The CirclePack Owl
Professor of Mathematics
Department of mathematics
University of Tennessee
Twisted Pent tiling,
glass by Ken Stephenson
||CirclePack (Java WebStart)
Pentagonal Quilt by
Mary Jo Wickliff
Dodecahedral tiling, circle
packing of Cannon, Floyd, Parry
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:
Here is a small menagerie of circle packings in the plane, the
hyperbolic plane, and the sphere.
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
by Ken Stephenson, Amer. Math. Soc. Notices (cover article), Dec. 2003.
Download the pdf file (13 pages, 4.2 Megabytes)
|Curvature Flow in
by Tobin Driscoll, Charles Collins, and Ken Stephenson
Download the PostScript file (23 pages, 3.75 Megabytes)
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
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
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);
|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
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
Circle Packing Bibliography:
I try to maintain a bibliography of papers on circle packing. Download the PostScript, pdf , 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