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VIRUS shells, atomic nuclei and cage-like carbon molecules may all be expressions of the same simple geometric principle. The finding might help explain why these structures choose the shapes they do.

Viruses and large carbon molecules generally adopt highly symmetric shapes, such as an icosahedron or the football shape of 60-carbon buckyballs. Similar shapes also crop up in a description of atomic nuclei known as the skyrmion model.

If viruses and atoms take on these shapes, it seems intuitive that they must in some sense be the most efficient ones available, and require the least energy. But previous attempts to work out why this should be have got bogged down in complicated equations. Now mathematicians Paul Sutcliffe of the University of Kent at Canterbury and Michael Atiyah of the University of Edinburgh have found a single description that works for all these shapes.

Atiyah and Sutcliffe made their discovery while studying something completely different: the quantum behaviour of a group of particles. They took a collection of particles in three-dimensional space, then used the directions of the lines connecting the particles to create a new collection of points in a much higher-dimensional space. For example, if they started with 15 particles, they ended up with 15 points in a 15-dimensional space.

They then calculated the volume that the points filled in this higher-dimensional space. They also devised an equation, or 鈥渆nergy function鈥�, that describes how much energy each configuration of points has. According to this energy function, the greater the volume that is encompassed by the points, the less energy the configuration has.

Although the researchers were initially interested in the maximum values of the energy, out of pure curiosity they wondered which configurations of points would minimise the energy. To their surprise, the shapes of carbon molecules and viruses emerged from their calculation. 鈥淭here is a simple geometry behind these shapes, but it鈥檚 hidden in higher dimensions,鈥� Sutcliffe says.

Richard Battye, a physicist at the University of Manchester who studies skyrmions, warns that the discovery of this simple geometry doesn鈥檛 necessarily explain the shapes of carbon molecules, skyrmions and virus particles in a fundamental way. It could simply demonstrate another connection between them, he cautions. But it 鈥渄oes suggest that the principle underlying them is the same鈥�, he says.

Sutcliffe hopes the result might help researchers understand how the pieces of a growing viral shell form in such a way that they eventually meet each other and close up into a sphere. It could also help explain why, out of the large family of possible shapes of carbon molecules, only a few seem to appear in nature. Writing down an energy function to tell you why you get buckyballs and not other configurations can be heavy going if you use chemistry alone, he says. 鈥淏ut using this simple geometry you get buckyballs instantly.鈥