{
  "id": "2505.09505",
  "version": "v1",
  "published": "2025-05-14T15:53:50.000Z",
  "updated": "2025-05-14T15:53:50.000Z",
  "title": "Regular 3-polytopes of type $\\{n,n\\}$",
  "authors": [
    "Mingchao Li",
    "Wei-Juan Zhang"
  ],
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "For each integer \\( n \\geq 3 \\), we construct a self-dual regular 3-polytope \\( \\mathcal{P} \\) of type \\( \\{n, n\\} \\) with \\( 2^n n \\) flags, resolving two foundamental open questions on the existence of regular polytopes with certain Schl\\\"afli types. The automorphism group \\( \\operatorname{Aut}(\\mathcal{P}) \\) is explicitly realized as the semidirect product \\( \\mathbb{F}_2^{n-1} \\rtimes D_{2n} \\), where \\( D_{2n} \\) is the dihedral group of order \\( 2n \\), with a complete presentation for \\( \\operatorname{Aut}(\\mathcal{P}) \\) is provided. This advances the systematic construction of regular polytopes with prescribed symmetries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-14T15:53:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regular polytopes",
      "foundamental open questions",
      "dihedral group",
      "self-dual regular",
      "semidirect product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}