{
  "id": "math/0207211",
  "version": "v1",
  "published": "2002-07-23T16:07:49.000Z",
  "updated": "2002-07-23T16:07:49.000Z",
  "title": "McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions",
  "authors": [
    "R. H. Hardin",
    "N. J. A. Sloane"
  ],
  "comment": "16 pages, 1 figure",
  "journal": "Discrete Computational Geometry, 15 (1996), 429-441",
  "categories": [
    "math.CO"
  ],
  "abstract": "Evidence is presented to suggest that, in three dimensions, spherical 6-designs with N points exist for N=24, 26, >= 28; 7-designs for N=24, 30, 32, 34, >= 36; 8-designs for N=36, 40, 42, >= 44; 9-designs for N=48, 50, 52, >= 54; 10-designs for N=60, 62, >= 64; 11-designs for N=70, 72, >= 74; and 12-designs for N=84, >= 86. The existence of some of these designs is established analytically, while others are given by very accurate numerical coordinates. The 24-point 7-design was first found by McLaren in 1963, and -- although not identified as such by McLaren -- consists of the vertices of an \"improved\" snub cube, obtained from Archimedes' regular snub cube (which is only a 3-design) by slightly shrinking each square face and expanding each triangular face. 5-designs with 23 and 25 points are presented which, taken together with earlier work of Reznick, show that 5-designs exist for N=12, 16, 18, 20, >= 22. It is conjectured, albeit with decreasing confidence for t >= 9, that these lists of t-designs are complete and that no others exist. One of the constructions gives a sequence of putative spherical t-designs with N= 12m points (m >= 2) where N = t^2/2 (1+o(1)) as t -> infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-07-23T16:07:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B11",
      "05B30"
    ],
    "keywords": [
      "spherical designs",
      "dimensions",
      "regular snub cube",
      "12m points",
      "accurate numerical coordinates"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......7211H"
    }
  }
}