Updated: 2016 January 24
Annotated Bibliography by Subject
Geometric Knot Theory  
Foams and CMC Surfaces
Sphere Eversions and Willmore Energy  
Discrete Differential Geometry and Meshing
Math Visualization, Art and Polyhedra  
Miscellany

Geometric Knot Theory
 [CFKS]
 Ropelength Criticality,
with Jason Cantarella, Joe Fu and Rob Kusner.
ArXiv 1102.3234
[math.DG].
Geometry and Topology 18 (2014) pp. 1973–2043.
(Published online 3 Oct. 2014 as DOI:10.2140/gt.2014.18.1973.)
The ropelength problem asks for the minimumlength configuration of a
knotted tube embedded with fixed diameter. The core curve of such a tube
is called a tight knot, and its length is a knot invariant measuring
complexity. In terms of the core curve the thickness constraint has two
parts: an upper bound on curvature and a selfcontact condition. We give
a set of necessary and sufficient conditions for criticality with respect
to this constraint, based on a version of the Kuhn–Tucker theorem that
we established [CFKSW]. The key technical difficulty is to
compute the derivative of thickness under a smooth perturbation. This
is accomplished by writing thickness as the minimum of a
C^{1}compact
family of smooth functions in order to apply a theorem of Clarke. We give
a number of applications, including a classification of critical curves
with no selfcontacts (constrained by curvature alone), a characterization
of helical segments in tight links, and an explicit but surprisingly
complicated description of tight clasps.
 [BCSvdM]
 Geometric Knot Theory,
organizer, with Dorothy Buck, Jason Cantarella and Heiko von der Mosel.
Oberwolfach Reports
10:2, 2013, pp 1313–1358.
(DOI: 10.4171/OWR/2013/22)
This report collects extended abstracts from
the workshop held April 28–May 4, 2013,
with an introduction from the organizers.
 [S21]
 Curves of finite total curvature.
In Discrete Differential Geometry [BSSZ],
Birkhäuser, 2008, pp. 137–161.
ArXiv math.GT/0606007.
We consider the class of curves of finite total curvature,
as introduced by Milnor. This is a natural class for variational problems
and geometric knot theory, and since it includes both smooth and
polygonal curves, its study shows us connections between
discrete and differential geometry. To explore these ideas,
we consider theorems of Fáry/Milnor, Schur, Chakerian and Wienholtz.
 [DS2]
 The Distortion of a Knotted Curve,
with Elizabeth Denne.
Proc. Amer. Math. Soc. 137 (2009) pp 1139–1148.
Published online 29 Sep. 2008.
ArXiv math.GT/0409438.
The distortion of a curve measures the maximum arc/chord length
ratio. Gromov showed any closed curve has distortion at least pi/2
and asked about the distortion of knots. Here, we use the existence of
a shortest essential secant to show that any nontrivial tame knot
has distortion at least 5pi/3; examples show that
distortion under 7.16 suffices to build a trefoil knot.
(This 2007 version is a thorough revision of the original,
where the lower bound was only 3.99 and applied only to FTC curves.)
 [DS1]
 Convergence and isotopy type for graphs of finite total curvature,
with Elizabeth Denne.
In Discrete Differential Geometry [BSSZ],
Birkhäuser, 2008, pp. 163–174.
ArXiv math.GT/0606008.
Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result for essential subarcs of a knot.
 [CFKSW]
 Criticality for the Gehring Link Problem,
with Jason Cantarella, Joe Fu, Rob Kusner and Nancy Wrinkle.
Geometry and Topology 10 (2006) pp. 2055–2115.
(Published online 14 Nov. 2006 as DOI:10.2140/gt.2006.10.2055.)
ArXiv math.DG/0402212.
Zbl 1129.57006
In 1974, Gehring posed the problem of minimizing the length of two
linked curves separated by unit distance. This constraint can be
viewed as a measure of thickness for links, and the ratio of length
over thickness, as the (Gehring) ropelength. In this paper we refine
Gehring's problem to deal with links in a fixed linkhomotopy class:
we prove ropelength minimizers exist and introduce a theory of
ropelength criticality.
Our balance criterion is a set of necessary and sufficient conditions
for criticality, based on our strengthened, infinitedimensional version
of the KuhnTucker theorem. We use this to prove that every critical
link is C^{1} with finite total curvature.
The balance criterion also allows us to explicitly describe critical
configurations (and presumed
minimizers) for many links including the Borromean rings. We also
exhibit a surprising critical configuration for two clasped ropes:
near their tips the curvature is unbounded and a small gap appears between
the two components. These examples reveal the depth and richness
hidden in Gehring's problem and our natural extension of it.
 [DDS]
 Quadrisecants Give New Lower Bounds for the Ropelength of a Knot,
with Elizabeth Denne and Yuanan Diao.
Geometry and Topology 10 (2006) pp 1–26.
(Published online 25 Feb. 2006 as DOI:10.2140/gt.2006.10.1.)
ArXiv math.DG/0408026.
MR 2006m:58015
Zbl 1108.57004
Using the existence of a special quadrisecant line, we show the ropelength
of any nontrivial knot is at least 15.66. This improves the previously
known lower bound of 12. Numerical experiments have found a trefoil with
ropelength less than 16.372, so our new bounds are quite sharp.
 [SW]
 Some RopelengthCritical Clasps,
with Nancy Wrinkle.
