Generalised symmetries for viruses, fullerenes and quasicrystals

What is it about?

Symmetry occurs widely in nature, such as in salt crystals and snowflakes. The polyhedral symmetries of these two objects (cubic and hexagonal) are special in that they can be extended to a crystal, or lattice. This is not the case for many other interesting naturally occurring polyhedral objects such as viruses, fullerenes (football-shaped carbon molecules) and quasicrystals. These symmetries are used to describe individual `blobs' rather than crystals of identical objects.

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Why is it important?

However, in this paper we show that the same mathematical process of affinisation which yields the crystals in the former cases can still be applied to the non-crystallographic cases: the affine symmetry then relates different features within the same object, e.g. the outer capsid shell of a virus to its inner nucleic acid organisation, or carbon cages of different size within a carbon onion. We are the first to construct these generalisations and classify these in a mathematical framework.