Polynomial spline surfaces with rational linear transitions

We explore a class of polynomial tensor-product spline surfaces on 3-6 polyhedra, whose vertices have valence n = 3 or n = 6. This restriction makes it possible to exclusively use rational linear transition maps between the pieces: transitions between the bi-cubic tensor-product spline pieces are either C1 or they are G1 (tangent continuous) based on one single rational linear reparameterization. The simplicity of the transition functions yields simple formulas for a hierarchy of splines on subdivided 3-6 polyhedra.