  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.
