The Kissing Number Problem in Three Dimensions: Geometric and Linear Programming Methods

The kissing number problem asks for the largest number of congruent, nonoverlapping spheres that can touch a fixed sphere of the same radius. This literature review studies the three-dimensional case and asks how geometric and optimization-based methods establish the equality k3=12. The analysis first converts a kissing configuration into a spherical code with minimum angular separation 60. It then distinguishes the lower-bound construction supplied by the face-centered cubic lattice from the substantially harder upper-bound problem. The geometric approach is summarized through spherical caps, contact graphs, and the Schütte-van der Waerden exclusion of a thirteen-point code. The linear-programming approach is developed from Schoenberg positivity, the Delsarte bound, and Musin’s degree-nine auxiliary polynomial together with its required local spherical analysis. The comparison shows that geometric proofs expose the low-dimensional structure of extremal configurations, whereas linear and semidefinite programming provide a systematic framework that extends more effectively across dimensions. The principal finding is that neither packing density nor an explicit twelve-sphere construction alone proves optimality; the decisive step is an upper-bound that combines global algebraic positivity with local geometric constraints.