Contact numbers for congruent sphere packings in Euclidean 3-space

TL;DR

This paper investigates the maximum number of contacts in finite and lattice-based sphere packings in three-dimensional Euclidean space, combining analytic and combinatorial methods to derive bounds and estimates.

Contribution

It introduces new bounds for contact numbers in finite and lattice sphere packings, extending previous work with combined analytic and combinatorial approaches.

Findings

Derived bounds for maximum contact numbers in finite packings

Estimated contact numbers for lattice-based packings

Extended previous results on sphere packing contact graphs

Abstract

Continuing the investigations of Harborth (1974) and the author (2002) we study the following two rather basic problems on sphere packings. Recall that the contact graph of an arbitrary finite packing of unit balls (i.e., of an arbitrary finite family of non-overlapping unit balls) in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have in Euclidean 3-space. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. Finally, we are interested also in the following more special version of the above problem. Namely, let us imagine that we are given a lattice unit sphere packing with the center points forming the lattice L in Euclidean 3-space (and with certain pairs of unit balls touching each other) and then let us generate packings of n unit balls such that each and every center of the n unit balls is chosen from L. Just as in the general case we are interested in finding good estimates for the largest contact number of the packings of n unit balls obtained in this way.