{
  "id": "1804.06692",
  "version": "v1",
  "published": "2018-04-18T12:51:11.000Z",
  "updated": "2018-04-18T12:51:11.000Z",
  "title": "Platonic solids, Archimedean solids and semi-equivelar maps on the sphere",
  "authors": [
    "Basudeb Datta",
    "Dipendu Maity"
  ],
  "comment": "11 pages, 3 figure",
  "categories": [
    "math.CO",
    "math.GT"
  ],
  "abstract": "A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${\\rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. In particular, there are semi-equivelar maps on the torus, on the Klein bottle and on the surface of Euler characteristic $-1$ which are not vertex-transitive. Here we show that every semi-equivelar map on $\\mathbb{S}^2$ and $\\mathbb{RP}^2$ are vertex-transitive. In particular, we show that a semi-equivelar map on $\\mathbb{S}^2$ is the boundary of a Platonic solid, an Archimedean solid, the prism, a drum or an antiprism (Section 2).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-18T12:51:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C20",
      "52B70",
      "51M20",
      "57M60"
    ],
    "keywords": [
      "semi-equivelar map",
      "archimedean solid",
      "platonic solid",
      "vertex-transitive map",
      "automorphism group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}