Layered circlepackings and the type problem

We study the geometric type of a surface packed with circles. For circles packed in concentric layers of uniform degree, the circlepacking is specified by this sequence of degrees. We write an infinite sum whose convergence discerns the geometric type: if layers of degree follow the th layer of degree , and the th layer of degree has circles, then converges/diverges as the circlepacking is hyperbolic/Euclidean. We illustrate a hyperbolic circlepacking with surprisingly few layers of degree .