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Layered circlepackings and the type problem


Author: Ryan Siders
Journal: Proc. Amer. Math. Soc. 126 (1998), 3071-3074
MSC (1991): Primary 52C15
DOI: https://doi.org/10.1090/S0002-9939-98-04472-4
MathSciNet review: 1459150
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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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Additional Information

Ryan Siders
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: rcsiders@math.princeton.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04472-4
Keywords: Circlepacking, electric network
Received by editor(s): November 28, 1995
Received by editor(s) in revised form: February 28, 1997
Additional Notes: This work was done under Dr. Phil Bowers of Florida State University during FSU’s 1994 Research Experience for Undergraduates. Dr. Bowers was an inspiring mentor. I will treasure what I learned from our conversations. The program was sponsored by the NSF
Communicated by: James West
Article copyright: © Copyright 1998 American Mathematical Society

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