Updated: 2016 January 24

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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 minimum-length 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 self-contact 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 C¹-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 self-contacts (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 link-homotopy 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, infinite-dimensional version of the Kuhn--Tucker theorem. We use this to prove that every critical link is C¹ 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 Ropelength-Critical Clasps, with Nancy Wrinkle. In Physical and Numerical Models in Knot Theory (MR 2006h:00009 Zbl 1085.57002), World Sci., 2005, pp. 565-580. 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 1335-1348. 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 (ropelength-minimizing) 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 257-286. (Published online 17 Jun. 2002 as DOI:10.1007/s00222-002-0234-y.) 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 181--186. 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 p-energies, 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 237-238. (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, 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].

[KS4]

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. ArXiv dg-ga/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 non-zero for polygonal curves, and thus are easier to measure in computer simulations of knots minimizing their ropelength (length divided by thickness).

[KS3]

Möbius-invariant Knot Energies, with Rob Kusner, in Ideal Knots (MR 2000j:57018), World Scientific, 1998, pp 315-352.    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 570-604.    MR 98d:57014 Zbl 0888.57012
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 crossings.

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Foams and CMC Surfaces

[GKKRS]

Coplanar k-unduloids are nondegenerate, with Karsten Große-Brauckmann, 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 square-integrable solutions to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surfaces is a real-analytic 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 Große-Brauckmann 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 three-space 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 n-space 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 237-245.
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 484-490. (Published online 1 Aug. 2004 as DOI:10.1209/epl/i2003-10295-7.)
We present a mean-field theory for the diffusive coarsening of three-dimensional 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 F-faced 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 569-579. (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 equal-pressure bubbles.

[GKS3]

Triunduloids: Embedded Constant Mean Curvature Surfaces with Three Ends and Genus Zero, with Karsten Große-Brauckmann and Rob Kusner. J. reine angew. Math. 564, 2003, pp 35-61. (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 Große-Brauckmann and Rob Kusner, Proc. Natl. Acad. Sci. 97:26, 2000 Dec 19, pp 14067-14068. (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 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.

[AST]

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.

[S10]

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.

[S9]

Foams and Bubbles: Geometry and Simulation, Intl. J. Shape Modeling, 5:1, 1999, pp 101-114. (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 379-402.    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 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.

[OS]

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.

[GKS1]

Constant Mean Curvature Surfaces with Cylindrical Ends, with Karsten Große-Brauckmann and Rob Kusner, in Mathematical Visualization (MR 99k:65005 Zbl 0899.00041), Springer, 1998, pp 107-116.    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 95-117.    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 Weaire-Phelan foam, which is a better partition of space into equal-volume cells (see [KS2]).

[SM]

Open Problems in Soap-Bubble Geometry, editor, with Frank Morgan, International J. Math. 7:6, 1996, pp 833-842. (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 Soap-Bubble Geometry at the AMS MathFest in Burlington in 1994, as well as further problems suggested by the editors.

[KS2]

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.    MR 99e:52031 Zbl 1017.52502
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 Kelvin's foam.

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Sphere Eversions and Willmore Energy

[FS]

Visualizing a Sphere Eversion, with George Francis. IEEE Transactions on Visualization and Computer Graphics, 10:5, 2004, pp 509-515. (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 inside-out allowing self-intersections) 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 12-foot Scale, with Alex Kozlowski, Dan Schwalbe, Carlo H. Séquin and Stan Wagon. Proceedings of Bridges 2004, Southwestern Coll., Kansas, 2004, pp 27-36. At the 2004 International Snow Sculpting Championships in Breckenridge, we carved a 12-foot 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 large-scale mathematical visualization.

[FLS2]

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 ( Zbl 1137.00002), Zbl 1137.00002 Springer, 2003, pp 39-52.
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 201-212 and 311-313.
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 37-51.
An Italian translation of [FLS2].

[S8]

"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 Visual Mathematics, 1:3, September 1999. ArXiv e-print 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, 7-minute 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 237-255.    MR 1677 675 Zbl 0931.68128
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.

