Spiral hexagonal circle packings in the plane


We discuss an intriguing geometric algorithm which generates infinite spiral patterns of packed circles in the plane. Using Kleinian group and covering theory, we construct a complex parametrization of all such patterns and characterize those whose circles have mutually disjoint interiors. We prove that these ‘coherent’ spirals, along with the regular hexagonal packing, give all possible hexagonal circle packings in the plane. Several examples are illustrated.

This is a preview of subscription content, access via your institution.


  1. [B]

    Beardon, Alan F.,The Geometry of Discrete Groups, GTM 91, Springer-Verlag, New York, Heidelberg, Berlin, 1983.

    Google Scholar 

  2. [BSt]

    Beardon, Alan F. and Stephenson, Kenneth, ‘The uniformization theorem for circle packings’,Indiana Univ. Math. J. 39 (1990), 1383–1425.

    Google Scholar 

  3. [F]

    Ford, L. R.,Automorphic Functons, 2nd edn, Chelsea, New York, 1951.

    Google Scholar 

  4. [L]

    Lockwood, E. H.,A Book of Curves, Cambridge University Press, Cambridge, 1961.

    Google Scholar 

  5. [M]

    Maskit, B., ‘On Poincaré's theorem for fundamental polygons’,Adv. Math. 7 (1971), 219–230.

    Google Scholar 

  6. [RK]

    Rothen, F. and Koch, A.-J., ‘Phyllotaxis or the properties of spiral lattices. II. Packing of circles along logarithmic spirals’,J. Phys. France 50 (1989), 1603–1621.

    Google Scholar 

  7. [RS]

    Rodin, Burt and Sullivan, Dennis, ‘The convergence of circle packings to the Riemann mapping’,J. Diff. Geom. 26 (1987), 349–360.

    Google Scholar 

  8. [Sc]

    Schramm, Oded, ‘Rigidity of infinite (circle) packings’,J. Amer. Math. Soc. 4 (1991), 127–149.

    Google Scholar 

  9. [T]

    Thurston, William, ‘The geometry and topology of 3-manifolds’, preprint, Princeton University Notes.

Download references

Author information



Additional information

The last two authors gratefully acknowledge support of the National Science Foundation and the Tennessee Science Alliance.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Beardon, A.F., Dubejko, T. & Stephenson, K. Spiral hexagonal circle packings in the plane. Geom Dedicata 49, 39–70 (1994). https://doi.org/10.1007/BF01263534

Download citation


  • Complex Parametrization
  • Kleinian Group
  • Circle Packing
  • Spiral Pattern
  • Geometric Algorithm