Mathematisches Forschungsinstitut OberwolfachReport No. 02/2009DOI: 10.4171/OWR/2009/02Discrete Differential GeometryOrganised byAlexander I. Bobenko, BerlinRichard Kenyon, Providence, RIJohn Sullivan, BerlinGünter M. Ziegler, BerlinJanuary 11th – January 17th, 2009Abstract. This is the collection of extended abstracts for the 26 lectures andthe open problems session at the second Oberwolfach workshop on DiscreteDifferential Geometry.Mathematics Subject Classification (2000): 52-xx, 53-xx, 57-xx.Introduction by the OrganisersDiscrete Differential Geometry is an active mathematical terrain where differen-tial geometry (the theory of smooth manifolds, providing notions of curvature,flows, integrability, etc.) interacts with discrete geometry (concerned with poly-hedral surfaces, frameworks and their rigidity, polytopes and their subdivisions,etc.), using tools and ideas from all parts of mathematics, including, for example,conformal geometry, integrable systems, algebraic combinatorics, mathematicalphysics (discrete electrodynamics, hydrodynamics and elasticity), computationalgeometry, and geometry processing.In view of two books entitled “Discrete Differential Geometry” – the proceed-ings of the 2004 Oberwolfach Seminar (Birkhäuser 2008) and the recent volume byBobenko & Suris (AMS 2008) – as well as a number of workshops in recent years(St. Petersburg 2003, Oberwolfach 2006, Berlin 2007) treating the subject, it canbe said that Discrete Differential Geometry has started to be a well-establishedmathematical discipline. Nevertheless, the boundaries of the field are not fixed;76 Oberwolfach Report 02quite to the contrary, it is branching out into new directions, and new and some-times unexpected connections and lines of development appear frequently – as canalso be seen in the following pages.The present collection of extended abstracts documents the lectures and theopen problems session at the second Oberwolfach workshop on Discrete DifferentialGeometry. It records a successful and very active workshop, and thus presents amulti-facetted picture of the field. We are grateful to all the participants of theWorkshop for their manifold contributions, and to the institute in Oberwolfachand all its staff for providing, once again, a perfect setting for this.Discrete Differential Geometry 77Workshop: Discrete Differential GeometryTable of ContentsJohannes Wallner (joint with Christian Müller and Helmut Pottmann)Semidiscrete Surface Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Yuri B. Suris (joint with Alexander I. Bobenko)Discrete Differential Geometry: Integrable structure . . . . . . . . . . . . . . . . . . 81Yann OllivierDiscrete Ricci curvature for metric spaces and Markov chains . . . . . . . . . 84Ivan IzmestievThe discrete Hilbert-Einstein functional: History and applications . . . . . 86Jürgen Richter-Gebert, Martin von GagernHyperbolization of ornaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Jean-Marc Schlenker (joint with Francesco Bonsante and Kirill Krasnov)Earthquakes on hyperbolic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Igor PakDiscrete square peg problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Feng LuoVolume and angle structures on 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . . 94Ileana Streinu (joint with Ciprian S. Borcea)Extremal configurations of revolute-jointed robot arms . . . . . . . . . . . . . . . . 95Richard KenyonDiscrete random interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Serge Tabachnikov (joint with Valentin Ovsienko, Richard Schwartz)The Pentagram map: a discrete integrable system . . . . . . . . . . . . . . . . . . . . 99Gaiane PaninaPiecewise linear saddle spheres on S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Boris Springborn (joint with A. I. Bobenko, U. Pinkall, P. Schröder)Discrete conformal equivalence for triangle meshes . . . . . . . . . . . . . . . . . . . 104Günter Rote (problem session chair)Open Problems in Discrete Differential Geometry . . . . . . . . . . . . . . . . . . . . 106Ulrich Brehm (joint with Wolfgang Kühnel)Lattice triangulations of 3-space and of the 3-torus . . . . . . . . . . . . . . . . . . 111Wolfgang Kühnel (joint with Ulrich Brehm)Lattice triangulations of 3-space, and PL curvature . . . . . . . . . . . . . . . . . . 11378 Oberwolfach Report 02Bernd SchulzeSymmetry as a sufficient condition for a finite flex . . . . . . . . . . . . . . . . . . . 115Ulrich Bauer (joint with Carsten Lange and Max Wardetzky)Persistence simplification of discrete Morse functions on surfaces . . . . . . 116Ulrike BückingApproximation of conformal mappings by circle patterns . . . . . . . . . . . . . . 119Herbert Edelsbrunner (joint with Dmitriy Morozov and Amit Patel)Stability of the fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Ethan D. BlochThe Angle Defect and Its Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Benjamin MatschkeSquare pegs and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Robert ConnellyOptimizing circle arrangements: questions and comments . . . . . . . . . . . . . 128Joseph O’Rourke (joint with Jin-ichi Itoh and Costin Vı̂lcu)Unfolding convex polyhedra via quasigeodesic source & star unfoldings . . 129Max Wardetzky (joint with M. Bergou, S. Robinson, B. Audoly andE. Grinspun)Geometric aspects of discrete elastic rods . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Mirela Ben-Chen (joint with Ofir Weber, Craig Gotsman)Complex barycentric coordinates for shape deformation . . . . . . . . . . . . . . . 134Ken StephensonCircle packing sampler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Discrete Differential Geometry 79AbstractsSemidiscrete Surface RepresentationsJohannes Wallner(joint work with Christian Müller and Helmut Pottmann)A semidiscrete surface representation x(u, k) is a mapping x : R × Z → R3, i.e.,a bivariate function with one continuous and one discrete parameter. It can beinterpreted as the limit of discrete surfaces (εZ) × Z → R3 as ε tends to zero.There is actually an entire spectrum of mappings Zr × Rs → Rd of which thepurely discrete and the purely continuous surfaces are the extremal cases, andwhich are discussed in depth by [2]. A major point of that theory is that forr ≥ 1, s ≥ 2 they represent the classical topic of transformations of surfaces, andthat the limit viewpoint allows us to consider transformations and especially theirpermutability theorems within the theory of discrete integrable systems.The low-dimensional case r = d = 1 turned out to be very interesting in itsown right when in [4] the approximation of surfaces by a sequence of single-curvedstrips was discussed from the viewpoint of geometry processing, motivated byapplications in architectural design. Such ‘developable strip models’ are recognizedas semidiscrete conjugate nets, and their theory and specializations to circular andconical nets are developed by [4]. A general discussion of relations between discretedifferential geometry and architectural design is given by [5].In the following we report on results of [4] concerning conjugate nets, as wellas material not yet published [3, 7].Our basic entity is a net which can be discrete, semidiscrete, or continuous, andwhich is a mapping “x” defined in Z × Z, or R × Z, or R × R, having values inR3. We write x(j, k), x(u, k), and x(u, v), respectively. The dependence on thecontinuous parameters is understood to be C2. We write ∂ux for derivatives w.r.t.smooth variables, and ∆kx for discrete derivatives, i.e., forward differences.Conjugate nets. These are defined by planarity of elementary (infinitesimal)quadrilaterals, which is expressed as the linear dependence of three vectors, namely∆j∆kx, ∆jx, ∆kx, or ∆k∂ux, ∆kx, ∂ux, or ∂u∂vx, ∂ux, ∂vx,in the three respective categories. For semidiscrete x this means that the ruledstrip generated by curves x|R×{k} and x|R×{k+1}, viz.,s(k) : R × [0, 1] → R3, (u, v) 7→ (1 − v)x(u, k) + v x(u, k + 1),is developable. Circularity of the net means that each strip has a family of