Dept. of Mathematics,
Univ. of Illinois, Urbana
Annotated Chronological Bibliography
- The Second Hull of a Knotted Curve,
with Jason Catarella and Greg Kuperberg and Rob Kusner.
We define the second hull of a space curve, consisting of those
points which are doubly enclosed by the curve in a certain sense. We prove
that any knotted curve has nonempty second hull. We relate this to recent
results on thick knots, quadrisecants, and minimal surfaces.
- The Tight Clasp.
Electronic Geometry Model
This clasp is a numerical simulation of a tight
(ropelength-minimizing) configuration of two
linked arcs with endpoints in fixed parallel
planes. Surprisingly, the arcs are not semicircles through
each others' centers.
Embedded Constant Mean Curvature Surfaces with Three Ends and Genus Zero,
with Karsten Große-Brauckmann and Rob Kusner.
We classify complete, almost embedded surfaces of constant mean curvature,
with three ends and genus zero (called triunduloids):
they are classified by triples of points on the sphere
whose distances are the asymptotic necksizes of the three ends.
Since triunduloids are transcendental objects, and
are not described by any ordinary differential equation, it is
remarkable to have such a complete and explicit determination for
their moduli space.
- Cubic Polyhedra, with Chaim Goodman-Strauss.
In Discrete Geometry (Kuperberg festschrift),
Marcel Dekker, 2002, to appear.
A cubic polyhedron is a polyhedral surface whose edges
are exactly all the edges of the cubic lattice.
Every such polyhedron is a discrete minimal surface,
and it appears that many (but not all) of them
can be relaxed to smooth minimal surfaces
(under an appropriate smoothing flow, keeping their symmetries).
Here we give a complete classification of the cubic polyhedra.
Among these are five new infinite uniform polyhedra and an uncountable
collection of new infinite semi-regular polyhedra.
We also consider the somewhat larger class of
all discrete minimal surfaces in the cubic lattice.
- Approximating Ropelength by Energy Functions.
In Physical Knots (Las Vegas 2001),
AMS Contemp. Math., 2002, to appear.
The ropelength of a knot is the quotient of its length by its
thickness. We consider a family of energy functions for knots,
depending on a power p, which approach ropelength as p increases.
We describe a numerically computed trefoil knot which seems to
be a local minimum for ropelength; there are nearby critical points
for the p-energies,
which are evidently local minima for large enough p.
- On the Minimum Ropelength of Knots and Links,
with Jason Catarella and Rob Kusner.
Inventiones Math., 2002, to appear.
(Published online, 17 June 2002, as DOI 10.1007/s00222-002-0234-y.)
The ropelength of a knot is the quotient of its length and its thickness,
the radius of the largest embedded normal tube around the knot. We prove
existence and regularity for ropelength minimizers in any knot or link type;
these are C1,1 curves, but need not be smoother.
We improve the lower bound for the ropelength of a nontrivial knot,
and establish new ropelength bounds for small knots and links,
including some which are sharp.
- ALICE on the Eightfold Way:
Exploring Curved Spaces in an Enclosed Virtual Reality Theater,
with George Francis, Camille Goudeseune, Hank Kaczmarski and Ben Schaeffer.
In Visualization and Mathematics III, Springer, 2003, to appear.
We describe a collaboration between
mathematicians interested in visualizing curved three-dimensional spaces and
researchers building next-generation virtual-reality environments such as
ALICE, a six-sided, rigid-walled virtual-reality chamber.
This environment integrates active-stereo imaging, wireless position-tracking
and wireless-headphone sound. To reduce cost, the display
is driven by a cluster of commodity computers instead of a traditional
graphics supercomputer. The mathematical application tested in
this environment is an implementation of Thurston's eight-fold way;
these eight three-dimensional geometries are conjectured to suffice for
describing all possible three-dimensional manifolds or universes.
- The Aesthetic Value of Optimal Geometry.
In The Visual Mind II, MIT Press, 2003, to appear.
Geometric optimization problems arise physically in many situations:
material interfaces, for instance, usually minimize some surface energy.
Curves and surfaces which are optimal for geometric energies often
have aesthetically pleasing shapes. Computer simulation of such optimal
geometry can be useful for mathematicians seeking insight into the behavior
of minimizers, for designers looking for graceful shapes
and attractive graphics, and for scientists modeling nature.
- Making the Optiverse: A Mathematician's Guide to AVN,
a Real-Time Interactive Computer Animator,
with George Francis and Stuart Levy.
In Mathematics, Art, Technology, Cinema, Springer, 2003, to appear.
