  We study of the relation between the geometry of sets in complex hyperbolic
space and Hilbert spaces with complete Pick kernels. We focus on the geometry
associated with assembling sets into larger sets and of assembling Hilbert
spaces into larger spaces. Model questions include describing the possible
triangular faces of a tetrahedron in hyperbolic space and describing the three
dimensional subspaces of four dimensional Hilbert spaces with Pick kernels. Our
novel technical tool is a complex analog of the cosine of a vertex angle.