inscribedcircles which are tangent to the boundary curves in corresponding points. Inthis way a developable strip is a 1D Jonas transform, which becomes a Darbouxtransform if the strip is in addition circular. The net is conical, if for each vertexthere is a right circular cone tangent to the two adjacent strips. It turns out thatcircular and conical strips are convertible into each other and possess a nice focaltheory similar to [6].80 Oberwolfach Report 02We have further investigated the circular nets with regard to equivalence, iso-thermicity, the Koenigs property, and Christoffel duality. It is instructive todemonstrate how properties like the incidence-geometric characterization of dis-crete Koenigs nets manifest themselves in the semidiscrete category. First thediscrete version: A discrete conjugate net is Koenigs ⇐⇒ The diagonals of ele-mentary quadrilaterals intersect in the vertices of yet another conjugate net. Forthe semidiscrete case it is first necessary to consider the limit of a planar 4-gonabcd with points e = ac ∩ bd, f = bc ∩ ad as shown, as a → b and c → d:ab cde f x h x + ∆x rIt is not difficult to see that the limit points have cross ratio −1, and that theinterpretation of abcd as an infinitesimal quad in a net yields the following: Asemidiscrete conjugate net x is Koenigs ⇐⇒ The net h which is defined by thecondition that cr(x, h, x + ∆x, r) = −1 is itself a conjugate net, where r(u, k) isthe singular point of the strip s(k) located on the ruling x(u, k)x(u, k + 1).Asymptotic nets. Also here there is much analogy between the three categories.Asymptotic nets (A-nets) are defined by the condition that the second derivativesw.r.t. both parameters are co-planar with the first derivatives. In the semidiscretecase this amounts to the condition that∂x, ∂2x, ∆x, ∆−xare co-planar, where ∆− is the backwards difference, and where we dropped the in-dices after ∂, ∆. Consequently the union of ruled surface strips {s(k)(R×[0, 1])}k∈Zassociated with an A-net is a smooth surface, provided it does not overfold, withthe strip boundaries being asymptotic curves of each adjacent strip.Each semidiscrete A-net (possibly after changing handedness) has a semidiscreteLelieuvre vector field U with ∆x = U × ∆U and ∂x = −U × ∂U which is unique(like for continuous A-nets) and which fulfills a Moutard equation. In addition,the net Ũ(u, k) = (−1)kU(u, k) is a conjugate net (like in the discrete case): Theruled surface defined by the curves−U |R×{k}, U |R×{k+1},is developable, with its curve of regression exactly in between at ∆U(u, k)/2. Thismeans that Ũ is actually a T-net. Taking the definitionK = −‖U‖−4of Gauss curvature from the continuous category, we can show that A-nets ofconstant Gaussian curvature are characterized by the weak Chebyshev property:K = const. ⇐⇒ ∆ ‖∂x‖ = ∂ ‖∆x‖ = 0.Discrete Differential Geometry 81Further, both associated nets U , Ũ enjoy the weak Chebyshev property. Knowingthe relations between discrete and continuous A-nets of constant Gaussian curva-ture [1, 8], this theorem is to be expected. It is however interesting to observe therole of the T-net Ũ in the proof of this fact.This research is supported by the National Research Network Industrial Geom-etry (grant No. S92, Austrian Science Fund).References[1] A. Bobenko and U. Pinkall, Discrete Surfaces with constant negative Gaussian curvature andthe Hirota equation, J. Differential Geom. 43 (1996), 527–611.[2] A. Bobenko and Yu. Suris, Discrete differential geometry: Integrable structure, GraduateStudies in Math., vol. 98, American Math. Soc., 2008.[3] C. Müller, Semi-discrete isothermic surfaces, forthcoming paper, 2009.[4] H. Pottmann, A. Schiftner, P. Bo, H. Schmiedhofer, W. Wang, N. Baldassini, and J. Wallner.Freeform surfaces from single curved panels., ACM Trans. Graphics 27 (2008), 76:1–76:10.[Proceedings of ACM SIGGRAPH ’08][5] H. Pottmann, A. Schiftner, and J. Wallner, Geometry of architectural freeform structures,Int. Math. Nachr. 209 (2008), 15–28.[6] H. Pottmann and J. Wallner, The focal geometry of circular and conical meshes, Adv. Com-put. Math. 29 (2008), 249–268.[7] J. Wallner, Semidiscrete A-surfaces and K-surfaces, forthcoming paper, 2009.[8] W. Wunderlich, Zur Differenzengeometrie der Flächen konstanter negativer Krümmung,Österreich. Akad. Wiss. Math.-Nat. Kl. S.-B. IIa. 160 (1951), 39–77.Discrete Differential Geometry: Integrable structureYuri B. Suris(joint work with Alexander I. Bobenko)This talk was based on the textbook with A. Bobenko [1] recently published bythe AMS, and aimed at giving an overview of an (integrable part of) discretedifferential geometry. Discrete differential geometry (DDG) develops discrete ana-logues and equivalents of notions and methods of differential geometry of smoothcurves, surfaces etc. The smooth theory appears in a limit of the refinement ofdiscretizations. Integrable differential geometry deals with parametrized objects(surfaces and coordinate systems) described by integrable differential equations.As integrability attributes one counts traditionally: zero curvature representations,transformations with remarkable permutability properties, hierarchies of commut-ing flows etc. Development of DDG led, somewhat unexpectedly, among otherthings, to a better understanding of the very notion of integrability.The basic notion of DDG is that of a discrete net, i.e., a map f : Zm → X . HereX is some space; in the most straightforward examples, like Q-nets (or discreteconjugate nets), X is just the ambient space of the underlying geometry, likeX = RPN , however more intricate mathematical models require for other spacesX , and a good deal of this talk was devoted to these less trivial situations. Notationfor discrete nets: f = f(u), fi = f(u + ei), fij = f(u + ei + ej), etc.We started with the following definitions:82 Oberwolfach Report 02– A (hyperbolic) 2d system (with fields assigned to vertices) is a geometric con-dition, equation, etc., which allows to determine the 4th point of an elementarysquare of Z2 if other three are arbitrarily prescribed.– A (hyperbolic) 3d system (with fields assigned to vertices) is a geometric con-dition, equation, etc., which allows to determine the 8th point of an elementarycube of Z3 if other seven are arbitrarily prescribed.These notions are schematically represented in Figure 1. Natural boundary valueproblems for these systems are Goursat problems (prescribing initial data alongcoordinate axes for a 2d system, resp. along coordinate planes for a 3d system)and Cauchy problems (prescribing initial data along a non-characteristic staircaseline, resp. along a non-characteristic stepped surface).ff2f1f12ff3f12f1f13f2f23 f123Figure 1. Elementary square of a 2d system (left) and an ele-mentary cube of a 3d system (right). Black circles mark the initialdata; white circles mark the vertices uniquely determined by theinitial data.One of the fundamental organizing principles of integrable DDG is the multi-dimensional consistency principle: discretizations of surfaces, coordinate systemsand other parametrized objects should be extendable to multidimensionally con-sistent nets. Multidimensional consistency is, in our understanding, synonymouswith integrability. For a 2d system, it is enough to establish its 3d consistency,and likewise for a 3d system it is enough to establish its 4d consistency. Thesenotions are schematically represented in Figure 2. Given a multidimensionallyconsistent discrete net, a smooth limit in some of the coordinate directions leadsto smooth surfaces with transformations possessing permutability properties. Asa rule, consistency is a consequence of elementary incidence theorems of geometry,which therefore constitute a true root of integrable differential geometry.Several classes of discrete nets were considered, with an emphasis on the situa-tions where a less simple choice of the space X becomes crucial.