Our 1998 video
illustrates an optimal sphere eversion, computed automatically
by minimizing an elastic bending energy for surfaces.
This paper describes AVN, the custom software program we wrote to
explore the computed eversion. Various special features allowed
us to use AVN also to produce our video: it controlled the camera path
throughout and even rendered most of the frames.
- Building space-time meshes over arbitrary spatial domains,
with Jeff Erickson and Damrong Guoy and Alper Üngör.
In Proceedings of the
Meshing Roundtable, Sandia, 2002, pp 391-402.
We present an algorithm to construct meshes suitable for
space-time discontinuous Galerkin finite-element methods. Our method
generalizes and improves the "Tent Pitcher" algorithm of
Üngör and Sheffer. Given an arbitrary simplicially
meshed spatial domain of any dimension and a time interval,
our algorithm builds a simplicial mesh of their space-time product domain,
in constant time per element. Our algorithm avoids the limitations
of previous methods by carefully adapting the durations of space-time
elements to the local quality and feature size of the underlying space mesh.
- Sphere Eversions: from Smale through "The Optiverse".
In Mathematics and Art:
Mathematical Visualization in Art and Education,
(Maubeuge 2000), Springer, 2002, pp 201-212 and 311-313.
This is a revised and updated version of [S8].
- The Optiverse: una guida ai matematici per AVN, programma
interattivo di animazione,
with George Francis and Stuart Levy.
In Matematica, arte, tecnologia, cinema, Springer, 2002, pp 37-51.
An Italian translation of [FLS2].
- Rescalable Real-Time Interactive Computer Animations.
Tools for Communicating Mathematics, Springer, 2002, pp 311-314.
Animations are one of the best tools for communicating
three-dimensional geometry, especially when it changes
in time through a homotopy. For special-purpose animations,
custom software is often necessary to achieve real-time performance.
This paper describes how, in recent years, computer hardware has improved,
and libraries have been standardized, to the point where such
custom software can be easily ported across all common platforms,
and the performance previously found only on high-end graphics
workstations is available even on laptops.
- Dynamic Skin Triangulation,
with Ho-Lun Cheng, Tamal K. Dey, and Herbert Edelsbrunner,
Discrete and Computational Geometry 25, 2001, pp 525-568.
CMP 1 838 419
This paper describes an algorithm for maintaining an
approximating triangulation of a deforming smooth surface in space.
The surface is the envelope of an infinite family of spheres defined
and controlled by a finite collection of weighted points. The triangulation
adapts dynamically to changing shape, curvature, and topology of the surface.
- Dynamic Skin Triangulation,
with Ho-Lun Cheng, Tamal K. Dey, and Herbert Edelsbrunner,
Proc. 12th Ann. ACM-SIAM Sympos. Discrete Alg., 2001 Jan, pp 47-56.
This is the 10-page announcement of [CDES2].
- Constant Mean Curvature Surfaces with Three Ends,
with Karsten Große-Brauckmann and Rob Kusner,
Proc. Natl. Acad. Sci. 97:26, 2000 Dec 19,
We announce the classification of triunduloids
given in [GKS3].
- Tomographic Imaging of Foam,
with Fetterman, Tan, Ying, Stack, Marks, Feller, Cull, Munson, Thoroddsen
Optics Express 7:5, 2000 Aug 28, pp 186-197.
We explore the use of visual-light tomography to create
three-dimensional volume images of small samples of soap foams.
We place the foam sample on a rotating stage, and acquire
a sequence of images. The tomographic algorithm corrects for
the distortion of the curved plexiglass container. Such reconstructions
allow comparison of physical foam experiments with computer simulations
of foam diffusion in the Surface Evolver.
- Foam Evolution: Experiments and Simulations,
with Hassan Aref and Sigurdur T. Thoroddsen,
in Proc. NASA 5th Microgravity Fluid Physics Conf.,
Aug 2000, pp 99-100.
This extended abstract describes relations between experimental observations,
mathematical models, and numerical simulations of foams, including
the dynamics of reconnection events, phase transitions in compressible foams,
tomographic reconstruction of foams, and the combinatorics of TCP foams.
- New Tetrahedrally Close-Packed Structures,
in Proc. Eurofoam 2000 (Delft), June 2000, pp 111-119.
This article in the proceedings of the Third Euroconference on Foams
describes a new construction for TCP structures and their associated foams.
This construction allows the creation of TCP triangulations of different
3-manifolds, which are convex combinations of the known basic TCP structures
exactly when the manifold is flat. This is a first step in understanding
the relation between combinatorics and topology for three-manifolds.
