Domain-filling circle packings

  • David Krieg
  • Elias WegertEmail author
Original Paper


According to a classical theorem of Constantin Carathéodory, any Jordan domain G admits a unique conformal mapping onto the unit disk \(\mathbb {D}\) such that three distinguished boundary points of G have prescribed images on \(\partial \mathbb {D}\). This result can be extended to general domains when the role of boundary points is taken over by prime ends. In this paper we prove a discrete counterpart of Carathéodory’s theorem in the framework of circle packing. A trilateral is a domain whose (intrinsic) boundary is decomposed into three arcs \(\alpha \), \(\beta \), and \(\gamma \) (of prime ends). Using Sperner’s lemma, we prove existence of circle packings and more general circle agglomerations filling arbitrary (bounded) simply connected trilaterals. In order to get complete results we have to admit (in some exceptional cases) that the packings contain degenerate circles. We study in detail which properties of the domain G and the underlying complex of the packing guarantee uniqueness and exclude degeneracy.


Circle packing Circle agglomeration Conformal mapping Conformal modulus Prime end Trilateral Tri-complex Discrete conformal geometry Sperner’s lemma 

Mathematics Subject Classification

52C26 30C80 30D40 05C10 



We would like to thank Gunter Semmler and Ken Stephenson for stimulating discussions.


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Copyright information

© The Managing Editors 2019

Authors and Affiliations

  1. 1.Institute of Applied AnalysisTU Bergakademie FreibergFreibergGermany

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