• The most fundamental class of discrete nets constitute Q-nets f : Zm → RPN ,characterized by the property that four vertices of each elementary quadrilaterallie in a plane. A generalization has been recently proposed in [2]: GrassmannianQ-nets are nets f : Zm → GNr with values in the Grassmannian GNr of r-planesin RPN , characterized by the property that for each elementary quadrilateral thefour r-planes f , fi, fj and fij assigned to its vertices span a (3r+2)-plane. (UsualDiscrete Differential Geometry 83ff3f12f1f13f2f23 f123f f1f2f3f12f13f23 f123f4f14f24f34f124f134f234 f1234Figure 2. 3d consistency of a 2d system (left) and 4d consistencyof a 3d system (right). Black circles mark the initial data; whitecircles mark the vertices uniquely determined by the initial data;white square marks the vertex where the consistency conditionappears.Q-nets correspond to r = 0.) It has been proven in [2] that Grassmannian Q-nets,like the usual ones, are described by a multidimensionally consistent 3d system.• Another remarkable class of discrete nets described by a 3d system featuringmultidimensional consistency constitute A-nets f : Zm → RP3, characterized bythe property that all neighbor points f(u± ei) of f(u) lie in a plane P(u) throughf(u). It turns out to be natural to consider A-nets as maps with values in thespace X of all pairs (f,P) satisfying f ∈ P , characterized by the condition thatfor two neighboring pairs (f,P) ∼ (fi,Pi) there holds f ∈ Pi and fi ∈ P . Theincidence theorem underlying the 4d consistency of this system is that of Möbiuson the pairs of mutually inscribed tetrahedra. The space X of contact elementsadmits a beautiful realization in the framework of Plücker line geometry: contactelements (f,P) are interpreted as sets of lines in P through f and are representedby isotropic lines in the Plücker quadric in P(R3,3). The characteristic propertyof A-nets reduces to the condition that any two neighboring isotropic lines ℓ, ℓiintersect. In other words, A-nets correspond to discrete congruences of isotropiclines in the Plücker quadric.• If one interprets contact elements (f,P) in the framework of Lie spheregeometry as the sets of all spheres through f in oriented contact with P , one arrivesat the notion of principal contact element nets, which serve as a discretization ofthe curvature line parametrized surfaces. The defining condition is that for twoneighboring pairs (f,P) ∼ (fi,Pi) there is a sphere through both f and fi inoriented contact with both P and Pi. The points of a principal contact elementnet build a circular net, while the planes build a conical net. In the framework ofLie sphere geometry, the space X of contact elements admits a realization as the84 Oberwolfach Report 02space of isotropic lines in the Lie quadric in P(R4,2). The characteristic property ofprincipal contact element nets translates to the condition that any two neighboringisotropic lines ℓ, ℓi intersect. Thus, principal contact element nets correspond todiscrete congruences of isotropic lines in the Lie quadric.In conclusion, it has been stressed that multidimensional consistency serves asthe organizing principle of integrable discrete differential geometry, and that inter-esting geometries appear as discrete nets in some less trivial spaces X . Presumably,many important examples still wait to be discovered.References[1] A. I. Bobenko, Yu. B. Suris, Discrete Differential Geometry: Integrable Structure. GraduateStudies in Mathematics, Vol. 98. AMS, 2008. xxiv+404 pp.[2] V. E. Adler, A. I. Bobenko, Yu. B. Suris. Integrable discrete nets in Grassmannians. arXiv:0812.5102 [math.DG].Discrete Ricci curvature for metric spaces and Markov chainsYann OllivierWe define the coarse Ricci curvature of metric spaces in terms of how much smallballs are closer (in Wasserstein transportation distance) than their centers are.This definition naturally extends to any Markov chain on a metric space. For aRiemannian manifold this gives back, after scaling, the value of Ricci curvatureof a tangent vector. Examples of positively curved spaces for this definition in-clude graphs such as the discrete cube. Moreover this generalization is consistentwith the Bakry–Émery Ricci curvature for Brownian motion with a drift on aRiemannian manifold.Positive Ricci curvature implies a spectral gap, a Lévy–Gromov–like Gaussianconcentration theorem and a kind of modified logarithmic Sobolev inequality. Thebounds obtained are sharp in a variety of examples.Our starting point is the following: Is there a common geometric feature be-tween the N -dimensional sphere SN , the discrete cube {0, 1}N , and the spaceRN equipped with a Gaussian measure? For a start, all three spaces exhibit theconcentration of measure phenomenon [Led01]; moreover, by the Dvoretzky the-orem random small-dimensional sections of the cube are close to a sphere, andsmall-dimensional projections of either the sphere or the cube give rise to nearly-Gaussian measures.So one can wonder whether there exists a common underlying geometric prop-erty. A hint is given by the Gromov–Lévy theorem [Gro86], which states thatGaussian concentration occurs not only for the N -dimensional sphere, but for allRiemannian manifolds of positive curvature in the sense that their Ricci curvatureis at least that of the sphere. In Riemannian geometry, Ricci curvature is therelevant notion in a series of positive-curvature theorems.One is left with the problem of finding a definition of Ricci curvature valid forspaces more general than Riemannian manifolds. Moreover, the definition shouldDiscrete Differential Geometry 85be local and not global, since the idea of curvature is to seek local propertiesentailing global constraints. A first step in this direction is provided by Bakry–Émery theory [BE85], which allows to define the Ricci curvature of a diffusionprocess on a Riemannian manifold (or equivalently, of a second-order differentialoperator); when the diffusion is ordinary Brownian motion, this gives back usualRicci curvature. When applied to the natural process on RN associated with theGaussian measure, this yields a positive curvature for the Gaussian space.Because the Bakry–Émery definition involves differential calculus, it is not read-ily adaptable to discrete spaces. If one wants to deal with the third basic example,the discrete cube, one has to drop the continuity aspect and deal with more “ro-bust” or “coarse” notions that forget the small-scale properties of the underlyingspace. This is similar in spirit to what has been done for a long time in the(very different) world of negative curvature, for which coarse notions such as δ-hyperbolicity and CAT(0) spaces have been developed.Such a notion can be summarized as follows [Oll07, Oll09]: a metric space haspositive curvature if small balls are closer than their centers are. Here one usestransportation distances [Vil03] to measure the distance between balls.It is possible to put emphasis on a random process (consistently with Bakry–Émery theory) and replace the ball centered at a point with the transition proba-bility of a random walk. Doing so, one finds that the property above is equivalentto a property first introduced by Dobrushin [Dob70, DS85] for Markov fields, andstill known in the Ising community as the “Dobrushin criterion” (several variantsof which are in use). The 1970 Dobrushin paper was actually the one to maketransportation distances known to a wider audience.Examples. Here are some spaces for which coarse Ricci curvature can be easilycomputed:– Riemannian manifolds: we get usual Ricci curvature up to some scaling.– The discrete cube {0, 1}N : coarse Ricci curvature is positive.– Zn with its lattice metric: coarse Ricci curvature is 0.– Diffusions on a manifold: we recover the Bakry–Émery curvature, in a morevisual way.– Multinomial distributions, waiting queues, and similar examples.– Ising model and its variants: coarse Ricci curvature depends on temperature, inan explicit way.– δ-hyperbolic groups: coarse Ricci curvature is negative.Results. Here are some of the results obtained when coarse Ricci curvature ispositive:– Concentration results (Gaussian or exponential concentration) as in theGromov–Lévy theorem.