In Physical and Numerical Models in Knot Theory
(MR 2006h:00009
Zbl 1085.57002),
World Sci., 2005, pp. 565580.
ArXiv math.DG/0409369.
MR 2006j:58017
Zbl 1104.57006
We describe several configurations of clasped ropes which are balanced and thus critical for the Gehring ropelength problem of [CFKSW].
 [S16]
 Distortion of Knotted Curves,
pp 2515–2517 in Geometrie, organized by Bangert, Burago and Pinkall,
Oberwolfach Reports 1:4 (2004) pp 2493–2538.
This report on a lecture at the 2004 Oberwolfach "Geometrie" meeting
gives a short summary of the results of [DS2],
including the necessary ingredients from [DDS].
In particular, we show that the distortion of a knotted curve
(the maximum arc/chord length ratio) is at least pi, twice that
of a round circle.
 [CKKS]
 The Second Hull of a Knotted Curve,
with Jason Catarella, Greg Kuperberg and Rob Kusner.
Amer. J. Math. 125:6, Dec 2003, pp 13351348.
ArXiv math.GT/0204106.
MR 2004k:57004
Zbl 1054.53003
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.
 [S14]
 The Tight Clasp.
Electronic Geometry Model
2001.11.001, May 2003.
This clasp is a numerical simulation of a tight
(ropelengthminimizing) configuration of two
linked arcs with endpoints in fixed parallel
planes. Surprisingly, the arcs are not semicircles through
each others' centers.
 [CKS2]
 On the Minimum Ropelength of Knots and Links,
with Jason Catarella and Rob Kusner.
Inventiones Math. 150:2, 2002, pp 257286.
(Published online 17 Jun. 2002 as DOI:10.1007/s002220020234y.)
ArXiv math.GT/0103224.
MR 2003h:58014
Zbl 1036.57001
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 C^{1,1} 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.
 [S12]
 Approximating Ropelength by Energy Functions.
In Physical Knots (Las Vegas 2001)
(MR 2003h:57001),
AMS Contemp. Math., 2002, pp 181186.
ArXiv math.GT/0203205.
MR 2003j:58020
Zbl 1014.57007
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 penergies,
which are evidently local minima for large enough p.
 [CKS1]
 Tight Knot Values Deviate from Linear Relations,
with Jason Cantarella and Rob Kusner,
Nature 392:6673, 1998, pp 237238.
(Published online 19 Mar. 1998 as DOI:10.1038/32558.)
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.
 [S6]
 Knot Energies,
in VideoMath Festival at ICM'98
(MR 2000j:01030/2006i:00009
Zbl 0909.00003/1071.00001),
Springer, 1998, 3minute video.
This video, selected by an international jury to be shown at ICM'98,
shows examples of Möbiusenergy minimization for knots and links,
as described in [KS1].
 [KS4]
 On the Distortion and Thickness of Knots,
with Rob Kusner, in Topology and Geometry in Polymer Science
(IMA volume 103), Springer, 1998, pp 6778.
ArXiv dgga/9702001.
MR 99i:57019
Zbl 0912.57006
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 nonzero for polygonal curves,
and thus are easier to measure in computer simulations of
knots minimizing their ropelength (length divided by thickness).
 [KS3]
 Möbiusinvariant Knot Energies,
with Rob Kusner, in Ideal Knots
(MR 2000j:57018),
World Scientific, 1998, pp 315352.
MR 1702 037
Zbl 0945.57006
This is an updated reprinting of [KS1],
as an invited contribution to volume 19 in the "Series on Knots and Everything",
edited by Stasiak, Katritch, and Kauffman.
 [KS1]
 Möbius Energies for Knots and Links,
Surfaces and Submanifolds,
with Rob Kusner, in
Geometric Topology,
International Press, 1996, pp 570604.
MR 98d:57014
Zbl 0888.57012
In this paper, we give a nicer explanation of the Möbiusinvariance of
the knot energy studied by Freedman, He and Wang, and extend it
to higherdimensional 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
energyminimizing configurations of all knots and links through eight
crossings.

Foams and CMC Surfaces
 [GKKRS]
 Coplanar kunduloids are nondegenerate,
with Karsten GroßeBrauckmann, Nick Korevaar,
Rob Kusner and Jesse Ratzkin.
Intl. Math. Res. Not. 2009:18 (2009), pp 3391–3416.
(Published online 5 Jun. 2009 as DOI:10.1093/imrn/rnp058.)
ArXiv 0712.1865 [math.DG].
We consider constant mean curvature (CMC) surfaces in Euclidean space,
and prove that each embedded surface with genus zero and finitely many
coplanar ends is nondegenerate: it has no nontrivial squareintegrable
solutions to the Jacobi equation, the linearization of the CMC
condition. This implies that the moduli space of such coplanar surfaces is
a realanalytic manifold and that a neighborhood of these in the full CMC
moduli space is also a manifold. It also implies (infinitesimal and local)
rigidity in the sense that the asymptotes map is an analytic immersion
on these spaces. We also show the classifying map
of [GKS4] is a diffeomorphism.
 [GKS4]
 Coplanar constant mean curvature surfaces,
with Karsten GroßeBrauckmann and Rob Kusner.
Comm. Anal. Geom. 15:5 (2008) pp. 985–1023.
