Voronoi and Delone cells of root lattices

What is it about?

Symmetries of the Root lattices An and Dn are described by the affine Coxeter groups. Their Voronoi and Delone cells are interesting as they tile the n- dimensional space. They are polytopes with interesting point symmetries and their 2-dimensional faces turn out to be interesting. When 2-dimensional faces are projected onto a plane they reproduce the prototiles of the aperiodic tilings of the plane which are useful for the classification of the quasicrystals. It was proved that the 2-dimensional faces of the Voronoi cell of An lattice are identical rhombuses. Similarly, the 2-dimensional faces of the Voronoi cell of Dn lattice are isosceles triangles. 2-dimensional faces of the Delone cells of both lattices are equilateral triangles. Cassification of the Delone and Voronoi polytopes of An and Dn lattices are presented. Examples of the aperiodic tilings obtained by projections are illustrated.

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Why is it important?

It turns out to be useful in quasicrystallography when they are projected in lower dimensions.