– Spectral gap estimate as in the Lichnerowicz theorem.– Logarithmic Sobolev inequality and gradient contraction by the heat kernel, asin Bakry–Émery theory.86 Oberwolfach Report 02– Convergence rates of the underlying Markov chain; in particular, explicit con-vergence rates for the Markov chain Monte Carlo (MCMC) method.– Good behavior in Gromov–Hausdorff topology.References[BE85] D. Bakry, M. Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX,1983/84. Lecture Notes in Math. 1123, Springer, Berlin (1985), 177–206.[Dob70] R. L. Dobrušin, Definition of a system of random variables by means of conditional dis-tributions (Russian), Teor. Verojatnost. i Primenen. 15 (1970), 469–497. English trans-lation: Prescribing a system of random variables by conditional expectations, Theory ofProbability and its Applications 15 (1970) nr. 3, 458–486.[DS85] R. L. Dobrushin, S. B. Shlosman, Constructive criterion for the uniqueness of Gibbsfield, in J. Fritz, A. Jaffe and D. Szász (eds), Statistical physics and dynamical systems,papers from the second colloquium and workshop on random fields: rigorous resultsin statistical mechanics, held in Köszeg, August 26–September 1, 1984, Progress inPhysics 10, Birkhäuser, Boston (1985), 347–370.[Gro86] M. Gromov, Isoperimetric inequalities in Riemannian manifolds, in V. Milman,G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Lecture Notesin Mathematics 1200, Springer, Berlin (1986), 114–129.[Led01] M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys andMonographs 89, AMS (2001).[Oll07] Y. Ollivier, Ricci curvature of metric spaces, C. R. Math. Acad. Sci. Paris 345 (2007),nr. 11, 643–646.[Oll09] Y. Ollivier, Ricci curvature of Markov chains on metric spaces, J. Funct. Anal. 256(2009), nr. 3, 810–864.[Vil03] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics 58,American Mathematical Society, Providence (2003).The discrete Hilbert-Einstein functional: History and applicationsIvan Izmestiev1. The total mean curvatureJacob Steiner found in 1840 that the volume of the parallel bodyKt = {x | dist(x, K) ≤ t}to a convex body K ⊂ R3 can be expanded as(1) vol(Kt) = vol(K) + t · area(∂K) + t2 · S(K) + t3 ·4π3,provided that K is either a polytope or a body with smooth boundary. If K is apolytope, then the coefficient at t2 is(2) S(K) =12∑eℓe(π − θe),Discrete Differential Geometry 87where the sum is taken over all edges of K, and ℓe is the length of, θe is thedihedral angle at the edge e. If K is a body with smooth boundary, then we have(3) S(K) =∫∂Kk1 + k22d area,where k1 and k2 are the principal curvatures of the boundary of K. This is areason to call the sum (2) the total mean curvature of the polyhedral surface ∂K.Later, Steiner’s argument war reinforced by Minkowski who proved that theexpansion (1) holds for an arbitrary convex body K, and its coefficients dependcontinuously on K in the Hausdorff metric. In particular, if a convex body withsmooth boundary is approximated by convex polytopes, then (2) converges to (3).2. The total scalar curvature, or the Hilbert-Einstein functionalLet M be a closed manifold equipped with a Riemannian metric g. The Hilbert-Einstein functional is the integral of the scalar curvature(4) S(g) =∫Msg d volg .Theorem 1 (Hilbert). For a fixed manifold M , consider the space of all unitvolume metrics g. Then the critical points of S correspond to Einstein metrics.Unfortunately, the functional S is neither bounded from above, nor from below,so that critical points are hard to find. But there are good news too.Theorem 2 (Yamabe, see [6]). On the space of the unit volume metrics confor-mally equivalent to g{u · g | u : M → R+, volu·g(M) = 1},the functional S can be minimized. The point of minimum is a metric of constantscalar curvature.The discrete Hilbert-Einstein functional for dim M = 3 is defined as follows.Call a discrete Riemannian metric on M a pair (T, ℓ) of a triangulation T of Mand an assignment ℓ : e 7→ ℓe of a positive number to every edge of T such thatthe tetrahedra of T can be realized as Euclidean ones with edge lengths ℓe. Let(5) S(T, ℓ) =∑eℓe(2π − ωe),where ωe is the angle around e. The functional (5) is also known as the Reggefunctional, and can be defined with hyperbolic or spherical tetrahedra as well.Similarly to Section 1, (5) converges to (4) (up to a constant factor), when thediscrete Riemannian metric (T, ℓ) converges to the Riemannian metric g, see [3].Theorem 3. The discrete Hilbert-Einstein functional (5) has the property(6)∂S∂ℓe= 2π − ωe =: κeThus, the critical points of S correspond to Euclidean metrics on M , subdividedby the triangulation T .88 Oberwolfach Report 02Equation (6) is a direct consequence of the Schläfli formula.Consider the matrix of the second partial derivatives of S(7) Hess(S) =(∂2S∂ℓi∂ℓj)=(∂κi∂ℓj).• If Hess(S) is positive (negative) semidefinite, then S is convex (concave).• If detHess(S) 6= 0, then the metric (T, ℓ) is infinitesimally rigid, that is a non-zero first-order change in ℓ entails a non-zero first-order change in κ.3. Partial results on the signature of Hess(S)A discrete Riemannian metric (T, ℓ) is called a ball packing metric, if there existsa function r : i 7→ ri on the set of vertices of T such that ℓij = ri + rj holds for alledges ij.Theorem 4 (Cooper, Rivin [4]). The functional S is convex on the space of all ballpacking metrics. The nullspace of the Hessian is one-dimensional and correspondsto scaling.A consequence of Theorem 4 is the infinitesimal rigidity of ball packings in3–dimensional manifolds.A change of ℓ defined by ℓ′ij = ℓij +ui +uj can be viewed as a discrete analog ofa conformal deformation; then Theorem 4 says that S is convex on the conformalclass of an “equilateral” metric {ℓij = 1 for all ij}.In the next two theorems the discrete Hilbert-Einstein functional of manifoldswith boundary is considered. It is obtained by adding to (5) a boundary termsimilar to (2). We consider only deformations that preserve the metric on theboundary. The property (6) remains valid for deformations of the interior edges.Theorem 5 (I., Schlenker [5]). Let M ≈ B3, and let the metric (T, ℓ) on B3 besuch that ωe = 2π for all interior edges and θe ≤ π for all boundary edges. Thatis, let (T, ℓ) be a triangulation of a convex polytope P . Leti = the number of the interior vertices of T ;f = the number of the boundary vertices of T not in the 1–skeleton of P.Then the Hessian (7) has corank 3i + f and exactly i positive eigenvalues.In particular, the Hessian is negatively definite, if there are no interior verticesand no vertices inside the faces of P . This fact was used in [5] to prove theinfinitesimal rigidity for a class of non-convex polytopes.Note that the number of positive eigenvalues in Theorem 5 is equal to thedimension of the space of conformal deformations. However, we were unable todescribe an i–dimensional space of deformations on which the Hessian would bepositively definite.Theorem 6 (Bobenko, I. [2]). Let M ≈ B3, let T be the cone over a triangulationof S2, and let θe ≤ π for all boundary edges. The interior edges are then inone-to-one correspondence with the boundary vertices.Discrete Differential Geometry 89• Assume that φi < ωi < 2π, where φi is the angle around the vertex i in themetric on S2. Then Hess(S) is non-degenerate and has at least one positiveeigenvalue. Namely, dℓi =1ℓiis a positive direction for Hess(S).• Assume that ωi = 2π for all i, so that (T, ℓ) is a triangulation of a convexpolytope. Then Hess(S) has corank 3 and exactly one positive eigenvalue. Thevectors dℓi =1ℓiand dℓi = 1 both belong to its positive cone.The first statement of Theorem 6 suggests that (ℓ′ij)2 = ℓ2ij + ui + uj could beconsidered as a discrete analog of a conformal deformation.The main result of [2] is a new proof of Alexandrov’s theorem on the existence ofa convex polytope with a given metric on the boundary. Note that for its smoothanalog, Weyl’s problem, a variational approach based on the Hilbert-Einstein func-tional was proposed by Blaschke and Herglotz in [1].References[1] W. Blaschke and G. Herglotz. Über die Verwirklichung einer geschlossenen Fläche mitvorgeschriebenem Bogenelement im Euklidischen Raum. Sitzungsber. Bayer. Akad. Wiss.,Math.