The class of TCP triangulations is of further interest because most can
be made with all dihedral angles acute. This property is important
for many meshing applications, for good numerical analysis, but methods
of constructing acute triangulations were previously unknown.
- Foams and Bubbles: Geometry and Simulation,
Intl. J. Shape Modeling, 5:1, 1999, pp 101-114.
This invited contribution to a special issue edited
by Michele Emmer is adapted and updated from
- "The Optiverse" and Other Sphere Eversions,
in ISAMA 99, Univ. Basque Country, 1999, pp 471-479.
Reprinted in Bridges 1999,
Southwestern Coll., Kansas, 1999, pp 265-274.
Full-color version published in the online journal
1:3, September 1999.
Also available in an HTML version.
For decades, the sphere eversion has been a classic subject for
mathematical visualization. Our 1998 video "The Optiverse"
geometrically optimal eversions created by minimizing elastic bending
energy. This paper contrasts these minimax eversions with earlier ones,
including those by Morin, Phillips, Max, and Thurston. Our minimax
eversions were automatically generated by flowing downhill in
energy using Brakke's Evolver.
- Crossing Numbers of Tight Knots,
with Jason Cantarella and Rob Kusner,
Nature 392:6673, 1998, pp 237-238.
This note shows that, contrary to a conjecture published by
some biophysicists a year earlier in Nature,
there is not a linear relation between the minimum crossing number
of a knot and its minimal ropelength. Instead, we construct
examples where the crossing number grows like the 4/3 power
of ropelength, the optimum possible.
- The Optiverse,
with George Francis and Stuart Levy,
in VideoMath Festival at ICM'98, Springer, 1998, 7-minute video.
This video shows the minimax sphere eversions described in
[FSH]. These are geometrically optimal ways to turn a sphere inside out,
computed by minimizing Willmore's elastic bending energy for surfaces.
The video was chosen for the exclusive Electronic Theater at SIGGRAPH 98,
and was selected by the jury for presentation at ICM'98. It has been the
subject of an article in Science and others in magazines
on three continents.
- The Geometry of Bubbles and Foams,
in Foams and Emulsions (NATO ASI volume E 354),
Kluwer, 1999, pp 379-402.
This survey records my invited series of lectures at
an interdisciplinary NATO school on foams (Cargèse, 1996) organized by
J.F. Sadoc and N. Rivier. It reviews the theory of
constant-mean-curvature surfaces, the combinatorics of foams
and their dual triangulations, their relation to crystal
structures, and the current status of the Kelvin problem
and related results.
- Knot Energies,
in VideoMath Festival at ICM'98, Springer, 1998, 3-minute video.
This video, selected by an international jury to be shown at ICM'98,
shows examples of Möbius-energy minimization for knots and links,
as described in [KS1].
- The beta-Sn Dual Structure: A 4-Connected Net
Based on a Packing of Simple Polyhedra with 18 Faces,
with Michael O'Keeffe,
Z. Kristallographie 213, 1998, pp 374-376.
This crystallography paper describes a new three-dimensional
structure arising out of discussions on foam structures and
their relations to crystals.
- On the Distortion and Thickness of Knots,
with Rob Kusner, in Topology and Geometry in Polymer Science
(IMA volume 103), Springer, 1998, pp 67-78.
We formulate and compare different rigorous definitions for the thickness of a
space curve, that is, the diameter of the thickest tube that can be embedded
around the curve. One definition involves Gromov's notion of the
distortion of the embedding of the curve. Our definitions are
especially useful because they are non-zero for polygonal curves,
and thus are easier to measure in computer simulations of
knots minimizing their ropelength (length divided by thickness).
- Möbius-invariant Knot Energies,
in Ideal Knots
World Scientific, 1998, pp 315-352.
CMP 1 702 037
This is an updated reprinting of [KS1],
as an invited contribution to a volume in the "Series on Knots and Everything",
edited by Stasiak, Katritch, and Kauffman.
- Computing Sphere Eversions,
with George Francis and Chris Hartman,
in Mathematical Visualization
Springer, 1998, pp 237-255.
CMP 1 677 675
This paper describes how to adapt the methods of
[FSK+] to compute
the minimax sphere eversions of higher-order symmetry which are
also shown in "The Optiverse" [SFL]. In particular, we must use
symmetry features of the evolver [BS] to perform the computations.
- Constant Mean Curvature Surfaces with Cylindrical Ends,
with Karsten Große-Brauckmann and Rob Kusner,
in Mathematical Visualization
Springer, 1998, pp 107-116.
CMP 1 677 699
Almost embedded CMC surfaces have ends asymptotic to
Delaunay unduloids; therefore they have finite total absolute
curvature if and only if all of their ends are asymptotic to cylinders.
