The CirclePack Owl Welcome to my homepage: Ken Stephenson Professor of Mathematics Department of mathematics University of Tennessee Knoxville Twisted Pent tiling, glass by Ken Stephenson
 Research Interests: Classical Analytic Function Theory Circle Packing (see below) and Discrete Analytic Function Theory Numerical Conformal Mapping/Structures Brain Mapping CirclePack (Java WebStart)
 Pentagonal Quilt by  Mary Jo Wickliff Circle Packing Dodecahedral tiling, circle packing of Cannon, Floyd, Parry pattern

General Comments:

My primary research interests revolve around circle packing: connections to analytic function theory, Riemann surfaces, computational conformal structures, and applications.  (see my curriculum vita.)

A circle packing is a configuration of circles with a specified pattern of tangencies. See the examples below; the circles typically must assume a variety of different sizes in order to fit together in a prescribed tangency pattern. (It is easy to confuse this with the well-known topic of `sphere packing'  -- how many ping-pong balls fit in a box car -- but there is almost no contact between these two topics!!)

Circle packings were introduced by William Thurston in his Notes. Maps between circle packings which preserve tangency and orientation act in many ways as discrete analogues of analytic functions. Moreover, work flowing from a 1985 conjecture of Thurston, proven by Burt Rodin and Dennis Sullivan, shows that classical analytic functions and more general classical conformal objects can be approximated using circle packings. Circle packings are computable, so they are introducing an experimental, and highly visual, component to research in conformal geometry and related areas. Circle packings are also useful in graph embedding and have interesting connections to random walks. Some links to related work:

The Zoo:

Here is a small menagerie of circle packings in the plane, the hyperbolic plane, and the sphere.

 The  CirclePack owl:  ----->

Research Papers: Recent papers for downloading:

 Cortical cartography using the discrete conformal approach of circle packings, by Monica Hurdal and Ken Stephenson Paper from Science Direct (2.5MB pdf file) Circle Packing: A Mathematical Tale, by Ken Stephenson, Amer. Math. Soc. Notices (cover article), Dec. 2003. Download the pdf file (13 pages, 4.2 Megabytes) Curvature Flow in Conformal Mapping, by Tobin Driscoll, Charles Collins, and Ken Stephenson Download the PostScript file (23 pages, 3.75 Megabytes) Approximation of Conformal Structures via Circle Packing,  by Ken Stephenson  Survey paper, Computational Methods and Function Theory 1997, Proceedings of the Third CMFT Conference, World Scientific, 1999, pp 551-582.  Download the postscript file (32 pages, 2 megabytes); view Illustrations Circle Packing and Discrete Analytic Function Theory, by Ken Stephenson Survey: Handbook of Complex Analysis, Vol. 1: Geometric Function Theory, Chapter 11, Editor Kuhnau, Elsevier, 2002  Download  the postscript file (40 pages, 2 megabytes); view  Illustrations Quasi-conformally flat mapping the human cerebellum, by P. Bowers and M. Hurdal and K. Rehm and D. Rottenberg and  K. Schaper and K. Stephenson and D. Sumners  Download the postscript file (8 pages, 1.5 megabytes) A Probabilistic Proof of Thurston's Conjecture on Circle Packings,  by Kenneth Stephenson  Rendiconti del Seminario Mate. e Fisico di Milano, LXVI (1996), pp. 201-291. Download the pdf file (66 pages, .28 megabytes); view Illustrations A "Regular" Pentagonal Tiling of the Plane,  by Philip L. Bowers and Ken Stephenson  Appeared in AMS e-journal Conformal Geometry and Dynamics,  Vol 1, (1997), 58-86.  Official download for subscribers; preprint download of PostScript file (30 page, .5 megabytes);  view Illustrations A Circle Packing Algorithm ,  by Chuck Collins and Kenneth Stephenson Computational Geometry: Theory and Applications, 25 (2003), pp. 233-256. Download , gzipped postscript file (24 pages, .5 megabytes) Geometric Sequences of Discs in the Apollonian Packing,  by Dov Aharonov and Kenneth Stephenson  Algebra i Analiz, Russian Academy of Sciences, dedicated to Goluzin, Vol 9 (1997), pp 104-140.  Download the postscript file (42 pages, .19 megabytes); view Illustrations Circle packing: experiments in discrete analytic function theory, by Tomasz Dubejko and Kenneth Stephenson Experimental Mathematics, Vol 4 (1995), pp. 307-348. Download the postscript file (50 pages, .8 megabytes); view Illustrations

Circle Packing Software:

The Java Version 1.0 of CirclePack is now available and should run on any platform with Java 1.6++. Information, CirclePac.jar (8.7 Mgb), prepared "scripts" for single-click experiments, and a number of circle packing data sets can be found at CirclePack

Circle Packing Bibliography:

I try to maintain a bibliography of papers on circle packing. Download the PostScriptpdf , or CP-bib.bib version (as of December 2002). I would appreciate any suggestions for corrections and/or additions and any references to related work.

Ways to contact me:

Kenneth Stephenson, (kens "at" math.utk.edu)
Department of Mathematics, Univ. of Tenn., Knoxville, TN 37996-1300
Phone: (865) 974-4330
Fax: (865) 974 6576

Last Modified: May 2007.