Coincidences of a Shifted Hexagonal Lattice and the Hexagonal Packing

Arias, Jeanine Concepcion H.; Gabinete, Evelyn D.; Loquias, Manuel Joseph C.

doi:10.12693/aphyspola.126.516

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Shifted Point Packings1

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2020

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“…This observation leads us to investigate translates of point packings, which will be referred to as shifted point packings, and their similarity isometries. The same idea was employed by Arias et al (2014) to find the coincidence isometries of point packings about points different from the origin.…”

Section: Shifted Point Packingsmentioning

confidence: 99%

Similarity isometries of point packings

Arias

Loquias

2020

Acta Cryst Sect A

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A linear isometry R of {\bb R}^{d} is called a similarity isometry of a lattice \Gamma\subseteq{\bb R}^{d} 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.

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Section: Shifted Point Packingsmentioning

confidence: 99%