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A natural way to describe the Penrose tiling employs the projection method onthe basis of the root lattice A4 or its dual. Properties of these lattices arethus related to properties of the Penrose tiling. Moreover, the root lattice A4appears in various other contexts such as sphere packings, efficient codingschemes and lattice quantizers.  Here, the lattice A4 is considered within the icosian ring, whose richarithmetic structure leads to parametrisations of the similar sublattices andthe coincidence rotations of A4 and its dual lattice. These parametrisations,both in terms of a single icosian, imply an index formula for the correspondingsublattices. The results are encapsulated in Dirichlet series generatingfunctions. For every index, they provide the number of distinct similarsublattices as well as the number of coincidence rotations of A4 and its dual.

Similar Sublattices and Coincidence Rotations of the Root Lattice A4 and its Dual