ArXiv math.DG/0509210.
We consider constant mean curvature surfaces of finite topology, properly
embedded in threespace in the sense of Alexandrov. Such surfaces with three
ends and genus zero were constructed and completely classified by the authors
in [GKS3]. Here we extend the arguments to the case of an
arbitrary number of ends, under the assumption that the asymptotic axes of the
ends lie in a common plane: we construct and classify the entire family of
these coplanar constant mean curvature surfaces.
 [AMS]
 When Soap Bubbles Collide,
with Colin Adams and Frank Morgan.
Amer. Math. Monthly, 114:4, April 2007, pp. 329–337.
ArXiv math.DG/0412020.
Can you fill nspace with a froth of "soap bubbles"
that meet at most n at a time? Not if they have bounded diameter,
as follows from Lebesgue's Covering Theorem.
We provide some related results and conjectures.
 [S19]
 A Complete Family of CMC Surfaces.
In Integrable Systems, Geometry and Visualization, 2005, pp 237245.
This is a summary of the results
of [GKS3] and [GKS4]—in
particular, the classification of embedded CMC surfaces with coplanar
ends—published in the proceedings of the conference at Kyushu Univ.,
Fukuoka in November 2004.
 [HKRS]
 The structure of foam cells: Isotropic Plateau polyhedra,
with Sascha Hilgenfeldt, Andrew M. Kraynik and Douglas A. Reinelt.
Europhys. Lett., 67:3, 2004, pp 484490.
(Published online 1 Aug. 2004 as DOI:10.1209/epl/i2003102957.)
We present a meanfield theory for the diffusive coarsening of
threedimensional foams, based on idealized foam cells with
F faces. Although these exist only for values of F
corresponding to the Platonic solids, they seem to give surprisingly
good predictions about the average behavior of Ffaced
cells in real foams.
 [WKC+]
 Pressures in Periodic Foams,
with Denis Weaire, Norbert Kern, Simon J. Cox and Frank Morgan.
Proc. R. Soc. London A 460:2042, February 2004, pp 569579.
(Published online 8 Dec. 2003 as
DOI:10.1098/rspa.2003.1171.)
MR 2004k:74072
Zbl 1041.74049
We show that pressures are periodic for any periodic foam,
and that any planar foam with congruent bubbles is a (possibly sheared)
hexagonal honeycomb with equalpressure bubbles.
 [GKS3]
 Triunduloids:
Embedded Constant Mean Curvature Surfaces with Three Ends and Genus Zero,
with Karsten GroßeBrauckmann and Rob Kusner.
J. reine angew. Math. 564, 2003, pp 3561.
(DOI:10.1515/crll.2003.093)
ArXiv math.DG/0102183.
MR 2005a:53009
Zbl 1058.53005
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.
 [GKS2]
 Constant Mean Curvature Surfaces with Three Ends,
with Karsten GroßeBrauckmann and Rob Kusner,
Proc. Natl. Acad. Sci. 97:26, 2000 Dec 19,
pp 1406714068.
(DOI:10.1073/pnas.97.26.14067)
ArXiv math.DG/9903101.
MR 2001j:53009
Zbl 0980.53011
We announce the classification of triunduloids
given in [GKS3].
 [FTY+]
 Tomographic Imaging of Foam,
with Fetterman, Tan, Ying, Stack, Marks, Feller, Cull, Munson, Thoroddsen
and Brady,
Optics Express 7:5, 2000 Aug 28, pp 186197.
We explore the use of visuallight tomography to create
threedimensional 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.
 [AST]
 Foam Evolution: Experiments and Simulations,
with Hassan Aref and Sigurdur T. Thoroddsen,
in Proc. NASA 5th Microgravity Fluid Physics Conf.,
Aug 2000, pp 99100.
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.
 [S10]
 New Tetrahedrally ClosePacked Structures,
in Proc. Eurofoam 2000 (Delft), June 2000, pp 111119.
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
3manifolds, 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 threemanifolds.
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.
 [S9]
 Foams and Bubbles: Geometry and Simulation,
Intl. J. Shape Modeling, 5:1, 1999, pp 101114.
(Available online as DOI:10.1142/S0218654399000101.)
This invited contribution to a special issue edited
by Michele Emmer is adapted and updated from
[S7].
 [S7]
 The Geometry of Bubbles and Foams,
in Foams and Emulsions (NATO ASI volume E 354)
(MR 2000a:76006
Zbl 0954.76002),
Kluwer, 1999, pp 379402.
MR 2000b:53015
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
constantmeancurvature 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.
 [OS]
 The betaSn Dual Structure: A 4Connected Net
Based on a Packing of Simple Polyhedra with 18 Faces,
with Michael O'Keeffe,
Z. Kristallographie 213, 1998, pp 374376.
This crystallography paper describes a new threedimensional
structure arising out of discussions on foam structures and
their relations to crystals.
 [GKS1]
 Constant Mean Curvature Surfaces with Cylindrical Ends,
with Karsten GroßeBrauckmann and Rob Kusner,
in Mathematical Visualization
(MR 99k:65005
Zbl 0899.00041),
Springer, 1998, pp 107116.
MR 1677 699
Zbl 0940.68150
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.
 [BS]
 Using Symmetry Features of the
Surface Evolver to Study Foams, with Ken Brakke,
in Visualization and Mathematics
(MR 99g:68212
Zbl 0898.53001),
Springer, 1997, pp 95117.
