Symmetry relationships in icosahedral capsids
Viruses of the morphologically icosahedral group have proteinous shells of 60 alike subunits. The 60 subunits are arranged in the most symmetric way possible, forming a sphere made up of pentagons and hexagons. This may be seen by reducing the subunits to the corners of the sphere they form . All elements of this structure are related by spherical symmetry to a counterpart on the 'other side' of the sphere, of course. Additionally, there are symmetry relations important for the interaction of the protein subunits.
So, there are fivefold symmetry axes which require the subunits to interact with each other in a 72° pattern .
Another symmetry requirement is brought about by axes with threefold symmetry .
To complete the protein-protein interaction requirements, there are twofold symmetry axes as well .
The shape of the subunits has to meet the geometric restraints imposed by these three symmetry operations.
Now where does the icosahedron come in? By emphasizing the fivefold symmetry of the viral sphere and connecting all ends of the fivefold axes you obtain the lattice of a simpler sphere with 20 triangular surfaces , which obviously is an icosaeder. Symmetries remain the same as before , you still find threefold (10 of them!) and twofold axes (15 in all).
For the description of features of the capsids virologists tend to use 'road maps' in which they use just one (geometrically defined) subunit to exemplify their means. These subunits are oriented between one fivefold and two threefold axes (encompassing a twofold axis too) with the fivefold axis pointing north . Individual aminoacids in this region are listed in the road map with their positions projected into the plane of the triangel along with the physical, chemical or immmunological property in question.
The viral model shown here is no virus at all. The protein lattice may remind you of a soccer ball, and indeed the underlying atomic coordinates are of a substance named buckyball or in full length Buckminsterfullerene (chemical formula C60).