{
  "id": "2002.07460",
  "version": "v1",
  "published": "2020-02-18T10:00:49.000Z",
  "updated": "2020-02-18T10:00:49.000Z",
  "title": "Similarity Isometries of Point Packings",
  "authors": [
    "Jeanine Concepcion H. Arias",
    "Manuel Joseph C. Loquias"
  ],
  "comment": "13 pages, 6 figures",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "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. In particular, similarity isometries of a certain shifted hexagonal packing will be computed and compared with that of the hexagonal packing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-18T10:00:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C07",
      "52C05",
      "11H06",
      "82D25"
    ],
    "keywords": [
      "similarity isometry",
      "point packing",
      "hexagonal packing",
      "finite index",
      "similar sublattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}