-Naturwiss. Abt. 1937, 229–230 (No. 2) (1937).[2] A. I. Bobenko and I. Izmestiev. Alexandrov’s theorem, weighted Delaunay triangulations,and mixed volumes. Ann. Inst. Fourier (Grenoble), 58(2):447–505, 2008.[3] J. Cheeger, W. Müller, and R. Schrader. On the curvature of piecewise flat spaces. Comm.Math. Phys., 92(3):405–454, 1984.[4] D. Cooper and I. Rivin. Combinatorial scalar curvature and rigidity of ball packings. Math.Res. Lett., 3(1):51–60, 1996.[5] I. Izmestiev and J.-M. Schlenker. On the infinitesimal rigidity of polyhedra with vertices inconvex position. arXiv:0711.1981v1, 2007.[6] C. Lebrun. Einstein metrics and the Yamabe problem. In Trends in mathematical physics(Knoxville, TN, 1998), volume 13 of AMS/IP Stud. Adv. Math., pages 353–376. Amer.Math. Soc., Providence, RI, 1999.Hyperbolization of ornamentsJürgen Richter-Gebert, Martin von Gagern1. What is hyperbolization?While ornamental patterns based on the 17 Euclidean wallpaper groups arewidely present in art and architecture, designs of similar ornaments based on hy-perbolic symmetries are rare. The reason for this is twofold. First of all, the gener-ation of hyperbolic ornamental patterns requires already a substantial knowledgeor intuition in hyperbolic structures. Secondly, the technical process of renderingmay be tedious (at least if performed by manual work) since it in principle requiresthe generation of infinitely many smaller and smaller objects.We present a method that can be used to automatically transform an existingEuclidean ornamental pattern into a corresponding one with an underlying hy-perbolic symmetry group. During this process we want to preserve as much as90 Oberwolfach Report 02Figure 1. Transforming a scanned Euclidean input image to ahyperbolic ornamentpossible from the original artistic input. Our process can be roughly divided intothree steps:Pattern recognition: Based on auto-correlation methods calculated via fastFourier transformation we first analyze a Euclidean ornamental pattern. We ex-tract its symmetry group and a fundamental region (see [1]).Deformation: In a second step we deform the fundamental region such that weget a new region that could serve as fundamental region of a hyperbolic ornament.For this step we require only distortions of the pattern that are mathematicallymotivated, artistically reasonable and performable in reasonable time. We requirethe deformation to be a conformal map. This preserves the intersection angles ofobjects. Furthermore once the combinatorics is fixed a conformal map is uniquelydetermined. A detailed analysis is presented in [2].Rendering: Finally, the deformed fundamental region is used to create the entirehyperbolic picture. To get a perfect rendering a reverse pixel lookup strategy isused that associates to every pixel of the final image a preimage in the centralfundamental region (for details again see [2]).2. HyperbolizationWe will exemplify the concept of hyperbolization in the relatively simple case oftriangular reflection groups. Deformations of seven out of the 17 crystallographicgroups can be reduced to this case (namely those with orbifold symbols 442, ∗442,4∗2, 333, ∗333, 3∗3, 632, ∗632). The others groups have to be treated by slightlymore advanced methods. Assume that we have an ornamental kaleidoscopic pat-tern based on a triangular reflection group for instance with corner angles π2 ,π4and π4 . The fundamental region of this pattern is the triangle itself. The cornerangles imply 2-fold, 4-fold and 4-fold rotational symmetry at the corners.We want to conformally deform the triangle to a circular arc triangle (represent-ing a hyperbolic triangle in the Poincaré disk model) such that the corner anglesare other fractions of π (say π2 ,π4 andπ5 ). Riemann’s mapping theorem guaran-tees that the map relating the two triangles is uniquely defined by conformality. Inessence this map can be computed by a composition of Schwarz-Christoffel map(SCM) that maps the unit disk to the deformed triangle with an inverse SCMDiscrete Differential Geometry 91Figure 2. Deformation of a fundamental domain via discreteconformal mapsthat maps the original triangle to the unit disk. This map can be extended tothe Euclidean plane (punctured at the rotation centers) via the Schwarz reflectionprinciple applied to the boundaries of the triangle.3. Discrete conformal mapsUnfortunately, there is no known reasonably fast way to calculate the mapdescribed above. This is where Discrete Differential Geometry enters the stage.The concept of discrete conformal maps of a triangular mesh (as introduced byBobenko, Pinkall, Schröder and Springborn [3]) is perfectly suited to calculatean approximation of the above map in reasonably short time using a variationalprinciple. The method introduced there has several advantages that makes it per-fectly suitable for applications in computer graphics. A discrete conformal mapof a triangular mesh is defined by a collection of weights at the vertices. Theedge lengths of the image mesh are related to the original mesh by scaling themaccording to the weights of the incident endpoints. Under successive refinement ofthe mesh this method approximates a smooth conformal map. The specificationof target angle sums for the whole mesh allows an easy way to express the requiredchanges to corner angles and flatness conditions elsewhere. A discrete conformalmap can used as basis for a continuous interpolation of triangle interiors by assign-ing suitably chosen projective transformations to each of them. Here continuity isa consequence of the discrete conformality property. Figure 2 illustrates a discreteconformal map for a quadratic fundamental region from the Euclidean symmetrygroup 442 to the hyperbolic symmetry group 562. The Hyperbolic fundamentaldomain was approximated with a triangle mesh, which was then transformed tothe corresponding Euclidean mesh.4. The other groupsFor the remaining (low symmetry) groups where all centers of rotation areat most two-fold, the symmetry type does not automatically fix the shape ofthe fundamental region. Thus also in the hyperbolization the exact shape of thedeformed boundary is a priori unknown. As a consequence, the transformation hasto start with an Euclidean mesh, and be performed in hyperbolic geometry to yielda conformally equivalent hyperbolic mesh. In general, straight boundary edges of92 Oberwolfach Report 02Figure 3. Straight edged hyperbolic fundamental region (red)and deformation of a straight edged Euclidean region (blue) for alow symmetry groupthe Euclidean fundamental domain will not map to geodesics in the hyperbolicornament. In this general case, any distinguished boundary curve will be straightin at most one world, as Figure 3 illustrates. This problem can be avoided byperforming the deformation process directly on a triangle mesh of the orbifold, i.e.,with corresponding edges identified, (see [2]). This approach avoids the problemof distinguished non-reflecting lines altogether.References[1] M. von Gagern. Computergestütztes Zeichnen in den Symmetriegruppen der euklidischenEbene. Diplomarbeit, TU München, 2007.[2] J. Richter-Gebert and M. von Gagern. Hyperbolization of Euclidean Ornaments. sub-mitted.