A linear isometry R of is called a similarity isometry of a lattice if there exists a positive real number β such that βRΓ is a sublattice of (finite index in) Γ. The set βRΓ is referred to as a similar sublattice of Γ. A (crystallographic) point packing generated by a lattice Γ is a union of Γ with finitely many shifted copies of Γ. In this study, the notion of similarity isometries is extended to point packings. A characterization for the similarity isometries of point packings is provided and the corresponding similar subpackings are identified. Planar examples are discussed, namely the 1 × 2 rectangular lattice and the hexagonal packing (or honeycomb lattice). Finally, similarity isometries of point packings about points different from the origin are considered by studying similarity isometries of shifted point packings. In particular, similarity isometries of a certain shifted hexagonal packing are computed and compared with those of the hexagonal packing.

中文翻译：

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点填料的相似性等距

线性等距ř的被称为晶格的相似性等距如果存在β这样的正实数β - [R Γ是（在有限指数）Γ的亚晶格。这组β [RΓ称为Γ的相似子格。晶格Γ产生的（晶体学）点堆积是Γ与Γ的有限多个移位副本的并集。在这项研究中，相似性等距的概念扩展到点堆积。提供了点填充的相似性等距的表征，并标识了相应的相似子填充。讨论了平面示例，即1×2矩形格子和六边形堆积（或蜂窝格子）。最后，通过研究位移点堆积的相似性，考虑了关于与原点不同的点的点堆积的相似性。特别地，计算出一定位移的六边形填充物的相似性等距并将其与六边形填充物的相似性进行比较。