{
  "id": "math/0307171",
  "version": "v1",
  "published": "2003-07-11T16:24:55.000Z",
  "updated": "2003-07-11T16:24:55.000Z",
  "title": "Once more about the 52 four-dimensional parallelotopes",
  "authors": [
    "Michel Deza",
    "Viacheslav Grishukhin"
  ],
  "comment": "14 pages (submitted)",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "abstract": "There are several works \\cite{De} (and \\cite{St}), \\cite{En}, \\cite{Co} and \\cite{Va} enumerating four-dimensional parallelotopes. In this work we give a new enumeration showing that any four-dimensional parallelotope is either a zonotope or the Minkowski sum of a zonotope with the regular 24-cell $\\{3,4,3\\}$. Each zonotopal parallelotope is the Minkowski sum of segments whose generating vectors form a unimodular system. There are exactly 17 four-dimensional unimodular systems. Hence, there are 17 four-dimensional zonotopal parallelotopes. Other 35 four-dimensional parallelotopes are: the regular 24-cell $\\{3,4,3\\}$ and 34 sums of the regular parallelotope with non-zero zonotopal parallelotopes. For the nontrivial enumerating of the 34 sums we use a theorem discribing necessary and sufficient conditions when the Minkowski sum of a parallelotope with a segment is a parallelotope.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-07-11T16:24:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minkowski sum",
      "non-zero zonotopal parallelotopes",
      "four-dimensional zonotopal parallelotopes",
      "four-dimensional unimodular systems",
      "theorem discribing necessary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......7171D"
    }
  }
}