Packing twelve spherical caps to maximize tangencies

The maximum number of non-overlapping unit spheres in R3R3 that can simultaneously touch another unit sphere is given by the kissing number, k(3)=12k(3)=12. Here, we present a proof that the maximum number of tangencies in any kissing configuration is 24 and that, up to isomorphism, there are only two configurations for which this maximum is achieved. The result is motivated by a three-dimensional crystallization problem.