Abstract
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.
Similar content being viewed by others
References
Beardon, Alan F.,The Geometry of Discrete Groups, GTM 91, Springer-Verlag, New York, Heidelberg, Berlin, 1983.
Beardon, Alan F. and Stephenson, Kenneth, ‘The uniformization theorem for circle packings’,Indiana Univ. Math. J. 39 (1990), 1383–1425.
Ford, L. R.,Automorphic Functons, 2nd edn, Chelsea, New York, 1951.
Lockwood, E. H.,A Book of Curves, Cambridge University Press, Cambridge, 1961.
Maskit, B., ‘On Poincaré's theorem for fundamental polygons’,Adv. Math. 7 (1971), 219–230.
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.
Rodin, Burt and Sullivan, Dennis, ‘The convergence of circle packings to the Riemann mapping’,J. Diff. Geom. 26 (1987), 349–360.
Schramm, Oded, ‘Rigidity of infinite (circle) packings’,J. Amer. Math. Soc. 4 (1991), 127–149.
Thurston, William, ‘The geometry and topology of 3-manifolds’, preprint, Princeton University Notes.
Author information
Authors and Affiliations
Additional information
The last two authors gratefully acknowledge support of the National Science Foundation and the Tennessee Science Alliance.
Rights 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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01263534