{
  "id": "1609.06552",
  "version": "v1",
  "published": "2016-09-21T13:34:00.000Z",
  "updated": "2016-09-21T13:34:00.000Z",
  "title": "Distances from the vertices of a regular simplex",
  "authors": [
    "Mowaffaq Hajja",
    "Mostafa Hayajneh",
    "Bach Nguyen",
    "Shadi Shaqaqha"
  ],
  "categories": [
    "math.MG"
  ],
  "abstract": "If $S$ is a given regular $d$-simplex of edge length $a$ in the $d$-dimensional Euclidean space $\\mathcal{E}$, then the distances $t_1$, $\\cdots$, $t_{d+1}$ of an arbitrary point in $\\mathcal{E}$ to the vertices of $S$ are related by the elegant relation $$(d+1)\\left( a^4+t_1^4+\\cdots+t_{d+1}^4\\right)=\\left( a^2+t_1^2+\\cdots+t_{d+1}^2\\right)^2.$$ The purpose of this paper is to prove that this is essentially the only relation that exists among $t_1,\\cdots,t_{d+1}.$ The proof uses tools from analysis, algebra, and geometry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-21T13:34:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regular simplex",
      "dimensional euclidean space",
      "edge length",
      "arbitrary point",
      "elegant relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}