The CirclePack Owl
Welcome to my homepage:
Ken Stephenson
Professor of Mathematics
Department of mathematics
University of Tennessee

Twisted Pent tiling,
glass by Ken Stephenson
2010 Barrett Memorial Lectures Poster
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

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.


Research Papers: Recent papers for downloading:
cortical hemisphere
Cortical cartography using the discrete conformal approach of circle packings,
by Monica Hurdal and Ken Stephenson
Paper from Science Direct (2.5MB pdf file)
discrete welding
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)
flow picture
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"
   Department of Mathematics, Univ. of Tenn., Knoxville, TN 37996-1300
Phone: (865) 974-4330
Fax: (865) 974 6576

Last Modified: May 2007.