Triangulation Number of Icosahedral Capsids: Formula Variable Definitions

I'm trying to understand the definition of the Triangulation Number for Icosahedral Virus Capsids: In Alan J. Cann's "Principles of Molecular Virology 5th Edition", it is defined as

$$T=f^2\times P$$ where $$f$$ is the number of subdivisions of each side of the triangular face, and $$f^2$$ is the number of subtriangles on each face; $$P=h^2+hk+k^2$$, where $$h$$ and $$k$$ are any distinct, nonnegative integers

I have also seen the Triangulation Number defined as

The number of structural units that creates each [triangular] side

in Chapter 2 of Jennifer Louten's "Essential Human Virology". I have redrawn part of a diagram I am confused about:

Only one face shows the relevant subdivision, with sides drawn in yellow. Blue marks edge subdivision points, and red marks triangular face subdivisions. To my eye, the $$T=1$$ icosahedron has 1 edge subdivision per side, again marked by the blue dots. If $$h$$ and $$k$$ are $$1$$ and $$0$$, then it makes sense as to how that icosahedron has $$T=1$$, and similarly there is only one shape repeated 3 times comprising the triangular face, so Louten's stipulation makes sense to me so far (I don't understand where the 1 subtriangle for $$f^2$$ is, maybe its considering the whole face as a subtriangle). For the $$T=3$$ icosahedron though, it appears to me that there are two subdivisions per edge, and there's no $$h$$ and $$k$$ such that $$3=4\times P$$. Further, in "Principles of Molecular Virology", the $$h$$ and $$k$$ for $$T=3$$ are listed as $$(1, 1)$$, unless I'm misreading; How are those values distinct per the original definition?

My raw questions coalesce to this list:

• What is being counted in the "number of subdivisions of each side of the triangular face"?
• What does "distinct" mean when referring to $$h$$ and $$k$$?
• How is a "structural subunit" defined? For example, in "Essential Human Virology" Figure 2.8, why is the (A) $$T=1$$ tiling considered a single structural subunit while the $$T=4$$ tiling is not? The distinction of what is and is not a "structural subunit" seems nebulous to me. I could see it being the largest average number of non-overlapping triangular units excluding non-triangular subfaces, but I can't tell if that's just me grasping at straws
• I don't have Cann's virology on hand here, can you provide their figure that shows subdivisions? Or another redraw for that one? I suspect their use of the the terms "subunit" and "subdivision" is at least slightly different from Louten's. Aug 28, 2023 at 14:47
• The above diagrams are from Cann's, I can redraw the other capsid on the diagram too if that's what you're referring to (I think its T=4), in the originals the subdivisions are drawn on every triangular face but in my redraw I only drew the subdivision for the face facing the observer, to reduce visual clutter and because I am lazy; They repeat on each face in Cann's. I'd post the diagram itself but I'm worried about copyright. I can do another redraw once I'm back home at my drawing tablet, I just want to ensure I'm including the parts you're mentioning. Aug 28, 2023 at 18:18

I'm not familiar with the $$T = f^2 * P$$ formulation. In Caspar + Klug (1962) $$T$$ itself is equal to $$h^2+h*k+k^2$$, so I guess $$f$$ is assumed to always =1. So I don't really know the answer to your first two points.
As for the last point, "structural subunit" basically means an individual copy of the protein that makes up the viral shell. Icosahedral symmetry means there must be 60 identical "asymmetric units", regardless of the $$T$$ number. In these diagrams its usually presented as the 20 faces of an icosahedron, which are equilateral triangles and therefore can be divided into 3 identical parts, so $$3 * 20 = 60$$. (I find it more helpful to focus on the 12 vertices, where 5 faces come together, so $$12 * 5 = 60$$) The $$T$$ number for a particular icosahedral virus tells you how many copies of its protein are needed for its asymmetric unit. So in fig. 2.8 of Essential Human Virology, the structural subunits are the trapezoids, not the tiling of three of them in a face.
One of the important results from this theory is that only certain T numbers can exist. For example, you can't make an icosahedral shell out of 120 subunits ($$T=2$$).
• Thanks for your answer! Could you wager a guess as to why the T=3 Cann diagram splits the face into 4 unequal parts instead of 3 identical parts then? And defined like that can't you just take the average number of proteins on a face and divide it by 3 to get the triangulation number instead of the whole $T=f^2×P$ formulation? Aug 28, 2023 at 19:52