The raspberries in three dimensions with at most two sizes of berry

In three dimensional Euclidean space, a raspberry is defined to be an arrangement of spheres with pairwise disjoint interiors, where all spheres are tangent to a central unit sphere and such that the contact graph of the non-central spheres triangulates the central sphere. We discuss the relevance of these structures in related work. We present a catalog of all configurations of radii that permit the formation of raspberries that have at most two sizes of non-central spheres. Throughout, we discuss the construction of this catalog.