MR 1607 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 WeairePhelan foam, which
is a better partition of space into equalvolume cells (see [KS2]).
 [SM]
 Open Problems in SoapBubble Geometry,
editor, with Frank Morgan,
International J. Math. 7:6, 1996, pp 833842.
(DOI:10.1142/S0129167X9600044X)
MR 98a:53014
Zbl 0867.53009
This list collects and organizes a long list of open problems posed by
participants at a special session on SoapBubble Geometry at
the AMS MathFest in Burlington in 1994, as well as further
problems suggested by the editors.
 [KS2]
 Comparing the WeairePhelan EqualVolume Foam to Kelvin's Foam,
with Rob Kusner, Forma 11:3, 1996, pp 233242.
Reprinted in The Kelvin Problem,
Taylor & Francis, 1996, pp 7180.
MR 99e:52031
Zbl 1017.52502
Lord Kelvin conjectured a foam structure as the optimal partition
of space into equalvolume cells, with least surface area.
A century later, Weaire and Phelan discovered an equalvolume 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 WeairePhelan foam does beat
Kelvin's foam.

Sphere Eversions and Willmore Energy
 [FS]
 Visualizing a Sphere Eversion,
with George Francis.
IEEE Transactions on
Visualization and Computer Graphics, 10:5, 2004, pp 509515.
(Published online as DOI:10.1109/TVCG.2004.33
in the special issue on Mathematical Visualization, edited
by K. Polthier and H.C. Hege.)
The mathematical process of everting a sphere
(turning it insideout allowing selfintersections) is a grand
challenge for visualization because of the complicated, ever
changing internal structure. We have computed an optimal
minimax eversion, requiring the least bending energy. Here
we discuss techniques we used to help visualize this eversion
for visitors to virtual environments and viewers of our video
``The Optiverse'' [SFL].
 [KSS+]
 Turning a Snowball Inside Out:
Mathematical Visualization at the 12foot Scale,
with Alex Kozlowski, Dan Schwalbe, Carlo H. Séquin and Stan Wagon.
Proceedings of Bridges 2004,
Southwestern Coll., Kansas, 2004, pp 2736.
At the 2004 International Snow Sculpting Championships in Breckenridge,
we carved a 12foot tall representation of the Morin surface—the
halfway point of a classical sphere eversion process.
This paper describes the design and realization of this piece
of largescale mathematical visualization.
 [FLS2]
 Making the Optiverse: A Mathematician's Guide to AVN,
a RealTime Interactive Computer Animator,
with George Francis and Stuart Levy.
In Mathematics, Art, Technology, Cinema
(
Zbl 1137.00002),
Zbl 1137.00002
Springer, 2003, pp 3952.
Our 1998 video
``The Optiverse''
[SFL]
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.
 [S13]
 Sphere Eversions: from Smale through "The Optiverse".
In Mathematics and Art:
Mathematical Visualization in Art and Education
(Maubeuge 2000)
(MR 2003g:00017
Zbl ?),
Springer, 2002, pp 201212 and 311313.
This is a revised and updated version of [S8].
 [FLS1]
 The Optiverse: una guida ai matematici per AVN, programma
interattivo di animazione,
with George Francis and Stuart Levy.
In Matematica, arte, tecnologia, cinema
(MR 2004d:00018
Zbl 0988.00002),
Springer, 2002, pp 3751.
An Italian translation of [FLS2].
 [S8]
 "The Optiverse" and Other Sphere Eversions,
in ISAMA 99, Univ. Basque Country, 1999, pp 471479.
Reprinted in Bridges 1999,
Southwestern Coll., Kansas, 1999, pp 265274.
Fullcolor version
published in
Visual
Mathematics, 1:3, September 1999.
ArXiv eprint math.GT/9905020;
also available in an HTML version.
(Zbl 0965.57028)
For decades, the sphere eversion has been a classic subject for
mathematical visualization. Our 1998 video "The Optiverse"
[SFL] shows
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.
 [SFL]
 The Optiverse,
with George Francis and Stuart Levy,
in VideoMath Festival at ICM'98
(MR 2000j:01030/2006i:00009
Zbl 0909.00003/1071.00001),
Springer, 1998, 7minute video.
This video shows the minimax sphere eversions described in
[FSK+] and
[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.
 [FSH]
 Computing Sphere Eversions,
with George Francis and Chris Hartman,
in Mathematical Visualization
(MR 99k:65005
Zbl 0899.00041),
Springer, 1998, pp 237255.
MR 1677 675
Zbl 0931.68128
This paper describes how to adapt the methods of
[FSK+] to compute
the minimax sphere eversions of higherorder symmetry which are
also shown in "The Optiverse" [SFL]. In particular, we must use
symmetry features of the evolver [BS] to perform the computations.
 [FSK+]
 The Minimax Sphere Eversion,
with George Francis, Rob B. Kusner, Ken A. Brakke,
Chris Hartman, and Glenn Chappell,
in Visualization and Mathematics
(MR 99g:68212
Zbl 0898.53001),
Springer, 1997, pp 320.
MR 1607 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.
 [HKS]
 Minimizing the Squared Mean Curvature Integral for Surfaces in
Space Forms, with Lucas Hsu and Rob Kusner,
Experimental
Math. 1:3, 1992, pp 191207.
