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

A natural way to describe the Penrose tiling employs the projection method on the basis of the root lattice A4 or its dual. Properties of these lattices are thus related to properties of the Penrose tiling. Moreover, the root lattice A4 appears in various other contexts such as sphere packings, efficient coding schemes and lattice quantizers. Here, the lattice A4 is considered within the icosian ring, whose rich arithmetic structure leads to parametrisations of the similar sublattices and the coincidence rotations of A4 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 A4 and its dual.

💡 Research Summary

The paper investigates the similar sublattices (SSLs) and coincidence rotations (CRs) of the four‑dimensional root lattice A₄ and its dual A₄* by exploiting the arithmetic of the icosian ring, a non‑commutative quaternionic order that encodes the icosahedral symmetry. The authors first embed A₄ (and A₄*) as a left ideal of the icosian ring ℍ_I. Every non‑zero icosian q defines a scaled copy of the lattice through the map L ↦ q L q̄ (and similarly for the dual). The index of the resulting sublattice is |N(q)|², where N(q) is the quaternionic norm lying in the ring of integers ℤ