Similarity Isometries of Point Packings

Abstract

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 by studying similarity isometries of shifted point packings. In particular, similarity isometries of a certain shifted hexagonal packing will be computed and compared with that of the hexagonal packing.

16 pages; 7 figures; Theorem 3.1, Corollary 3.5 and Proposition 3.6 in original version were incorrect