MR 94a:53015
Zbl 0778.53001
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
lipid vesicles.

Discrete Differential Geometry and Meshing
 [S31]
 Lifting Spherical Cone Metrics.
In Discrete Differential Geometry,
organized by Bobenko, Kenyon, Schröder and Ziegler,
Oberwolfach Reports 9:3 (2012),
pp 2118–2121.
(DOI: 10.4171/OWR/2012/34)
This report on a lecture at the 2012 Oberwolfach Workshop
explains how to start with any metric of curvature at least 1
on S^{2} and find a Hopf lift, a metric on S^{3}
also with curvature at least 1. Similarly, we can lift a metric
on a "bad" pqorbifold to S^{3} along a Seifert
fibration. The examples arising from spherical cone metrics are
useful for understanding lowvalence triangulations of S^{3}.
 [IKRSS]
 There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems,
with Ivan Ismestiev, Rob Kusner, Günter Rote and Boris Springborn.
ArXiv 1207.3605
[math.CO]. Geometriae Dedicata, 166:1, 2013, pp 15–29.
(Published online 21 Sep. 2012 as DOI:
10.1007/s1071101297825)
There is no 5,7triangulation of the torus, that is, no triangulation with
exactly two exceptional vertices, of degree 5 and 7. Similarly, there is
no 3,5quadrangulation. The vertices of a 2,4hexangulation of the torus
cannot be bicolored. Similar statements hold for 4,8triangulations and
2,6quadrangulations. We prove these results, of which the first two are
known and the others seem to be new, as corollaries of a theorem on the
holonomy group of a euclidean cone metric on the torus with just two cone
points. We provide two proofs of this theorem: One argument is metric in
nature, the other relies on the induced conformal structure and proceeds
by invoking the residue theorem. Similar methods can be used to prove a
theorem of Dress on infinite triangulations of the plane with exactly two
irregular vertices. The nonexistence results for torus decompositions
provide infinite families of graphs which cannot be embedded in the torus.
 [JLSS]
 Triangulations,
organizer, with Bus Jaco, Frank Lutz and Paco Santos.
Oberwolfach Reports
9:2, 2012, pp 1405–1486.
(DOI: 10.4171/OWR/2012/24)
This report collects extended abstracts from
the workshop held April 29–May 5, 2012,
with an introduction from the organizers.
 [BSSZ]
 Discrete Differential Geometry,
editor, with Alexander I. Bobenko, Peter Schröder
and Günter M. Ziegler.
Oberwolfach Seminars 38, Birkhäuser, 2008, x+341 pp.
This book documents the lecture courses at the Oberwolfach
Seminar "Discrete Differential Geometry" held in May–June 2004.
 [S23]
 Curvatures of discrete curves and surfaces.
In Géométrie discrète,
Société Mathématique de France,
Journée Annuelle, 2009, pp. 45–58.
The basic notions in differential geometry are curvatures, for instance
those of a smooth curve or surface in space. If we approximate the
smooth object by a polygonal curve or polyhedral surface, then there
are many possible discretizations of curvature, all of which converge to
the smooth notion. The idea behind the relatively new field of Discrete
Differential Geometry is that one should pick a discretization
which—even at the discrete level—captures some of the properties of
the smooth notion, such as integral relations. We consider the simplest
possible examples—the curvature of a curve and the mean and Gauss
curvatures of a surface—and use these to show that there is no one
best notion of discrete curvature. Instead, the proper discretization
depends on which features of the smooth picture one wants to see at the
discrete level.
These notes for a lecture in Montpellier are based on [S21]
and [S22].
 [S22]
 Curvatures of smooth and discrete surfaces.
In Discrete Differential Geometry [BSSZ],
Birkhäuser, 2008, pp. 175–188.
ArXiv 0710.4497 [math.DG].
We discuss notions of Gauss curvature and mean curvature
for polyhedral surfaces. The discretizations are guided
by the principle of preserving integral relations for curvatures,
like the Gauss/Bonnet theorem and the meancurvature force balance equation.
 [BKSZ2]
 Discrete Differential Geometry,
organizer, with Alexander I. Bobenko, Richard W. Kenyon
and Günter M. Ziegler.
Oberwolfach Reports 6:1, 2009, pp 75–144.
(DOI: 10.4171/OWR/2009/02)
This report collects extended abstracts from
the workshop held January 11–17, 2009,
with an introduction from the organizers.
 [BKSZ1]
 Discrete Differential Geometry,
organizer, with Alexander I. Bobenko, Richard W. Kenyon
and Günter M. Ziegler.
Oberwolfach Reports 3:1, 2006, pp 653727.
MR 2278 898
This report collects extended abstracts from
the workshop held March 511, 2006,
with an introduction from the organizers.
 [EGSU2]
 Building spacetime meshes over arbitrary spatial domains,
with Jeff Erickson and Damrong Guoy and Alper Üngör.
Engineering with Computers 20:4 (2005) pp. 342353.
(Published online 25 May 2005 as DOI:10.1007/s0036600503030.)
ArXiv cs.CG/0206002.
We present an algorithm to construct meshes suitable for
spacetime discontinuous Galerkin finiteelement 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 spacetime product domain,
in constant time per element. Our algorithm avoids the limitations
of previous methods by carefully adapting the durations of spacetime
elements to the local quality and feature size of the underlying space mesh.
