Construction of continuum from a discrete surface by its iterated subdivisions

TL;DR

This paper introduces a subdivision method for discrete surfaces represented by trivalent graphs in 3D space, analyzes the convergence of these subdivided surfaces, and explores the resulting continuum geometric limit.

Contribution

It presents a novel subdivision approach using Goldberg-Coxeter subdivision to derive continuum objects from discrete surfaces.

Findings

Convergence of subdivided discrete surfaces is established.

The limit set as a continuum geometric object is characterized.

The method bridges discrete and continuous surface representations.

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 subdivision method 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.