{
"id": "1803.04948",
"version": "v1",
"published": "2018-03-13T17:31:32.000Z",
"updated": "2018-03-13T17:31:32.000Z",
"title": "Hyperball packings related to octahedron and cube tilings in hyperbolic space",
"authors": [
"JenÅ‘ Szirmai"
],
"comment": "33 pages, 12 figures. arXiv admin note: text overlap with arXiv:1510.03208, arXiv:1505.03338",
"categories": [
"math.MG"
],
"abstract": "In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular octahedron and cube tilings. These are derived from the Coxeter simplex tilings $\\{p,3,4\\}$ $(7\\le p \\in \\mathbb{N})$ and $\\{p,4,3\\}$ $(5\\le p \\in \\mathbb{N})$ in $3$-dimensional hyperbolic space $\\mathbb{H}^3$. We determine the densest hyperball packing arrangement and its density with congruent and non-congruent hyperballs related to the above tilings in $\\mathbb{H}^3$. We prove that the locally densest congruent or non-congruent hyperball configuration belongs to the regular truncated cube with density $\\approx 0.86145$. This is larger than the B\\\"or\\\"oczky-Florian density upper bound for balls and horoballs. Our locally optimal non-congruent hyperball packing configuration cannot be extended to the entire hyperbolic space $\\mathbb{H}^3$, but we determine the extendable densest non-congruent hyperball packing arrangement related to a regular cube tiling with density $\\approx 0.84931$.",
"revisions": [
{
"version": "v1",
"updated": "2018-03-13T17:31:32.000Z"
}
],
"analyses": {
"subjects": [
"52C17",
"52C22",
"52B15"
],
"keywords": [
"hyperbolic space",
"cube tiling",
"non-congruent hyperball packing configuration",
"optimal non-congruent hyperball packing",
"octahedron"
],
"note": {
"typesetting": "TeX",
"pages": 33,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}