On discrete constant principal curvature surfaces

TL;DR

This paper develops a discrete surface theory on 3-ary trees to study and construct discrete constant principal curvature surfaces, with applications to nanocarbon materials.

Contribution

It introduces a new discrete surface theory on 3-ary trees and constructs examples of discrete CPC surfaces, including tori, relevant to nanocarbon materials.

Findings

Defined discrete principal directions on 3-ary trees

Constructed examples of discrete CPC surfaces

Developed a theory applicable to nanocarbon geometries

Abstract

Recently, it is discovered that a certain class of nanocarbon materials has geometrical properties related to the discrete geometry, pre-constant discrete principal curvature [9] based on the discrete surface theory proposed on trivalent graphs by Kotani, Naito and Omori [10]. In this paper, with the aim of an application to the nanocarbon materials, we will study discrete constant principal curvature (CPC) surfaces. Firstly, we developed the discrete surface theory on a full 3-ary oriented tree so that we define a discrete analogue of principal directions on them to investigate it. We also construct some interesting examples of discrete constant principal curvature surfaces, including discrete CPC tori.