[3] B. Springborn, P. Schröder and U. Pinkall. Conformal equivalence of triangle meshes.ACM SIGGRAPH, 2008.Earthquakes on hyperbolic surfacesJean-Marc Schlenker(joint work with Francesco Bonsante and Kirill Krasnov)Consider a closed surface S, of genus at least 2. A weighted multicurve on S is adisjoint union of simple closed curves c1, . . . , cn, each with a positive number wi.Given a hyperbolic metric g on S, each ci can be uniquely realized as a simpleclosed geodesic and those geodesics are disjoint. The fractional Dehn twist alonga weighted multicurve is a surgery on hyperbolic metrics on S: one cuts (S, g)open along each of the ci, rotates the right-hand side by wi, and glues back (thisdoes not depend on the orientation chosen for the ci). This defines a map fromthe Teichmüller space T of S to itself.Thurston showed that the space of weighted multicurves on S has a naturalcompletion, the space ML of measured laminations on S. A geodesic measuredlamination on (S, g) is a closed set which is a disjoint union of complete geodesics,endowed with a “transverse measure”. Thurston also proved that map defined byDiscrete Differential Geometry 93the notion of fractional Dehn twist extends continuously to the notion of earth-quake on a measured lamination, therefore defining a mapE : ML× T → T .A neat property of this map is that it can be used to parameterize the Teich-müller space of S from a fixed point by the space of measured lamination, thanksto Thurston’s Earthquake Theorem:Theorem (Thurston). Let g0, g1 ∈ T , there exists a unique λ ∈ ML such thatE(λ, g0) = g1.A proof was given in [Ker83] and another one, based on anti-de Sitter geometry,in [Mes07].A first extension concerns hyperbolic surfaces with conical singularities of angleless than π. For those surfaces the notion of measured lamination still makesperfect sense, and we proved with Francesco Bonsante [BS06] that given any twohyperbolic metrics with cone angles (with the same angles) there is a uniquemeasured lamination λ such that the right earthquake on λ transforms one intothe other.A second extension, obtained with Francesco Bonsante and Kirill Krasnov[BKS06], applies to hyperbolic surfaces with geodesic boundary. Let TS,N bethe Teichmüller space of hyperbolic metrics with geodesic boundary on S with Ndisks removed (note that the length of the boundary components is not fixed).Let MLS,N be the space of measured geodesic laminations on the interior of S forsuch a metric (that is, the weight of the lamination can accumulate close to theboundary). Then, given g0, g1 ∈ TS,N , there are exactly 2N elements λ ∈ MLS,Nsuch that the right earthquake on λ sends g0 to g1.The proof of both results uses the arguments of Mess’ proof in [Mes07], butnew twists are necessary. Instead of globally hyperbolic AdS manifolds as thoseconsidered in [Mes07], it is necessary to use corresponding manifolds with “parti-cles” (cone singularities along time-like geodesics) for the statement on hyperbolicsurfaces with cone singularities, and on multi-black holes (see e.g. [ÅBB+98]) forthe statement on surfaces with geodesic boundary. In both cases, it is necessaryto understand the moduli spaces of the 3-dimensional metrics involved, as well asproperties of their “convex cores”.References[ÅBB+98] Stefan Åminneborg, Ingemar Bengtsson, Dieter Brill, Sören Holst, and Peter Peldán.Black holes and wormholes in 2+1 dimensions. Classical Quantum Gravity, 15(3):627–644, 1998.[BKS06] Francesco Bonsante, Kirill Krasnov, and Jean-Marc Schlenker. Multi black holes andearthquakes on Riemann surfaces with boundaries. arXiv:math/0610429 [math.GT],2006.[BS06] Francesco Bonsante and Jean-Marc Schlenker. AdS manifolds with particles andearthquakes on singular surfaces. Geom. Funct. Anal., to appear. arXiv:math/0609116 [math.GT], 2006.94 Oberwolfach Report 02[Ker83] Steven P. Kerckhoff. The Nielsen realization problem. Ann. of Math. (2), 117(2):235–265, 1983.[Mes07] Geoffrey Mess. Lorentz spacetimes of constant curvature. Geom. Dedicata, 126:3–45,2007.Discrete square peg problemIgor PakThe square peg problem is beautiful and deceptively simple. It asks whether everyJordan curve C ⊂ R2 has four points which form a square. We call such squaresinscribed into C.The problem goes back to Toeplitz (1911), and over almost a century has beenrepeatedly rediscovered and investigated, but never completely resolved. By nowit has been established for convex curves and curves with various regularity con-ditions, including the case of piecewise linear curves. While there are severalsimple and elegant proofs of the convex case, the piecewise linear case is usuallyobtained as a consequence of results proved by rather technical topological and an-alytic arguments. We present a simple proof in the piecewise linear case followingSchnirelmann’s ideas in [3]. Our proof follows [1, 2].References[1] I. Pak, Lectures on Discrete and Polyhedral Geometry, monograph to appear.[2] I. Pak, Discrete Square Peg Problem, arXiv:0804.0657.[3] L. G. Shnirelman, On certain geometrical properties of closed curves (in Russian), UspehiMatem. Nauk 10 (1944), 34–44.Volume and angle structures on 3-manifoldsFeng LuoWe propose an approach to find constant curvature metrics on triangulated closed3-manifolds using a finite dimensional variational method whose energy functionis the volume. The concept of an angle structure on a tetrahedron and on atriangulated closed 3-manifold is introduced. This follows the work of Casson,Murakami and Rivin who introduced the similar concept for ideal triangulations ofcompact 3-manifolds with torus boundary. It is proved by A. Kitaev and the authorthat any closed 3-manifold has a triangulation supporting an angle structure. Thespace of all angle structures on a triangulated 3-manifold is a bounded open convexpolytope in a Euclidean space. The volume of an angle structure is defined bygeneralizing the Schläfli formula. Both the angle structure and the volume arenatural generalizations of that of tetrahedra in the constant sectional curvaturespaces and their volume. It is shown that the volume functional can be extendedcontinuously to the compact closure of the moduli space, answering affirmativelya question of Milnor. In particular, the maximum point of the volume functionalalways exists in the compact closure of the space of angle structures. The mainresult shows that if the volume function on the space of angle structures has a localDiscrete Differential Geometry 95maximum point, then either the manifold admits a constant curvature Riemannianmetric or the triangulation supports a normal 2-sphere which intersects each edgein at most one point.Extremal configurations of revolute-jointed robot armsIleana Streinu(joint work with Ciprian S. Borcea)A fundamental question in Robotics is the Reach Problem: given a 3Drevolute-jointed robot, compute the extremal values of the endpoint distance func-tion and the corresponding extremal configurations.The most general type of revolute-jointed robot arms are the body-and-hingechains: rigid bodies connected serially by hinges, as in Figure 1. Panel-and-hingechains are a special case, with 2D panels instead of 3D bodies, i.e., consecutivehinges are coplanar. Polygonal chains with fixed edge lengths and fixed anglesbetween consecutive edges can also be viewed as panel-and-hinge chains. A hingeis understood here as an entire line or turning axis, constraining the relative motionof two connected bodies to a rotation around their common hinge. We mark twopoints: the origin or start point s on the first body and the terminus t on thelast body. We ask what are the extremal values of the distance function from sto t, as the chain takes all possible configurations (the boundaries of the bodiesare immaterial and self-intersections are permitted). The problem was intenselystudied in the 1980’s, when a necessary condition for extremal configurations wasrecognized and proven in several papers [2, 3, 4, 6]: the line joining the markedpoints must intersect all the hinges. This incidence is understood projectively, thatis, includes the possibility of parallelism, and the hinges are thought of as lines(not line segments).Figure 1. A body-and-hinge chain in R3 with n = 3 hinges, fourbodies (visualized as tetrahedra) and two marked points s and ton the end-bodies. In a maximum reach position, the axes meetthe oriented segment st in the natural order.96 Oberwolfach Report 02This talk presented our recent results, giving an intrinsic, combinatorial char-acterization of the configurations attaining the global extremal values. The proofswork in arbitrary dimension d. The results are somewhat different for the