Similar sublattices and coincidence rotations of the root lattice A ₄ and its dual

Abstract

Abstract.A natural way to describe the Penrose tiling employs the projection method on the basis of the root lattice A 4 or its dual. Properties of these lattices are thus related to properties of the Penrose tiling. Moreover, the root lattice A 4 appears in various other contexts such as sphere packings, efficient coding schemes and lattice quantizers. Here, the lattice A 4 is considered within the icosian ring, whose rich arithmetic structure leads to parametrisations of the similar sublattices and the coincidence rotations of A 4 and its dual lattice. These parametrisations, both in terms of a single icosian, imply an index formula for the corresponding sublattices. The results are encapsulated in Dirichlet series generating functions. For every index, they provide the number of distinct similar sublattices as well as the number of coincidence rotations of A 4 and its dual.

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