Similarity Isometries of Point Packings

Jeanine Concepcion H. Arias, Manuel Joseph C. Loquias

Published 2020-02-18Version 1

A linear isometry $R$ of $\mathbb{R}^d$ is called a similarity isometry of a lattice $\Gamma \subseteq \mathbb{R}^d$ if there exists a positive real number $\beta$ such that $\beta R\Gamma$ is a sublattice of (finite index in) $\Gamma$. The set $\beta R\Gamma$ is referred to as a similar sublattice of $\Gamma$. A (crystallographic) point packing generated by a lattice $\Gamma$ is a union of $\Gamma$ with finitely many shifted copies of $\Gamma$. In this study, the notion of similarity isometries is extended to point packings. We provide a characterization for the similarity isometries of point packings and identify the corresponding similar subpackings. Planar examples will be discussed, namely, the $1 \times 2$ rectangular lattice and the hexagonal packing (or honeycomb lattice). Finally, we also consider similarity isometries of point packings about points different from the origin. In particular, similarity isometries of a certain shifted hexagonal packing will be computed and compared with that of the hexagonal packing.