A conjecture due to Rick Schoen had been that the cylinder should
be the only such surface, but here we give good numerical evidence
against that conjecture. By gluing together truncated triunduloids,
we construct surfaces with, say, thirty ends, all cylindrical.
- Comparing the Weaire-Phelan Equal-Volume Foam to Kelvin's Foam,
with Rob Kusner, Forma 11:3, 1996, pp 233-242.
Reprinted in The Kelvin Problem,
Taylor & Francis, 1996, pp 71-80.
Lord Kelvin conjectured a foam structure as the optimal partition
of space into equal-volume cells, with least surface area.
A century later, Weaire and Phelan discovered an equal-volume foam
which numerically seemed better than Kelvin's candidate.
Our contribution to this special volume edited by Denis Weaire
shows how to rigorously prove that the Weaire-Phelan foam does beat
- In Memoriam Frederick J. Almgren Jr., 1933-1997:
On Being a Student of Almgren's, with Frank Morgan,
Experimental Math. 6:1, 1997, pp 8-10.
CMP 1 464 578
These descriptions of what is was like to be Almgren's student,
published alongside reminiscences by mathematicians who knew
him in other ways, show the evolution of Almgren's work over
the course of a decade, as he grew to appreciate the value of
computers in solving geometric problems in pure mathematics.
- The Minimax Sphere Eversion,
with George Francis, Rob Kusner et al,
in Visualization and Mathematics
Springer, 1997, pp 3-20.
CMP 1 607 221
Here we explain the mathematical theory behind the geometrically optimal
minimax sphere eversion shown in "The Optiverse" [SFL]. This
eversion is accomplished by numerically modeling the gradient flow
for the Willmore energy, starting from the lowest saddle point
and flowing down to the round sphere.
- Using Symmetry Features of the
Surface Evolver to Study Foams, with Ken Brakke,
in Visualization and Mathematics
Springer, 1997, pp 95-117.
CMP 1 607 360
This report describes how certain new features we have added to
the Surface Evolver can be used to take advantage of symmetries
of a surface being modeled. As a test case, we describe how to
accurately model the Kelvin foam and the Weaire-Phelan foam, which
is a better partition of space into equal-volume cells (see [KS2]).
- Open Problems in Soap-Bubble Geometry,
editor, with Frank Morgan,
International J. Math. 7:6, 1996, pp 833-842.
This list collects and organizes a long list of open problems posed by
participants at a special session on Soap-Bubble Geometry at
the AMS MathFest in Burlington in 1994, as well as further
problems suggested by the editors.
- Möbius Energies for Knots and Links,
Surfaces and Submanifolds,
with Rob Kusner, in
International Press, 1996, pp 570-604.
In this paper, we give a nicer explanation of the Möbius-invariance of
the knot energy studied by Freedman, He and Wang, and extend it
to higher-dimensional submanifolds. We also give the first examples
of knot and link types with several distinct critical points for this
energy. We include a table and illustrations of numerically computed
energy-minimizing configurations of all knots and links through eight
- Elliptic and Parabolic Methods in Geometry,
editor, with Ben Chow, Bob Gulliver and Silvio Levy. Published by AK
This book is the proceedings volume from a five-day workshop we organized,
held in Minneapolis in 1994. Twelve contributions by outstanding
geometers convey the potential of using computers in studying a wide
range of open questions in geometry. Topics include curvature flows, harmonic
maps, liquid crystals, and CMC surfaces.
- Sphere Packings Give an Explicit Bound for the Besicovitch
J. Geometric Analysis 4:2, 1994, pp 219-231.
This paper, which arose from a lemma used in my dissertation,
examines a standard proof of the Besicovitch Covering Theorem
from the point of view of finding the optimal constant,
which turns out to also be the answer to a sphere-packing problem:
how many unit spheres fit into a ball of radius five?
In high dimensions, I review the best asymptotic bounds known.
In two dimensions, I show the answer is 19, while in three dimensions,
I give the best upper and lower bounds known.
- Monotonicity Theorems for Two-Phase Solids,
with Frank Morgan, Francis Larché,
Arch. Rat. Mech. Anal. 124:4, 1994, pp 329-353.
Here we give a rigorous mathematical proof of some observations
about metal alloy systems at concentrations for which two phases coexist.
If there were no cost involved in mixing the phases, each phase would be
at a fixed concentration, even as the overall concentration c of the two
metals in the alloy varied. Here we explain, using techniques of
convex analysis, the counterintuitive fact that, with a mixing cost,
the individual concentrations vary inversely with c. Along
the way, we find several interesting lemmas about minima of functions
of several variables and parameters.
