The following is compiled largely from my "Applications of Group Theory to Virology" module I took at The University of York as an undergraduate back in 2012.
The icosahedral group $I$ with identity $e$ is given by a two-fold rotation $R_2$ and a three-fold rotation $R_3$ subject to the following presentation:
$$I\cong\langle R_2, R_3\mid R_2^2=R_3^3=(R_2R_3)^5=e\rangle.$$
Thus $I$ is isomorphic to the alternating group $\mathcal{A}_5$ and has $60$ elements.
Definition: A fundamental domain of a symmetry group of an object $O$ is a part that determines the whole object based on the symmetry and is as small or irredundant as possible. [NB: In the biological literature the fundamental domain is also called the asymmetric unit.]
Caspar-Klug's Quasi-Equivalence Theory:
(This, again, is taken from the notes for the module ibid.)
Viruses are composed of a protective shell of proteins called a capsid which encloses the viral genome. Most viruses display icosahedral symmetry [. . .]. If we place a subunit on a face of the icosahedral surface, then (assuming it doesn't lie on a symmetry axis) $n$ identical copies are to be created by a full rotation about an $n$-fold symmetry axis of rotation. Thus, in our example [i.e., the one given above], each equilateral triangle has three asymmetrical subunits on its face. Extending to the icosahedron, this is a surface with $60$ equivalent asymmetrical protein subunits. Not all viruses however contain only $60$ subunits. Some form larger structures whilst keeping the overall icosahedral symmetry. The theory of how the protein subunits can be arranged on the larger shells and also of the overall symmetry was given by Caspar and Klug in 1962 and is now known by the name of Quasi-Equivalence Theory.
[We] need to think of a way that the protein subunits can occupy quasi-equivalent positions on the viral capsid, i.e., the individual subunits can retain their basic bonding properties, but occupy slightly different environments. This is realised by the sub-triangulation of each face of the icosahedron into smaller facets, thus creating additional local quasi-equivalent six-fold vertices elsewhere on the viral capsid. The triangulation can be defined by the number, $T$, of facets the original face has been split into.
This triangulation can be performed in many ways and if the facets are allowed to bend, there is no geometric reason why facet edges must be congruent with face edges. However, face corners must coincide with facet corners. The reason for needing the vertices to be congruent is a simple one; when forming the icosahedron from the net, if the vertices were not congruent, there would be inconsistencies where the triangles don't match up.
The result of this triangulation gives rise to a special group of polyhedra with icosahedral symmetry - called icosadeltahedra by Caspar and Klug - which can be derived from a sheet of hexagons in which pentagons are inserted in place of certain hexagons according to selection rules described by $T$. There are (sic) only a certain number of ways to define this $T$:
Proposition [. . .]: Permissible triangulations are parameterised by $T = H^2 +HK +K^2$, where $H,K \in \Bbb N \cup \{0\}$.
Proof (summary):
By embedding an icosahedron net into a hexagonal lattice, we can think of two axes at an angle $\pi/3$, given by unit vectors $\hat{h}$ and $\hat{k}$. The main face size on the triangulated icosahedron can be determined by its edge length $S$. Thus each $T$-number allowed corresponds to a length $S$, considered as the length of the vector from the origin to $(H, K)$.
The infinite facet net has six-fold symmetry about the origin. Therefore, we only need to consider a sixth of the net.
Note that $S$ can be defined by the hypotenuse of the triangle with sides $H+\frac{K}{2}$ and $(\sqrt{3}/2)K$. Pythagoras' Theorem then gives
$$\begin{align} S^2&=\left(H+\frac{K}{2}\right)^2+((\sqrt{3}/2)K)^2\\ &=H^2+HK+K^2. \end{align}$$
The area of our large face is $(\sqrt{3}/4)S^2$; the size of our small triangle are unit length. Thus the triangulation number $T$ can be defined as the area of the large face over the area of the small triangle. Hence:
$$\begin{align} T&=\frac{(\sqrt{3}/4)S^2}{(\sqrt{3}/4)}\\ &=S^2. \end{align}$$
"$\square$"
Quoting the notes . . .
Note: All edge lengths which do not coincide with the $\pi/3$ sector edges or the line bisecting the two have a mirror image. This leads to two enantiomorphous tessellations of $T = 7, 13, 19, 21, \dots$ which are called the left-handed or leavo when $H > K > 0$, $(Tl)$, and right-handed or dextro when $K > H > 0$, $(Td)$.
[By Euler's formula and] following the rule that we place a protein subunit at each vertex of each triangle, and noting that we have each deltahedron has $20T$ facets, each with $3$ subunits, we can deduce that the allowed numbers of subunits for viral capsids are $60T = 60, 180, 240, 420,\dots$.
The notes then go on to tiling theory, with the caveat that not every virus follows C-K Theory. For example, there's rhomb tilings for MS2 and there's kite tilings for Polio.
The Question:
Does COVID-19 fit into the Caspar-Klug (Quasi-Equivalence) Theory for virus architecture?
Motivation:
Whilst, I suppose, the people in a position to do something about the virus at this level probably know this stuff already, it wouldn't hurt to make some of the theory I have learnt known to others.
Perhaps there's a particular tiling that COVID-19 exhibits outside of C-K Theory, too, which would be interesting in its own right.
Please help :)
The coronavirus picture that you see everywhere does not look to me like a regular tiling. I don't know how accurate it is, though.