{
  "id": "2408.07112",
  "version": "v1",
  "published": "2024-08-13T15:22:40.000Z",
  "updated": "2024-08-13T15:22:40.000Z",
  "title": "A generalization of the hexastix arrangement to higher dimensions",
  "authors": [
    "Jan Kristian Haugland"
  ],
  "comment": "5 pages",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "abstract": "Hexastix is an arrangement of non-overlapping infinite hexagonal prisms in four different directions that cover $\\frac{3}{4}$ of space. We consider a possible generalization to $n$ dimensions, based on the permutohedral lattice $A^*_n$. The central lines of the generalized prisms are going to be oriented in $n+1$ different directions (parallel to the shortest non-zero vectors of $A^*_n$). The projection of the lines oriented in any direction along that direction to a hyperplane perpendicular to it is required to be a translation of the corresponding projection of $A^*_n$, and the minimal distance between lines oriented in any two given directions should be maximal. It is shown that this is possible if $n$ is a prime power. Also, the proportion of $n$-space that is covered is calculated for $n \\in \\{4, 5\\}$, and an alternative generalization is briefly considered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-13T15:22:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B40"
    ],
    "keywords": [
      "higher dimensions",
      "hexastix arrangement",
      "generalization",
      "non-overlapping infinite hexagonal prisms",
      "shortest non-zero vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}