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

TL;DR

This paper explores the arithmetic and geometric properties of the root lattice A4 and its dual, focusing on their similar sublattices and coincidence rotations, with applications to Penrose tilings and lattice theory.

Contribution

It introduces parametrizations of sublattices and rotations of A4 using the icosian ring, leading to explicit index formulas and generating functions for counting these structures.

Findings

Derived index formulas for sublattices and rotations.

Established Dirichlet series generating functions.

Provided enumeration results for each index.

Abstract

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.