Icosadeltahedral geometry of fullerenes, viruses and geodesic domes

TL;DR

This paper explores the common icosadeltahedral symmetry in fullerenes, viruses, and geodesic domes, providing a unified geometric framework and detailed calculations based on Euler's theorem.

Contribution

It introduces a comprehensive icosadeltahedral model that unifies the symmetry analysis of fullerenes, viruses, and geodesic domes, and applies Euler's theorem for structural calculations.

Findings

Unified geometric framework for fullerenes, viruses, and geodesic domes.

Explicit calculations of vertices, edges, and faces using Euler's theorem.

Elaboration of Caspar-Klug virus classification within icosadeltahedral geometry.

Abstract

I discuss the symmetry of fullerenes, viruses and geodesic domes within a unified framework of icosadeltahedral representation of these objects. The icosadeltahedral symmetry is explained in details by examination of all of these structures. Using Euler's theorem on polyhedra, it is shown how to calculate the number of vertices, edges, and faces in domes, and number of atoms, bonds and pentagonal and hexagonal rings in fullerenes. Caspar-Klug classification of viruses is elaborated as a specific case of icosadeltahedral geometry.