Geometriae Dedicata

, Volume 174, Issue 1, pp 43–73 | Cite as

Lorentzian Coxeter systems and Boyd–Maxwell ball packings

  • Hao ChenEmail author
  • Jean-Philippe Labbé
Original Paper


In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In this paper, we show that the observed fractals are exactly the ball packings described by Boyd and Maxwell. This correspondence is a corollary of a more fundamental result: given a geometric representation of a Coxeter group in a Lorentz space, the set of limit directions of weights equals the set of limit roots. Additionally, we use Coxeter complexes to describe tangency graphs of the corresponding Boyd–Maxwell ball packings. Finally, we enumerate the Coxeter systems that generate Boyd–Maxwell ball packings.


Ball packing Infinite Coxeter groups Geometric representation Limit roots Limit sets Lorentz space 

Mathematics Subject Classification

52C17 20F55 05C30 



The authors are grateful to George Maxwell for his great availability to check the enumeration results. We thank Christian Stump for helpful discussions, and Christophe Hohlweg and Vivien Ripoll for valuable comments on a preliminary version of this article. We also thank the anonymous referee for careful reading and for pointing us to the work of Calabi and Markus.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Institut für MathematikFreie Universität BerlinBerlinGermany

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