Last week I was at a conference on mathematical biology in Durham. I will get to the biology in a bit, but first the mathematics, in particular geometry. You can make precisely five (not four or six) different regular solid shapes using only regular polygons, and where at every corner of the shape the same number of these polygons meet. The cube is probably the most familiar one. This is made from six squares (a square is a four-sided polygon), and at each corner, three of these squares meet. There are exactly four other shapes, one of which is the icosahedron shown to the left. The icosahedron is made of triangles (three-sided polygons), not squares, and here at each corner five of these triangles meet. The icosahedron has 20 of these triangles.

I find it somehow continually surprising, that there are precisely five and only five such shapes possible. It just seems a bit weird to me. Even more weirdly, we have known that you can only make five of these solid shapes since the time of the ancient Greek philosopher Plato (born about 427 BC). These five shapes are the most symmetric, and so many people think, the most beautiful. They are called the Five Platonic Solids, because they are mentioned in works by Plato.

That was the maths, now for the biology. One of the Platonic Solids, the icosahedron, is very important in the study of viruses (the sort that infect us, and other animals and plants, not the computer sort). The reason is that many viruses have the shape of an icosahedron.

Viruses are the simplest form of life, many viruses consist only of their genetic material encapsulated inside a protein shell. The shell is called a capsid and many of them are beautifully symmetric, and have the symmetry of an icosahedron. You can see this in the electron micrographs of a virus called adenovirus

In the middle there is an illustration of an icosahedron, but on the left and right are actual (electron microscopy) images of the viruses. The icosahedral shape of the viruses is pretty clear, you should be able to see a clear triangle in the left-hand image of the virus. As the virus has the shape of an icosahedron, it has triangular facets like an icosahedron.

I don’t know when the first icosahedral viruses evolved, presumably billions of years ago, so while we may have have understood the maths of icosahedra for two and half thousand years, evolution appears to have taken advantage of icosahedra for a lot longer. We often think of life as messy and imprecise, in contrast to our precision technology, but this is often not the case. As you can see, even the simplest form of life, viruses, can demonstrate great geometrical precision.