Skip to main content
SpringerLink
Log in

Lorentzian Coxeter systems and Boyd–Maxwell ball packings

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. By using the term future- or past-directed, we are assuming that the hyperplane h(x)=0 intersects the light cone only at the origin. If this is not the case, one can replace h(x) with any positively weighted sum of the coordinates. The only requirement is that the hyperplane h(x)=1 is transverse to \(\upphi ^+\); see [20, Section 5.2]

References

  1. Abramenko, P., Brown, K.S.: Buildings, GTM, vol. 248. Springer, New York (2008)

    Book  Google Scholar 

  2. Bourbaki, N.: Groupes et Algèbres de Lie. Chapitre 4–6. Hermann, Paris (1968)

  3. Boyd, D.W.: The residual set dimension of the Apollonian packing. Mathematika 20, 170–174 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  4. Boyd, D.W.: A new class of infinite sphere packings. Pac. J. Math. 50, 383–398 (1974)

    Article  MATH  Google Scholar 

  5. Calabi, E., Markus, L.: Relativistic space forms. Ann. Math. 2(75), 63–76 (1962)

    Article  MathSciNet  Google Scholar 

  6. Cecil, T.E.: Lie Sphere Geometry, Universitext, 2nd edn. Springer, New York (2008)

    Google Scholar 

  7. Chein, M.: Recherche des graphes des matrices de Coxeter hyperboliques d’ordre \(\le \,10\). Rev. Française Informat. Recherche Opérationnelle 3(Ser. R-3), 3–16 (1969)

  8. Chen, H.: Ball packings and Lorentzian discrete geometry. Ph.D. Thesis, Freie Universität Berlin (2014)

  9. Chen, H.: Sage code for enumeration of level-2 Coxeter graphs. Zenodo (2014). doi:10.5281/zenodo.11115

  10. Chen, H.: Apollonian ball packings and stacked polytopes (December 2013). Preprint, arXiv:1306.2515v2 [math.MG]

  11. Chen, H., Labbé, J.P.: Limit directions for Lorentzian Coxeter systems (March 2014). Preprint, arXiv:1403.1502v1 [math.GR]

  12. Dyer, M.: Imaginary cone and reflection subgroups of Coxeter groups (April 2013). Preprint, arXiv:1210.5206v2 [math.RT]

  13. Dyer, M., Hohlweg, C., Ripoll, V.: Imaginary cones and limit roots of infinite Coxeter groups (April 2013). Preprint, arXiv:1303.6710v2 [math.GR]

  14. Geck, M., Hiss, G., Lübeck, F., Malle, G., Pfeiffer, G.: CHEVIE: a system for computing and processing generic character tables. Appl. Algebra Eng. Commun. Comput. 7(3), 175–210 (1996)

    Article  MATH  Google Scholar 

  15. Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A.R., Yan, C.H.: Apollonian circle packings: number theory. J. Number Theory 100(1), 1–45 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A.R., Yan, C.H.: Apollonian circle packings: geometry and group theory. I. The Apollonian group. Discret. Comput. Geom. 34(4), 547–585 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A.R., Yan, C.H.: Apollonian circle packings: geometry and group theory. III. Higher dimensions. Discret. Comput. Geom. 35(1), 37–72 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hertrich-Jeromin, U.: Introduction to Möbius Differential Geometry, London Mathematical Society Lecture Note Series, vol. 300. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  19. Higashitani, A., Mineyama, R., Nakashima, N.: Distribution of accumulation points of roots for type \((n-1,1)\) Coxeter groups (July 2014). Preprint, arXiv:1212.6617v5 [math.GR]

  20. Hohlweg, C., Labbé, J.P., Ripoll, V.: Asymptotical behaviour of roots of infinite Coxeter groups. Can. J. Math. 66(2), 323–353 (2014)

    Article  MATH  Google Scholar 

  21. Hohlweg, C., Préaux, J.P., Ripoll, V.: On the limit set of root systems of Coxeter groups and Kleinian groups (July 2013). Preprint, arXiv:1305.0052v2 [math.GR]

  22. Humphreys, J.E.: Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  23. Krammer, D.: The conjugacy problem for Coxeter groups. Group Geom. Dyn. 3(1), 71–171 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Maxwell, G.: Hyperbolic trees. J. Algebra 54(1), 46–49 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  25. Maxwell, G.: Sphere packings and hyperbolic reflection groups. J. Algebra 79(1), 78–97 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  26. Maxwell, G.: Wythoff’s construction for Coxeter groups. J. Algebra 123(2), 351–377 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  27. Maxwell, G.: Euler characteristics and imbeddings of hyperbolic Coxeter groups. J. Austr. Math. Soc. Ser. A 64(2), 149–161 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. McMullen, C.T.: Hausdorff dimension and conformal dynamics. III. Computation of dimension. Am. J. Math. 120(4), 691–721 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  29. Stein, W.A., et al.: Sage Mathematics Software (Version 6.2). The Sage Development Team (2014). http://www.sagemath.org

  30. Stephenson, K.: Circle packing: a mathematical tale. Not. Am. Math. Soc. 50(11), 1376–1388 (2003)

    MATH  MathSciNet  Google Scholar 

  31. Vinberg, E.B.: Discrete linear groups that are generated by reflections. Izv. Akad. Nauk SSSR. Ser. Mat. 35, 1072–1112 (1971)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

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.

Author information

Authors and Affiliations

  1. Institut für Mathematik, Freie Universität Berlin, Arnimallee 2, 14195, Berlin, Germany

    Hao Chen & Jean-Philippe Labbé

Authors
  1. Hao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Philippe Labbé
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hao Chen.

Additional information

J.-P. Labbé is supported by a FQRNT Doctoral scholarship and SFB Transregio “Discretization in Geometry and Dynamics” (TRR 109).

H. Chen is supported by the Deutsche Forschungsgemeinschaft within the Research Training Group “Methods for Discrete Structures” (GRK 1408). An alternative version of this paper appeared in the Ph.D. thesis of the author [8].

Appendix

Appendix

See Figs. 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 and 19.

Fig. 6
figure 6

Graphs constructed from \(K_4\)

Full size image
Fig. 7
figure 7

Graphs constructed from \(K_4-e\)

Full size image
Fig. 8
figure 8

Graphs constructed from \(K_{23}\)

Full size image
Fig. 9
figure 9

Graphs with two cycles

Full size image
Fig. 10
figure 10

Cycles

Full size image
Fig. 11
figure 11

Cycles with one tail of length \(1\) (\(5\) vertices)

Full size image
Fig. 12
figure 12

Cycles with one tail of length \(1\) (\(>5\) vertices)

Full size image
Fig. 13
figure 13

Cycles with one tail of length \(2\)

Full size image
Fig. 14
figure 14

Cycles with two tails of length \(1\)

Full size image
Fig. 15
figure 15

Trees (\(5\) vertices)

Full size image
Fig. 16
figure 16

Trees (\(6\) vertices)

Full size image
Fig. 17
figure 17

Trees (\(7\) vertices)

Full size image
Fig. 18
figure 18

Trees (\(8\) or \(9\) vertices)

Full size image
Fig. 19
figure 19

Trees (\(10\) or \(11\) vertices)

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, H., Labbé, JP. Lorentzian Coxeter systems and Boyd–Maxwell ball packings. Geom Dedicata 174, 43–73 (2015). https://doi.org/10.1007/s10711-014-0004-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-014-0004-1

Keywords

Mathematics Subject Classification

Search

Navigation