A new link between the geometries of three and four dimensions

What is it about?

The Platonic solids are the regular bodies in three dimensions, such as the cube and the icosahedron, and have been known for millennia; they feature prominently in the natural world wherever geometry and symmetry are important, for instance in lattices and quasi-crystals, as well as fullerenes and viruses. The Platonic solids have six counterparts in four dimensions, of which five have very strange symmetries, and three are `exceptional', as there are only three in any dimension higher than 4.

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Why is it important?

This paper shows how these analogues of the Platonic solids in four dimensions can be constructed from the Platonic solids alone. The rotations of the Platonic solids (spinors) can be naturally interpreted as 4-dimensional polytopes, and in turn generate symmetry (Coxeter) groups in four dimensions. This spinorial construction explains all `exceptional objects' in 4D as well as their mysterious symmetries convincingly for the first time, despite many great mathematicians' efforts.