CapsidNet

Many viruses keep their genes in a shell made of many copies of one protein. The shell is a closed ball of flat triangles, like a geodesic dome. How many proteins does it take?

Toy model A simplified picture to show one idea. Not a measurement, and not a prediction. Of a virus shell: flat triangles on paper, where the real shell is many folded copies of one protein.

Pick a shell, then predict

On a sheet of triangles, take h steps one way, turn 60°, and take k steps. Where you land is the next five-fold corner of the shell.

Pick an h and a k.

The rule

Caspar and Klug (1962): those two numbers set the triangulation number T = h² + hk + k². The shell then needs 60 × T copies of the protein, arranged as 12 groups of five (the corners, always twelve) and 10 × (T − 1) groups of six.

h = 1, k = 0: T = 1, 60 proteins (12 fives, 0 sixes)
h = 1, k = 1: T = 3, 180 proteins (12 fives, 20 sixes)
h = 2, k = 0: T = 4, 240 proteins (12 fives, 30 sixes)
h = 2, k = 1: T = 7, 420 proteins (12 fives, 60 sixes)

Real shells: satellite tobacco necrosis virus is a T = 1 shell (60 copies); the hepatitis B virus core is mostly T = 4 (240 copies), with some T = 3 shells; the bacteriophage HK97 head is T = 7 (420 copies).

Print, cut, fold: the T = 1 shell

Twenty triangles. Cut round the outside, crease every inner line, and tape the edges into a ball. Each triangle holds three copies of the protein: 20 × 3 = 60.

Honest notes

Some shells break the rule: the polyomavirus shell sits on a T = 7 lattice but is built only from groups of five, and the HIV capsid is a cone, not a ball.
These shells are known from experiments on whole assemblies (crystallography and cryo-EM). An AlphaFold DB entry is one chain on its own and does not show the shell.
More paper models: PDB-101's paper models, including icosahedral viruses and the HIV capsid.