Layered circlepackings and the type problem

Siders, Ryan

doi:10.1090/S0002-9939-98-04472-4

Layered circlepackings and the type problem
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Proc. Amer. Math. Soc. 126 (1998), 3071-3074 Request permission

Abstract:

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 $h_i$ layers of degree $6$ follow the $i$th layer of degree $7$, and the $i$th layer of degree $7$ has $c_i$ circles, then $\sum \log (h_i)/c_i$ converges/diverges as the circlepacking is hyperbolic/Euclidean. We illustrate a hyperbolic circlepacking with surprisingly few layers of degree $>6$.

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