberg-Coxeter subdivision and discuss the convergence of a sequence of discrete surfaces defined inductively by the subdivision. We also study the limit set as the continuum geometric object associated with the given discrete surface.

KW - convergence theory

KW - discrete curvature

KW - Discrete geometry

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DO - 10.2748/tmj.20201225

M3 - Article

AN - SCOPUS:85135199858

SN - 0040-8735

VL - 74

SP - 229

EP - 252

JO - Tohoku Mathematical Journal

JF - Tohoku Mathematical Journal

IS - 2

ER -

CONSTRUCTION OF CONTINUUM FROM A DISCRETE SURFACE BY ITS ITERATED SUBDIVISIONS

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