A preliminary version appeared as [EGSU1].
 [ACE+]
 Spacetime meshing with adaptive refinement and coarsening,
with Reza Abedi, ShuoHeng Chung, Jeff Erickson, Yong Fan, Michael Garland,
Damrong Guoy, Robert Haber, Shripad Thite, and Yuan Zhou.
Proceedings of the 20th Annual ACM Symposium on Computational Geometry,
2004, pp 300309. (Published online as
DOI:10.1145/997817.997863.)
 [ESU]
 Tiling space and slabs with acute tetrahedra,
with David Eppstein and Alper Üngör.
Comput. Geom.: Theory and Appl. 27:3, 2004, pp 237255.
(Published online 18 Feb. 2004 as DOI:10.1016/j.comgeo.2003.11.003.)
ArXiv cs.CG/0302027.
MR 2004k:52029
Zbl 1054.65020
We show it is possible to tile threedimensional space using
only tetrahedra with acute dihedral angles.
We present several constructions to achieve this, including one
in which all dihedral angles are less than 78 degrees,
another which tiles a slab in space. Several of our examples
come from tetrahedrally closepacked (TCP) crystal structures.
 [EGSU1]
 Building spacetime meshes over arbitrary spatial domains
(extended abstract),
with Jeff Erickson and Damrong Guoy and Alper Üngör.
In Proceedings of the
11th International
Meshing Roundtable, Sandia, 2002, pp 391402.
ArXiv cs.CG/0206002.
We present an algorithm to construct meshes suitable for
spacetime discontinuous Galerkin finiteelement methods.
The full version of this paper appeared by invitation as
[EGSU2].
 [CDES2]
 Dynamic Skin Triangulation,
with HoLun Cheng, Tamal K. Dey, and Herbert Edelsbrunner,
Discrete and Computational Geometry 25, 2001, pp 525568.
MR 2002e:52018
Zbl 0984.68172
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.
 [CDES1]
 Dynamic Skin Triangulation (extended abstract),
with HoLun Cheng, Tamal K. Dey, and Herbert Edelsbrunner,
Proc. 12th Ann. ACMSIAM Sympos. Discrete Alg.
(MR 2003i:68002),
0988.65016
2001 Jan, pp 4756.
MR 1958 391
Zbl 0980.53011
This is the 10page announcement of [CDES2].

Math Visualization, Art and Polyhedra
 [S34]
 Diagrams and Visualization in Mathematics.
In Sichtbarmachen: Praktiken visuellen Denkens,
Diaphenes, Zürich, to appear.
We consider various uses of diagrams and visualization in
mathematics from the point of view of a practicing mathematician.
This article is based on my presentation at the session on
"Practices of Visual Thinking: Proving – Demonstration" at the conference
Sichtbarmachung: Praktiken visuellen Denkens held in Berlin
in November 2012.
 [S33]
 Blasencluster und Polyeder.
In Alles Mathematik: Von Pythagoras zu Big Data,
4th expanded edition, Springer Spektrum, 2016.
(DOI: 10.1007/9783658099909)
This is a German writeup of a generalinterest talk I have often given,
about the relationshiop between convex deltahedra and candidates for
soapfilm singularities.
 [S32]
 Nonspherical Bubble Clusters.
Bridges Proceedings (Seoul), 2014, pp. 453–456.
Soap bubbles have always captured the imagination of artists as well as
of children. We present computer graphics renderings of some small
bubble clusters of mathematical interest. A single soap bubble is
a perfectly round sphere; it seems that the soap films in (stable) clusters
of small numbers of bubbles are always pieces of spheres. We focus on a cluster
of six bubbles where this is not the case  in particular its central film
is a saddleshaped minimal surface. My computergraphics rendering of
this cluster dates from 1990. After it was featured in
Ziegler's 2013 book of mathematical pictures, I returned to it,
printing it for exhibition for the first time and describing it here.
 [SPP]
 Visualization,
with Ulrich Pinkall and Konrad Polthier.
In MATHEON – Mathematics for Key Technologies,
EMS, 2014, pp. 381–392.
We outline the research and outreach activities in
mathematical visualization carried out within
the DFG Research Center MATHEON.
We focus on discrete conformal maps, discrete smoke rings,
domain coloring, geometric threemanifolds, and virtual reality.
 [PSZH]
 Mathematical Visualization,
with Konrad Polthier, Günter M. Ziegler and HansChristian Hege.
In MATHEON – Mathematics for Key Technologies,
EMS, 2014, pp. 335–339.
We give a short overview of the research carried out in
application area F "Visualization"
of the DFG Research Center MATHEON.
 [S30]
 Mathematical Pictures: Visualization, Art and Outreach.
In Raising Public Awareness of Mathematics,
Springer, 2012, pp. 279–293.
(DOI: 10.1007/9783642257100_21)
Mathematicians have used pictures for thousands of years,
to aid their own research and to communicate their results
to others. We examine the different types of pictures
used in mathematics, their relation to mathematical art
and their use in outreach activities.
(This article is based on a talk at the conference in Óbidos.)
 [S29]
 Pleasing Shapes for Topological Objects.
In Mathematics and Modern Art,
Springer, Proc. in Math. 18, 2012, pp. 153–165.
