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Onion-Like Inorganic Fullerenes from a Polyhedral Perspective
Published in Klaus D. Sattler, st Century Nanoscience – A Handbook, 2020
Ch. Chang, A. B. C. Patzer, D. Sülzle, H. Bauer
The isolated pentagon rule (IPR) established for the Cn fullerenes provides a criterion to predict, which system will have icosahedral symmetry, if not distorted by an electronic Jahn–Teller effect.5 Those cluster that maintain Ih symmetry are special cases of polyhedra known as Goldberg polyhedra (Goldberg 1937). The number of vertices, faces, and edges of a Goldberg polyhedron can be calculated from T = m2 + mn + n2 = (m + n)2 − mn, where m, n are non-negative integers. It follows that their number of vertices is v = 20T, their number of edges is e = 30T, and their number of faces is f = 10T + 2, which comprises 12 pentagons and 10(T — 1) hexagons. Goldberg polyhedra correspond to hexagonal close packing on the surface of an icosahedron. If m = n and mn = 0 then the undistorted Goldberg polyhedra will have Ih symmetry otherwise the symmetry is reduced to its rotational subgroup I.
HIV-1 immature virion network and icosahedral capsids self-assembly with patchy spheres
Published in Molecular Physics, 2023
Brian Ignacio Machorro-Martínez, Anthony B. Gutiérrez, Jacqueline Quintana, Julio C. Armas-Pérez, Paola Mendoza-Espinosa, Gustavo A. Chapela
Kaspar and Klug [19] developed an icosahedral capsid of different sizes classifying method. They assigned them a triangulation number , where a larger number implies a bigger capsid. This classification is in accordance with the Goldberg polyhedra [20], characterised to be composed only by pentagons and hexagons, just like these types of capsids. This triangulation number provides information about the total number of proteins P = 60T and capsomers ; with hexamers and 12 pentamers. Triangulation number is defined as , where h and k are positive integers that correspond to the number of hexagons to cross from one pentagon to the other within the same capsid. Traveling h hexagons in a straight line and k hexagons, to the right or left, to get to the next pentagon.