The number of touching pairs of congruent sphere packings in Euclidean 3-space

@misc{strachan2026number,
  title = {{The number of touching pairs of congruent sphere packings in Euclidean 3-space}},
  author = {Cameron Strachan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31331},
  eprint = {2609.31331},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31331},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - The number of touching pairs of congruent sphere packings in Euclidean 3-space
AU  - Cameron Strachan
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31331
UR  - https://arxiv.org/abs/2609.31331
AB  - A packing of n congruent balls in 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 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√[3]2n^/23 when n=16k^3−33k^2+24k−6 where k∈ 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)≤ 6n−6/√[6]2n^/23 for all n, and determine the asymptotics of c_A(n) to be c_A(n)=6n−(1+o(1))6√[3]2n^/23.
ER  -