Applications of aesthetic pentagon-shaped stereo tiling employing pentagraphene carbon—star walls and embossment design

Katsumi Hagita,Yoshiyuki Kawazoe,Masao Ogino

doi:10.1063/1.5085641

Katsumi Hagita, Yoshiyuki Kawazoe + Show 1 more

Open Access

https://doi.org/10.1063/1.5085641

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Abstract

A three-dimensional pentagon-shaped stereo-tiling concept has been realized through use of penta-graphene carbon structures, although mathematically, regular planar pentagon shapes cannot be used to completely tile the Euclidean plane. Two applications of the findings of this study have been considered from the mathematical and engineering viewpoints. First, the proposed discovery facilitates math-rule-based generation of beautiful designs comprising star shapes formed using regular pentagons. The underlying mathematical logic and hexagon-division rules have been deduced to obtain the proposed pentagonal stereo-tiling pattern comprising equal-length bonds, and two aesthetic designs have been derived using the deduced logic. A fence-like structure exclusively comprising pentagram stars has been designed and given the name “Star Walls.” Emphasis has also been laid on application of the proposed stereo-tiling concept in industrial design operations, such as emboss manufacturing. To this end, finite element analysis of embossed steel sheets has been performed to verify the feasibility of the said industrial application.

Highlights

Recent developments in computing technology and everincreasing computational power have revolutionized modern-day applications of pure and applied mathematics
Extant studies performed on carbon-based materials, such as C60 fullerene,3–5 carbon nanotubes (CNTs),6,7 and graphene,8 have been based on the isolated pentagon rule (IPR), which essentially explains why none of the pentagons on the periphery of a soccer ball make contact with neighboring pentagons
We studied pentagon-shaped stereo tiling employing the PG carbon structure

Summary

INTRODUCTION

Recent developments in computing technology and everincreasing computational power have revolutionized modern-day applications of pure and applied mathematics. Convex polyhedrons with equal-length bonds and polyhedral symmetry can be classified into four types—the Platonic (regular) and Archimedean polyhedrons (such as the soccer ball) reported by ancient Greeks in the 17th century; rhombic polyhedral reported by Johannes Kepler; and lastly, Michael Goldberg (1937) proposed extensions to fullerene-based polyhedrons with non-planar quasi-hexagons, each of which possess six equallength bonds.. Convex polyhedrons with equal-length bonds and polyhedral symmetry can be classified into four types—the Platonic (regular) and Archimedean polyhedrons (such as the soccer ball) reported by ancient Greeks in the 17th century; rhombic polyhedral reported by Johannes Kepler; and lastly, Michael Goldberg (1937) proposed extensions to fullerene-based polyhedrons with non-planar quasi-hexagons, each of which possess six equallength bonds.16 This last type of convex polyhedron is referred to as the “Goldberg polyhedron.”. It has been observed that a hexagon comprising four pentagons (Figure 2(b)) is similar to the Cairo pentagon-tiling pattern

RESULTS AND DISCUSSION

Mathematical rule

Application 1

Application 2

SUMMARY