(DOI: 10.1007/9783642244971_13)
This is the English original of [S27],
an expanded version of my talk at Michele Emmer's conference
in Venice in 2010, about using geometric optimization to find
pleasing forms for topological objects.
 [S28]
 Conformal Tiling on a Torus.
Bridges Proceedings (Coimbra), 2011, pp 593–596.
Given a regular tiling of the torus, this paper describes how to depict it
on a torus in space with as much conformal symmetry as possible, using
Pinkall's Hopf tori.
 [S27]
 Affascinanti forme per oggetti topologici.
In Matematica e cultura 2011,
Springer Italia, 2011, pp. 145–158.
This is an expanded version of my talk at Michele Emmer's conference
in Venice in 2010, about using geometric optimization to find
pleasing forms for topological objects.
 [S26]
 Minimal Flowers.
Bridges Proceedings (Pécs), 2010, pp 395–398.
This paper describes my sculptures Minimal Flower 3, an
homage to Brent Collins, and its new cousin, Minimal Flower 4. They are
both constructed as minimal surfaces spanning certain knotted boundary
curves, with threefold and fourfold rotational symmetry, respectively.
 [GS3]
 The Borromean Rings: A video about the new IMU logo,
with Charles Gunn,
Bridges Proceedings (Leeuwarden), 2008, pp. 63–70.
This paper describes our video
The Borromean Rings: A new logo for the IMU [GS2],
which was premiered at the opening ceremony of the last International Congress.
The video explains some of the mathematics behind the logo of
the International Mathematical Union,
which is based on the tight configuration of the Borromean rings.
This configuration has pyritohedral symmetry, so the video
includes an exploration of this interesting symmetry group.
 [GS2]
 The Borromean Rings:
A new logo for the IMU,
with Charles Gunn,
in MathFilm Festival 2008,
Springer, 2008, 5minute video.
This video, premiered at ICM 2006, explains the mathematics behind the
new IMU logo. The logo depicts the Borromean rings (three linked rings with
the property that no pair is linked) in the form they have when tied tight.
This tight configuration has pyritohedral symmetry, with the rings
lying in orthogonal planes. The video starts with an exploration of this
symmetry group, featuring Fuller's "jitterbug". A fivecoloring of
the icosahedron edges shows how the pyritohedral group fits into the
icosahedral group. The Borromean rings then appear as three golden
rectangles, with pyritohedral symmetry.
After an interlude showing how the rings have been used in many cultures
as a symbol of interconnectedness, the video depicts a tightening process.
It preserves the symmetry and leads to the tight configuration, which is
explored with various rendering styles, including bubblelike transparency
and wovenrope textures.
 [S20]
 Spherical Duals and Minkowski Sums.
Bridges Proceedings (London), 2006, pp 117–122.
We examine the Gauss map of a polyhedron, giving a spherical dual network.
When this network is labeled with edge lengths, the original polyhedron
can be recovered. Following a suggestion of Zongker and Hart, we show
that the Minkowski sum of two polyhedra can be obtained simply by overlaying
their labeled spherical duals.
 [S18]
 The Aesthetic Value of Optimal Geometry.
In The Visual Mind II
(MR 2005m:00004
Zbl ?),
MIT Press, 2005, pp 547563.
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.
 [S17]
 Optimal Geometry as Art.
Symmetry: Art and Science 2004: 14, 2004, pp 234237.
This short summary of the material from [S18]
appears in the procedings of the ISIS Symmetry conference,
Budapest/Tihany, October 2004.
 [S15]
 Optimal Geometry as Art.
In Proceedings of ISAMA/Bridges 2003, Granada, 2003, pp 529532.
This essay was originally written,
by invitation, to appear on the web during Math Awareness Month, April 2004.
It considers various relations between art and mathematics,
especially the mathematics of optimization problems in geometry.
 [FGKSS]
 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
(MR 2004j:00028
Zbl 1014.00012),
Springer, 2003, pp 305315 and 429.
Zbl 1097.68650
We describe a collaboration between
mathematicians interested in visualizing curved threedimensional spaces and
researchers building nextgeneration virtualreality environments such as
ALICE, a sixsided, rigidwalled virtualreality chamber.
This environment integrates activestereo imaging, wireless positiontracking
and wirelessheadphone 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 eightfold way;
these eight threedimensional geometries are conjectured to suffice for
describing all possible threedimensional manifolds or universes.
 [GS1]
 Cubic Polyhedra,
with Chaim GoodmanStrauss,
in Discrete Geometry (Monogr. Textb. Pure Appl. Math. 253)
(MR 2004i:00017
Zbl 1034.52002),
Marcel Dekker, 2003, pp 305–330.
MR 2004k:52020
Zbl 1048.52008
Here, 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 semiregular polyhedra. We also consider the
somewhat larger class of all discrete minimal surfaces in the cubic lattice.
 [S11]
 Rescalable RealTime Interactive Computer Animations.
In
Multimedia
Tools for Communicating Mathematics
(MR 2003b:68205
Zbl 1083.00005),
Springer, 2002, pp 311314.
Animations are one of the best tools for communicating
threedimensional geometry, especially when it changes
in time through a homotopy. For specialpurpose animations,
custom software is often necessary to achieve realtime 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 highend graphics
workstations is available even on laptops.
 [AS]
 Visualization of soap bubble geometries,
with Fred Almgren, Leonardo 24:3/4, 1992, pp 267271.
