An alternative construction of generalised non-crystallographic symmetry

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 a previous paper we showed 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. In this paper we combine for the first time two known concepts (well-known lattices and a projection linking these to non-crystallographic symmetries) to arrive at an alternative derivation of this class of generalised symmetries.