globalmaximum and minimum, and for body-and-hinge as opposed to panel-and-hingestructures.The natural order of the hinges is 1, 2, 3, · · · as they appear on the chain. Thecomplement of a line segment is with respect to the projective line containing theaffine line segment, i.e., it goes through the point at infinity.Theorem 1. (Global Maximum) A body-and-hinge chain is in a global maxi-mum configuration if and only if the segment from the origin s to the terminus tintersects all hinges in their natural order. Generically, the global maximum isattained in a unique configuration.Indeed, given a configuration satisfying this condition, we will mark in red thesegment from s to t. In any other configuration of the chain, the red path appearsas a polygonal chain in 3D, hence the endpoint distance will be shorter thanin the straightened position. The necessity of the condition is obtained from acharacterization of the global maximum as a global minimum of another function:Theorem 2. (Global Maximum as a Global Minimum) The global maxi-mum of the endpoint distance function coincides with the length of the shortestpath from s to t which meets all hinges in their natural order.The treatment of the minimum endpoint distance and minimal configurationsis more specialized. When the endpoints can reach each other and the mini-mum distance is zero, an entire submanifold of critical values (rather than isolatedpoints) occurs. Under suitable genericity assumptions, the endpoint squared dis-tance function becomes a Morse-Bott function, leading to a formula for the Eulercharacteristic of the inverse kinematics solution space in terms of indices of criticalconfigurations.We derive an explicit formula for the Hessian, which is particularly useful forpanel-and-hinge chains. In this case, we prove that local extrema are already globalextrema, a fact which is not true for general body-and-hinge chains. Moreover:Theorem 3. (Global Non-Zero Minimum) A panel-and-hinge chain is in aglobal non-zero minimum configuration if and only if the complementary endpointline segment oriented from the start point to the terminus meets the hinges in thenatural order.Panel-and-hinge chains in critical configurations are subdivided into flat piecesconnected at fold points: the endpoint axis cuts across the hinges in the flat regions,and goes through two consecutive hinges at fold points. When a panel-and-hingechain is folded, the angles induced by the two incident hinges at a fold point andthe constrained shortest path between the endpoints will satisfy a simple conditionrelated to the triangle inequality on the sphere. At each fold point, the incidentpanels can be folded in two distinct ways. We obtain:Discrete Differential Geometry 97Figure 2. The shortest path between the chain endpoints givesthe maximum reach value. The turning points of the path becomethe fold points in a 3D maximum configuration.Theorem 4. (Number of Extremal Configurations) Generically, the numberof distinct extremal configurations is 2f , where f is the number of fold points.These characterizations lead directly to elementary, surprisingly efficient algo-rithms for the four fundamental problems of robot arm reachability listed below,in the case of the class of panel-and-hinge satisfying the local criterion at the foldpoints. This class includes robot arms presented as polygonal chains with equalobtuse angles, including the orthogonal case. Until now, only numerical methodsbased on standard gradient descent methods were known.(1) Extremal distances: compute the minimum and the maximum value of theendpoint distance function.(2) Extremal configurations: compute one (or all) of the configurations thatachieve the global minimum or maximum endpoint distance.(3) Motion Planning: given an arbitrary configuration of the chain, reconfigureit to an extremal position: i.e., compute a trajectory in configuration spaceending at an extremal configuration.(4) Optimized Motion Planning: given a flat non-extremal configuration, re-configure it in such a way that the distance function is monotone throughoutthe motion: decreases towards the minimum, or increases towards the maxi-mum.The first problem is solved using a shortest path calculation in an associatedflat polygonal region, as in Figure 2. Fold angles are then computed and a maxi-mum configuration is derived. To reconfigure from a flat position to a maximumconfiguration in a monotone fashion, we adapt Streinu’s pseudo-triangulation al-gorithm [5] for the planar Carpenter’s Rule Problem.A more detailed theoretical development is presented in an arXiv preprint [1].References[1] C. S. Borcea and I. Streinu. Extremal configurations of hinge structures. arXiv:0812.1375,Dec. 2008.[2] S. Derby. The maximum reach of revolute jointed manipulators. Mechanism and MachineTheory, 16(3):255–261, 1981.[3] A. Kumar and K. J. Waldron. The workspaces of a mechanical manipulator. Journal ofMechanical Design, 103(3):665–672, 1981.98 Oberwolfach Report 02[4] R. G. Selfridge. The reachable workarea of a manipulator. Mechanism and Machine Theory,18(2):131–137, 1983.[5] I. Streinu. Pseudo-triangulations, rigidity and motion planning. Discrete and ComputationalGeometry, 34(4):587–635, 2005.[6] K. Sugimoto and J. Duffy. Determination of extreme distances of a robot hand - Part 1: Ageneral theory. Journal of Mechanical Design, 103(3):631–636, 1981.Discrete random interfacesRichard KenyonWe discussed a model of discrete two-dimensional interfaces in R3. These arepiecewise-linear surfaces composed of translates of the faces of the unit cube, gluededge to edge, and with the property that the orthogonal projection to the planex+y +z = 0 is injective. These interfaces can be thought of as tilings of the planewith “lozenges”, that is, with 60◦ rhombi. Given a closed curve in Z3 which can bespanned by such an interface, consider the uniform probability measure on the setof all such spanning interfaces. There is a law of large numbers, proved by Cohn,myself and Propp [1], which says that for a fixed “wire frame” boundary curve γ,in the limit that the lattice spacing tends to zero the uniform random interface willconcentrate on a nonrandom limit shape, which is a continuous surface spanning γ.The limiting surface can be obtained through a variational principle, essentiallymaximizing a local entropy functional. Here the local entropy is a function onlyof the slope of the interface, and is just the exponential growth rate of interfaceshaving average slope in that direction.Techniques of Kasteleyn show how to compute explicitly the entropy functionas a function of slope, and the resulting formula is very simple: when the normal tothe interface has (positive) coordinates (px, py, pz), the growth rate is proportionalto the volume of the idea hyperbolic 3-simplex with dihedral angles proportionalto (px, py, pz).In work with Okounkov [2] we were able to solve explicitly the correspondingPDE for the limit interface, reducing it to the complex Burgers’ equation. Thispermits parameterization of solutions with analytic functions.Many related interface models, associated with underlying dimer models, havevery similar solutions [3, 2].References[1] H. Cohn, R. Kenyon, J. Propp, A variational principle for domino tilings, J. Amer. Math.Soc., 14 (2001), no. 2, 297–346.[2] R. Kenyon, A. Okounkov, Limit shapes and the complex Burgers equation, Acta Math. 199(2007), 263–302.