{
  "abstract": "A packing of $n$ congruent balls in $\\mathbb{R}^3$ is a family of interior-disjoint Euclidean balls all having the same radius. The contact number of a packing is the number of touching pairs of balls. In this paper we investigate the problem of determining the maximum contact number, $c(n)$, of a packing of $n$ congruent balls in $\\mathbb{R}^3$. We first show that all packings of $n$ congruent balls that have a contact number of $c(n)$ are minimally rigid. Furthermore, we show that $c(n)=3n-6$ for $n=6,7,8,$ and $9$. These two results resolve a conjecture of K. Bezdek and Khan. During the proof of the latter result, we also enumerate the contact structures of all packings of $n$ congruent balls with contact number $c(n)$ for $n=6,7,$ and $8$. Additionally, we provide a lower bound construction which shows $c(n)> 6n-6\\sqrt[3]{2}n^\\frac{2}{3}$ when $n=16k^3-33k^2+24k-6$ where $k\\in \\mathbb{N}$. We also look at the restricted problem where each ball is centered on the face-centered cubic lattice $A_3$. In this case let $c_{A}(n)$ denote the maximum contact number. We show that $c_{A}(n)\\leq 6n-\\frac{6}{\\sqrt[6]{2}}n^\\frac{2}{3}$ for all $n$, and determine the asymptotics of $c_{A}(n)$ to be $c_{A}(n)=6n-(1+o(1))6\\sqrt[3]{2}n^\\frac{2}{3}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31331",
  "author_total": 1,
  "authors": [
    {
      "name": "Cameron Strachan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eyze81dq2hmjw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31331",
  "fulltext_url": null,
  "id": "eyze81dq2hmjw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31331",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214659895",
  "pdf_url": "https://arxiv.org/pdf/2609.31331",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eyze81dq2hmjw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31331",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31331"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The number of touching pairs of congruent sphere packings in Euclidean 3-space",
  "title_text": "The number of touching pairs of congruent sphere packings in Euclidean 3-space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/eyze81dq2hmjw",
  "venue": "arXiv",
  "versions": []
}