Cosmos

Viruses and fullerenes: Platonic solids in nature

Many viruses build their shell as an icosahedron, and the carbon molecule C60 is a football of 12 pentagons and 20 hexagons. It is named after Buckminster Fuller.

The Platonic solids were long regarded as mental constructs. But nature does use them, above all the icosahedron, the solid of twenty triangles that Plato assigned to water.

The shells of viruses

A virus has to pack its genetic material into a shell, and it has little genetic material of its own to spend on it. The most economical solution is to assemble many identical protein building blocks into a symmetrical shell. The most symmetrical closed shape that can be built this way is the icosahedron. Many viruses, such as those causing herpes, polio and the common cold, have such an icosahedral shell.

Donald Caspar and Aaron Klug described the building principle in 1962: the shell consists of groups of five and of six building blocks. There are always exactly twelve groups of five, at the twelve corners of the icosahedron, and between them a number of groups of six that depends on the size, namely 10 × (T − 1). The possible sizes follow the formula $T = h^2 + hk + k^2$, that is 1, 3, 4, 7, 9, 12, 13 … Klug received the Nobel Prize in Chemistry in 1982.

The football molecule

In 1985 Harold Kroto, Robert Curl and Richard Smalley discovered a molecule of exactly 60 carbon atoms, C60. The atoms sit at the corners of a truncated icosahedron, the shape of the classic football: twelve pentagons and twenty hexagons. The three received the Nobel Prize in Chemistry for it in 1996. They named the molecule buckminsterfullerene, after Buckminster Fuller, whose geodesic domes are built on the same principle (see Space filling without gaps).

C60with icosahedron

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The football molecule C60: 60 atoms, 90 bonds, 12 pentagons (red) and 20 hexagons. It arises when the twelve corners of an icosahedron are cut off.

Always twelve pentagons

Viruses and fullerenes have something in common: however large they are, they always need exactly twelve pentagons. This follows from Euler’s formula, according to which vertices − edges + faces = 2 for every convex solid. Assemble a shell of pentagons and hexagons with three faces meeting at each corner, and the calculation necessarily gives twelve pentagons, while the number of hexagons is free. Hexagons alone give a flat surface, like a honeycomb or a sheet of graphite. Only the twelve pentagons curve it into a ball.

The calculation is short enough to show here. Let a shell have $F$ pentagons and $S$ hexagons, with three faces meeting at every corner. Each edge belongs to two faces, each corner to three. So there are $(5F+6S)/2$ edges and $(5F+6S)/3$ corners. Put this into Euler’s formula and multiply by 6, and the hexagons drop out:

$$2(5F+6S) - 3(5F+6S) + 6(F+S) = 12 \quad\Longrightarrow\quad F = 12$$

How many hexagons there are the formula does not say, but it does say that there must always be exactly twelve pentagons.

This fits an image that appears often on this website: hexagons fill the plane, the unlimited. Five breaks this order, and precisely by doing so it closes the form into a solid, the limited (see The ancient sources). The dodecahedron, the solid Plato gave to the whole, also has twelve pentagons.

Not everything is an icosahedron

But nature does not always use the Platonic solid. The tobacco mosaic virus, the first virus ever discovered, is a long rod in which the building blocks are wound helically around the genetic material. In the coronavirus too the genetic material lies in a helical shell, which in turn sits inside a soft envelope of fat. Form follows function, and the icosahedron is the best solution only for one particular task: building a closed capsule with little blueprint.

And the football molecule exists not only in the laboratory. In 2010 the Spitzer space telescope found C60 in the cloud of gas around a dying star, thousands of light years away. The figure of twelve pentagons and twenty hexagons thus arises even where no human builds it.

Sources and further reading

Sources

• Donald L. D. Caspar and Aaron Klug, Physical principles in the construction of regular viruses, Cold Spring Harbor Symposia on Quantitative Biology 27, 1962
• Harold W. Kroto et al., C60: Buckminsterfullerene, Nature 318, 1985
• Ernst Haeckel, Art Forms in Nature, Leipzig 1899 to 1904
• Jan Cami et al., Detection of C60 and C70 in a Young Planetary Nebula, Science 329, 2010