[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 3-20.    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 191-207.    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.

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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 S² and find a Hopf lift, a metric on S³ also with curvature at least 1. Similarly, we can lift a metric on a "bad" pq-orbifold to S³ along a Seifert fibration. The examples arising from spherical cone metrics are useful for understanding low-valence triangulations of S³.

[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/s10711-012-9782-5)
There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. 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 non-existence 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 mean-curvature 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 653-727.    MR 2278 898
This report collects extended abstracts from the workshop held March 5-11, 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. 342-353. (Published online 25 May 2005 as DOI:10.1007/s00366-005-0303-0.) ArXiv cs.CG/0206002.
We present an algorithm to construct meshes suitable for spacetime 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 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, Shuo-Heng 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 300-309. (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 237-255. (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 three-dimensional 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 close-packed (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 391-402. ArXiv cs.CG/0206002.
We present an algorithm to construct meshes suitable for spacetime discontinuous Galerkin finite-element methods. The full version of this paper appeared by invitation as [EGSU2].

[CDES2]

Dynamic Skin Triangulation, with Ho-Lun Cheng, Tamal K. Dey, and Herbert Edelsbrunner, Discrete and Computational Geometry 25, 2001, pp 525-568.    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 Ho-Lun Cheng, Tamal K. Dey, and Herbert Edelsbrunner, Proc. 12th Ann. ACM-SIAM Sympos. Discrete Alg. (MR 2003i:68002), 0988.65016 2001 Jan, pp 47-56.    MR 1958 391 Zbl 0980.53011
This is the 10-page announcement of [CDES2].

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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: 10.1007/978-3-658-09990-9)
This is a German write-up of a general-interest talk I have often given, about the relationshiop between convex deltahedra and candidates for soap-film 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 saddle-shaped minimal surface. My computer-graphics 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 three-manifolds, and virtual reality.

[PSZH]

Mathematical Visualization, with Konrad Polthier, Günter M. Ziegler and Hans-Christian 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: 10.1007/978-3-642-25710-0_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: 10.1007/978-3-642-24497-1_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 three-fold and four-fold 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, 5-minute 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 five-coloring 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 bubble-like transparency and woven-rope 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 547-563.
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: 1-4, 2004, pp 234-237.
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 529-532.
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 305-315 and 429.    Zbl 1097.68650
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.

[GS1]

Cubic Polyhedra, with Chaim Goodman-Strauss, 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 Real-Time Interactive Computer Animations. In Multimedia Tools for Communicating Mathematics  (MR 2003b:68205 Zbl 1083.00005), 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.

[AS]

Visualization of soap bubble geometries, with Fred Almgren, Leonardo 24:3/4, 1992, pp 267-271. Reprinted in The Visual Mind (MR 94h:00013 Zbl 1069.00009), MIT Press, 1993, pp 79-83.    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 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].

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Miscellany

[S25]

Knoten. In Besser als Mathe, Vieweg+Teubner, 2010, pp. 239–241. Reprinted in part as Knoten-Rätsel, Mitteilungen der DMV 18:1 (2010) p 58.
This presents (in German) an exercise on knot equivalence for high-school 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., 1933-1997: On Being a Student of Almgren's, with Frank Morgan, Experimental Math. 6:1, 1997, pp 8-10.    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 23-27 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 219-231.    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 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.

[MSL]

Monotonicity Theorems for Two-Phase Solids, with Frank Morgan, Francis Larché, Arch. Rat. Mech. Anal. 124:4, 1994, pp 329-353. (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 186-197.    MR 95h:49072 Zbl 0804.49034
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.

[S3]

Using Max-Flow/Min-Cut to Find Area-Minimizing Surfaces, in Computational Crystal Growers Workshop (MR 94f:58007/MR 94f:58052) 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.

[S2]

Crystalline Approximation: Computing Minimum Surfaces via Maximum Flows, in Computing Optimal Geometries (MR 93a:65021), 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.

[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, 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.

[ST]

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.

[LMS]

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].

[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 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], [S4], [S5], but the main section of the work has not yet been published elsewhere.