Abstract
The Caspar–Klug (CK) classification of viruses is discussed by parallel examination of geometry of icosahedral geodesic domes, fullerenes, and viruses. The underlying symmetry of all structures is explained and thoroughly visually represented. Euler’s theorem on polyhedra is used to calculate the number of vertices, edges, and faces in domes, number of atoms, bonds, and pentagonal and hexagonal rings in fullerenes, and number of proteins and protein–protein contacts in viruses. The T-number, the characteristic for the CK classification, is defined and discussed. The superposition of fullerene and dome designs is used to obtain a representation of a CK virus with all the proteins indicated. Some modifications of the CK classifications are sketched, including elongation of the CK blueprint, fusion of two CK blueprints, dodecahedral view of the CK shapes, and generalized CK designs without a clearly visible geometry of the icosahedron. These are compared to cases of existing viruses.
Highlights
The fullerene molecules were theoretically predicted and discussed [1], it was a surprise for the largest part of the scientific community when Kroto et al published a paper announcing their experimental discovery [2]
The basic geometry and symmetry behind such structures was known at least since 1937 when Goldberg discussed a class of polyhedra, often called Goldberg polyhedra, with only pentagonal and hexagonal faces [4]
A net of a geodesic dome would contain triangles of different sizes, but it is easier to think of a net of an icosahedron whose faces have been subdivided/triangulated prior to projecting them on the sphere—such a polyhedron has a simple net, with the triangles/faces that are all equal and equilateral and which can be grouped within 20 larger triangles, the faces of the starting icosahedron
Summary
The fullerene molecules were theoretically predicted and discussed [1], it was a surprise for the largest part of the scientific community when Kroto et al published a paper announcing their experimental discovery [2]. A net of a geodesic dome would contain triangles of different sizes, but it is easier to think of a net of an icosahedron whose faces have been subdivided/triangulated prior to projecting them on the sphere—such a polyhedron has a simple net, with the triangles/faces that are all equal and equilateral and which can be grouped within 20 larger triangles, the faces of the starting icosahedron (see Figure 1b) This is what is meant by the icosadeltahedral—the triangular subdivision of icosahedron faces that produces polyhedra with a larger number of triangular faces, which become unequal once the subdivided faces are projected on a sphere (Note here that the icosahedron with subdivided faces is not strictly convex and that there are neighboring triangular(sub)faces with a dihedral angle of π.).
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