Geometric constraints limit the ways that identical subunits can be arranged on a closed spherical surface with equivalent or nearly equivalent contacts between the subunits. By far, the most favored arrangement is based on a net of equilateral triangles. On a plane surface, these triangles will pack hexagonally with six fold vertices. Since the time of Plato, it has been appreciated that introducing vertices surrounded by three, four, or five triangles will cause such a network of triangles to pucker and, given an appropriate number of puckers, to close up into a complete shell. Four threefold vertices make a tetrahedron, six fourfold vertices make an octahedron, and 12 fivefold vertices make an icosahedron. Remarkably, no other ways of arranging triangles will complete a shell. In addition to threefold, fourfold, or fivefold vertices that introduce puckers, a closed polygon can contain additional triangular faces and sixfold vertices to expand the volume. The sixfold vertices can be placed symmetrically with respect to the fivefold vertices to produce a spherical shell or asymmetrically to form an elongated structure.
Most closed macromolecular assemblies in biology are polygons with fivefold vertices. An important reason for this is that most structures require some sixfold vertices to provide sufficient internal volume. This favors fivefold vertices for the puckers, as they require much less distortion of the subunits located on the triangular faces of the hexagonal plane sheet than do threefold or fourfold vertices. Further, the distortion in the contacts between the triangles is minimized if the fivefold vertices are in equivalent positions. Closed icosahedral shells can be assembled from any type of asymmetrical subunit given two provisions: (1) The subunit must be able to form bonds with like subunits in a triangular network; and (2) these subunits must be able to accommodate the distortion required to form both fivefold and sixfold vertices. Both fibrous and globular subunits can fulfill these criteria.
These considerations indicate that subunits in a closed macromolecular assembly must be arranged in rings of five or six. A simple variation has three like protein subunits on each face, but three different protein subunits, or more than three like subunits, can be used on each face to construct icosahedrons, The closest packing is achieved if the protein subunits form pentamers and hexamers, but other arrangements on the 20 faces of an icosahedron are possible.