- Computing Hypersurfaces Which Minimize Surface Energy
Plus Bulk Energy, in Motion by Mean Curvature and Related Topics,
de Gruyter, 1994, pp 186-197.
My dissertation proved an approximation theorem for area-minimizing
hypersurfaces in the context of geometric measure theory. This
kind of approximation is especially useful to prove the feasibility
of algorithms to find area-minimizing surfaces without a priori
knowing their topology. This paper (appearing in the proceedings of a 1992
conference in Trento) shows that the approximation theorem
and algorithms can be extended to the case where the minimization
involves not just a surface energy, but also bulk terms like volume or gravity.
- Minimizing the Squared Mean Curvature Integral for Surfaces in
Space Forms, with Lucas Hsu and Rob Kusner,
Math. 1:3, 1992, pp 191-207.
We report on the results of the first computer simulations of
Willmore surfaces, using Brakke's Evolver. The numerical evidence
supports Willmore's conjecture about the minimizing torus, and
suggests that certain Lawson surfaces minimize for higher genus.
These simulations have been of interest to biophysicists studying
- Using Max-Flow/Min-Cut to Find Area-Minimizing Surfaces,
in Computational Crystal Growers Workshop, AMS Sel. Lect. Math.,
1992, pp 107-110 plus video.
This video uses algorithm animation to illustrate how the algorithm
described in my dissertation uses max-flow/min-cut techniques to
find approximations to area-minimizing surfaces, without knowing
their topology in advance; it appears in the proceedings of a conference
organized by Jean Taylor.
- Visualization of soap bubble geometries,
with Fred Almgren, Leonardo 24:3/4, 1992, pp 267-271.
Reprinted in The Visual Mind
MIT Press, 1993, pp 79-83.
CMP 1 255 841
This paper, in a special volume edited by Michele Emmer,
surveys results on the geometry of bubble clusters,
and describes my rendering algorithm for photorealistic
computer graphics of soap film, also used later in "The Optiverse" [SFL].
- Crystalline Approximation: Computing Minimum
Surfaces via Maximum Flows,
in Computing Optimal Geometries, AMS Selected Lectures in Math.,
1991, pp 59-62 plus video.
This video shows, using a two-dimensional example, how the approximation
theorem proved in my dissertation works to find approximately
area-minimizing surfaces; it appears in the proceedings of an AMS
special session organized by Almgren and Taylor.
- Generating and Rendering Four-Dimensional Polytopes,
The Mathematica Journal 1:3, 1991, pp 76-85.
This expository paper shows a nice way to generate coordinates
for the regular polytopes in three and four dimensions, and describes
how to picture the four-dimensional polytopes via stereographic projection
as bubble clusters in three-space. It is illustrated with computer
graphics using the algorithm described in [AS].
- Computing Soap Films and Crystals,
with Fred Almgren, Ken Brakke, Jean Taylor,
in Computing Optimal Geometries,
AMS Selected Lectures in Math., 1991, 14-minute video.
This video, which we produced at the Geometry Supercomputer Center
while I was in graduate school, shows some early computations with
Brakke's evolver, computations done with my three-dimensional Voronoi
cell code, and crystalline minimal surfaces computed by Taylor.
- Animating the Heat Equation:
A Case Study in Mathematica Optimization,
with Matt Thomas, The Mathematica Journal 1:1, 1990, pp 80-84.
When I was asked to referee a submission by Thomas to the first issue
of The Mathematica Journal, I found I could optimize his code,
resulting in almost a thousand-fold speedup. The main thrust of the
published joint article became a description of these optimization techniques.
- Some Results on the Phase Behavior in Coherent Equilibria,
with Francis Larché, Frank Morgan,
Scripta Metallurgica 24:3, 1990, pp 491-493.
In some metal alloy systems two phases coexist for certain
concentrations. This metallurgy paper explains the counterintuitive fact
that the concentration in each phase varies inversely with the
overall concentration. The mathematical details are given in [MSL].
- A Crystalline Approximation Theorem for Hypersurfaces,
Princeton University Ph.D. thesis, 1990; Geometry Center report GCG 22.
My dissertation shows that any hypersurface can be approximated arbitrarily
well by polygons chosen from the finite set of facets of an appropriate
cell complex, with restricted orientation and positions. Thus we can
approximate the problem of finding the least-area surface on a given
boundary by a finite network-flow problem in linear programming.
This gives an effective algorithm for finding such surfaces,
without knowing their topology in advance.
Pieces of my dissertation, and related results,
appear in [S2], [S3],
but the main section of the work has not yet been published elsewhere.