Reprinted in The Visual Mind
(MR 94h:00013
Zbl 1069.00009),
MIT Press, 1993, pp 7983.
MR 1255 841
Zbl 0803.51021
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].
 [S1]
 Generating and Rendering FourDimensional Polytopes,
The Mathematica Journal 1:3, 1991, pp 7685.
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 fourdimensional polytopes via stereographic projection
as bubble clusters in threespace. It is illustrated with computer
graphics using the algorithm described in [AS].

Miscellany
 [S25]
 Knoten.
In Besser als Mathe,
Vieweg+Teubner, 2010, pp. 239–241.
Reprinted in part as KnotenRätsel,
Mitteilungen der DMV 18:1 (2010) p 58.
This presents (in German) an exercise on knot equivalence
for highschool students, originally used for Matheon's
mathematical advent calendar competition in 2006.
 [S24]
 Meeks' Proof of Osserman's Theorem,
in Arbeitsgemeinschaft: Minimal Surfaces, organized by Meeks and Weber,
Oberwolfach Reports 6:4 (2009), pp. 2561–2563.
(DOI:10.4171/OWR/2009/45)
This report on a lecture at the 2009 Oberwolfach Arbeitsgemeinschaft
"Minimal Surfaces" outlines Bill Meeks' 1995 proof of Osserman's 1964 theorem
on minimal surfaces with finite total curvature. Here we focus on two
lemmas which were left implicit in Meeks' paper and which were of interest
to the audience at Oberwolfach. One of these describes the limiting behavior
of certain light open maps from an open annulus to a closed surface.
 [MS]
 In Memoriam Frederick J. Almgren Jr., 19331997:
On Being a Student of Almgren's, with Frank Morgan,
Experimental Math. 6:1, 1997, pp 810.
MR 1464 578
Zbl 0803.51021
These descriptions of what is was like to be Almgren's
student—published alongside reminiscences by mathematicians
(David Epstein, Elliot H. Lieb, Jean Taylor, Robert Almgren,
Robert Kusner, Albert Marden) 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.
 [CGLS]
 Elliptic and Parabolic Methods in Geometry,
editor, with Ben Chow, Bob Gulliver and Silvio Levy. Published by AK
Peters, 1996.
MR 97f:58004
Zbl 0853.00042
This book is the proceedings volume from a workshop we organized,
held in Minneapolis 2327 May 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.
 [S5]
 Sphere Packings Give an Explicit Bound for the Besicovitch
Covering Theorem,
J. Geometric Analysis 4:2, 1994, pp 219231.
MR 95e:52038
Zbl 0797.52011
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 spherepacking 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.
 [MSL]
 Monotonicity Theorems for TwoPhase Solids,
with Frank Morgan, Francis Larché,
Arch. Rat. Mech. Anal. 124:4, 1994, pp 329353.
(DOI:10.1007/BF00375606)
MR 94m:73072
Zbl 0785.73006
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.
 [S4]
 Computing Hypersurfaces Which Minimize Surface Energy
Plus Bulk Energy, in Motion by Mean Curvature and Related Topics,
de Gruyter, 1994, pp 186197.
MR 95h:49072
Zbl 0804.49034
My dissertation proved an approximation theorem for areaminimizing
hypersurfaces in the context of geometric measure theory. This
kind of approximation is especially useful to prove the feasibility
of algorithms to find areaminimizing 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.
 [S3]
 Using MaxFlow/MinCut to Find AreaMinimizing Surfaces,
in Computational Crystal Growers Workshop
(MR 94f:58007/MR 94f:58052)
AMS Sel. Lect. Math.,
1992, pp 107110 plus video.
This video uses algorithm animation to illustrate how the algorithm
described in my dissertation uses maxflow/mincut techniques to
find approximations to areaminimizing surfaces, without knowing
their topology in advance; it appears in the proceedings of a conference
organized by Jean Taylor.
 [S2]
 Crystalline Approximation: Computing Minimum
Surfaces via Maximum Flows,
in Computing Optimal Geometries
(MR 93a:65021),
AMS Selected Lectures in Math.,
1991, pp 5962 plus video.
This video shows, using a twodimensional example, how the approximation
theorem proved in my dissertation works to find approximately
areaminimizing surfaces; it appears in the proceedings of an AMS
special session organized by Almgren and Taylor.
 [ABST]
 Computing Soap Films and Crystals,
with Fred Almgren, Ken Brakke, Jean Taylor,
in Computing Optimal Geometries
(MR 93a:65021),
AMS Selected Lectures in Math., 1991, 14minute 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 threedimensional Voronoi
cell code, and crystalline minimal surfaces computed by Taylor.
 [ST]
 Animating the Heat Equation:
A Case Study in Mathematica Optimization,
with Matt Thomas, The Mathematica Journal 1:1, 1990, pp 8084.
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 thousandfold speedup. The main thrust of the
published joint article became a description of these optimization techniques.
 [LMS]
 Some Results on the Phase Behavior in Coherent Equilibria,
with Francis Larché, Frank Morgan,
Scripta Metallurgica 24:3, 1990, pp 491493.
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].
 [S0]
 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 leastarea surface on a given
boundary by a finite networkflow 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],
[S4], [S5],
but the main section of the work has not yet been published elsewhere.