Layered circlepackings and the type problem
Ryan Siders · Proceedings of the American Mathematical Society · 1998
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 h i h_i layers of degree 6 6 follow the i i th layer of degree 7 7 , and the i i th layer of degree 7 7 has c i c_i circles, then ∑ log ( h i ) / c i \sum \log (h_i)/c_i converges/diverges as the circlepacking is hyperbolic/Euclidean. We illustrate a hyperbolic circlepacking with surprisingly few layers of degree > 6 >6 .