Selected Proofs on Sphere Packings

Abstract

Let B denote the unit ball centered at the origin o of $\mathbb{E}^3$ and let $\mathcal{P} : = {{\rm c}_1 + {\bf B}, {\rm c}_2 + {\bf c}, \cdots, {\rm c}_n + {\bf B}}$ denote the packing of n unit balls with centers c₁; c₂;… c_n in $\mathbb{E}^3$ having the largest number C(n) of touching pairs among all packings of n unit balls in $\mathbb{E}^3$. ($\mathcal{P}$ might not be uniquely determined up to congruence in which case $\mathcal{P}$ stands for any of those extremal packings.) First, observe that Theorem 1.4.1 and Theorem 2.4.3 imply the following inequality in a straightforward way.