[3] R. Kenyon, A. Okounkov, S. Sheffield, Dimers and amoebae. Ann. of Math. (2) 163 (2006),no. 3, 1019–1056.Discrete Differential Geometry 99The Pentagram map: a discrete integrable systemSerge Tabachnikov(joint work with Valentin Ovsienko, Richard Schwartz [2, 3])Given a convex n-gon P in the projective plane, let T (P ) be the convex hull ofthe intersection points of consecutive shortest diagonals of P . The map T is thepentagram map. The pentagram map commutes with projective transformations.Let Cn be the space of convex n-gons modulo projective transformations. Thenthe pentagram map induces a self-diffeomorphism T : Cn → Cn.T is the identity map on C5 and an involution on C6, see [5]. Experimentally, forn ≥ 7, the orbits of T on Cn exhibit the kind of quasi-periodic motion associatedto a completely integrable system: T preserves a certain foliation of Cn by roughlyhalf-dimensional tori, and the action of T on each torus is conjugate to a rotation.A conjecture in [6] that T is completely integrable on Cn is still open, but we arevery close to proving it.A twisted n-gon is a map φ : Z → RP2 such that φ(n+k) = M◦φ(k) for all k ∈ Zand some fixed element M ∈ PGL(3, R) called the monodromy. Set vi = φ(i) andassume that vi−1, vi, vi+1 are in general position for all i. Let Pn be the space oftwisted n-gons modulo projective equivalence. We show that the pentagram mapT : Pn → Pn is completely integrable in the classical sense of Arnold–Liouville.We give an explicit construction of a T -invariant Poisson structure and completelist of Poisson-commuting integrals for the map.We apply this result to universally convex twisted n-polygons. These are poly-gons for which the map φ is such that φ(Z) ⊂ R2 ⊂ RP2 is convex and containedin the positive quadrant, and the monodromy M : R2 → R2 is a diagonal lineartransformation with eigenvalues 0 < a < 1 < b. The image of φ looks some-what like a “polygonal hyperbola”. Denote by Un the space of universally convextwisted n-gons modulo projective equivalence.Theorem. Almost every point of Un lies on a smooth torus that has a T -invariantaffine structure. The orbit of almost every universally convex n-gon undergoesquasi-periodic motion under the pentagram map.We associate to every vertex vi two numbers:xi = [vi−2, vi−1, ((vi−2, vi−1) ∩ (vi, vi+1)) , ((vi−2, vi−1) ∩ (vi+1, vi+2))]yi = [((vi−2, vi−1) ∩ (vi+1, vi+2)) , ((vi−1, vi) ∩ (vi+1, vi+2)) , vi+1, vi+2],see Figure 1, where the bracket is the cross ratio of 4 points in RP1 given by[t1, t2, t3, t4] =(t1 − t2) (t3 − t4)(t1 − t3) (t2 − t4)in an affine parameter t. We call these coordinates the corner invariants.The pentagram map is described in these coordinates as follows:T ∗xi = xi1 − xi−1 yi−11 − xi+1 yi+1, T ∗yi = yi+11 − xi+2 yi+21 − xi yi.100 Oberwolfach Report 02ii+2vi+1vi−1vi−2vvFigure 1. Points involved in the definition of the corner invariants.Consider the rescaling operation given byRt : (x1, y1, ..., xn, yn) → (tx1, t−1y1, ..., txn, t−1yn).It follows that the pentagram map commutes with the rescaling operation. LetOn =n∏i=1xi , En =n∏i=1yi , On/2 =∏i evenxi +∏i oddxi , En/2 =∏i evenyi +∏i oddyi,the latter two when n is even. These functions are invariant under the penta-gram map. Let φ be a twisted n-gon with invariants x1, y1, ..., and let M be themonodromy. Lift M to an element of GL3(R). ThenΩ1 =trace3(M)det(M); Ω2 =trace3(M−1)det(M−1);depend only on the conjugacy class of M . Set: Ω̃1 = O2nEnΩ1, Ω̃2 = OnE2nΩ2.In [6] it is shown that Ω̃1 and Ω̃2 are polynomials in the corner invariants. Sincethe pentagram map preserves the monodromy, and On and En are invariants, thetwo functions Ω̃1 and Ω̃2 are also invariants. Consider the decomposition intohomogeneous components with respect to the scaling:Ω̃1 =[n/2]∑k=1Ok; Ω̃2 =[n/2]∑k=1Ekwhere Ok has weight k and Ek has weight −k. It follows that the functionsO1, E1, O2, E2, ... are integrals of the pentagram map. These are the monodromyinvariants. In [6] it is shown that the monodromy invariants are algebraicallyindependent.The Poisson bracket on Pn is as follows:{xi, xi±1} = ∓xi xi+1, {yi, yi±1} = ±yi yi+1,and all other brackets of coordinates functions vanish. The main lemmas concern-ing the Poisson bracket are:(1) The Poisson bracket is invariant with respect to the Pentagram map.(2) The monodromy invariants Poisson commute.Discrete Differential Geometry 101(3) The invariants On, En and, in the even case, On/2, En/2, are Casimir functions.(4) The Poisson bracket has corank 2 if n if odd and corank 4 if n is even.These results imply that Pn is foliated by symplectic leaves which carry a leaf-wise T -invariant Lagrangian foliation. The leaves of a Lagrangian foliation carry acanonical affine structure, in which T is a parallel translation, cf. [1]. For univer-sally convex twisted n-gons, the leaves are compact, and hence tori. This impliesthe theorem.We also consider the continuous limit n → ∞ of a twisted n-gon as a smoothtwisted non-degenerate parametrized curve γ : R → RP2 such that γ(x + 1) =M(γ(x)), for all x ∈ R and a fixed M ∈ PGL(3, R). As in the discrete case, weconsider classes of projectively equivalent curves.It is well known that the space of non-degenerate twisted curves is in one-to-onecorrespondence with the space of linear differential operatorsA =(ddx)3+ u(x)ddx+ v(x),where u and v are smooth periodic functions, see [4].The construction of a continuous analog of the map T is as follows. Given anon-degenerate curve γ(x), at each point x draw the chord (γ(x − ε), γ(x + ε))and obtain a new curve, γε(x), as the envelop of these chords. Let uε and vε bethe respective periodic functions. It turns out thatuε = u + ε2ũ + (ε3), vε = v + ε2ṽ + (ε3),giving the curve flow: u̇ = ũ, v̇ = ṽ. Excluding v yieldsü +(u2)′′6+u(IV )12= 0,which is the classical Boussinesq equation. Thus the pentagram map is a dis-cretization of the Boussinesq equation.References[1] V. Arnold. Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.[2] V. Ovsienko, R. Schwartz, S. Tabachnikov. The Pentagram Map: a discrete integrable sys-tem, preprint 2008.[3] V. Ovsienko, R. Schwartz, S. Tabachnikov. Quasiperiodic Motion for the Pentagram Map,preprint 2009.[4] V. Ovsienko, S. Tabachnikov, Projective differential geometry old and new, from Schwarzianderivative to the cohomology of diffeomorphism groups, Cambridge University Press, Cam-bridge, 2005.[5] R. Schwartz, The Pentagram Map, Experimental Math. 1 (1992), 71–81.[6] R. Schwartz, Discrete Monodromy, Pentagrams, and the Method of Condensation J. FixedPoint Theory Appl. 3 (2008), 379–409.102 Oberwolfach Report 02Piecewise linear saddle spheres on S3Gaiane PaninaBy S3 ⊂ R4 we denote the unit sphere centered at the origin O. A plane onthe sphere S3 is a plane in the sense of spherical geometry. A closed surface Γimmersed in S3 is called saddle if no plane intersects Γ locally at just one point.Definition. A piecewise linear saddle sphere (a PLS-sphere, for short) on S3 isan immersed piecewise linear surface which is homeomorphic to S2. We assume inaddition that a PLS-sphere does not coincide with a plane, that all its edges areshorter than π, and that its vertex-edge graph is 3-connected.A PLS-sphere is called elementary Barner if there is a point p ∈ S3 such thateach great semicircle with endpoints p and −p hits the surface exactly once.The interplay between PLS-spheres and smooth saddle spheres is not well un-derstood. On the one hand, it seems plausible that a piecewise linear saddlesphere can be approximated by a smooth saddle sphere and vice versa. On theother hand, there is just one proven result (see [7]). It asserts that an elementaryBarner PLS-sphere with a trivalent vertex-edge graph has a C∞-smooth saddleapproximation.Definition. A face f of a PLS-sphere Γ is called an inflexion face if(1) f is bounded by two convex broken lines (say, by L1 and L2) such that theconvexity directions look like in Figure 1,(2) all the edges of L1 are convex, whereas all the edges of L2 are concave.Figure 1. An inflexion face.Definition. A face f of a PLS-sphere Γ is called a reflex face if its supplementfits an open hemisphere.We announce here the following result:Theorem.(1) Each immersed saddle sphere Γ ⊂ S3 belongs to one of the following disjointclasses:(a) Γ has at least two reflex faces.(b) Γ has exactly one reflex face and at least two inflexion faces.Discrete Differential Geometry 103(c) Γ has no reflex faces and at least 4 inflexion faces.(2) There are no embedded PLS-spheres on S3 of type (a).(3) There are no embedded PLS-spheres on RP 3.(4) There exist immersed PLS-spheres on RP 3.(5) There exist elementary Barner PLS-spheres of type (c). Moreover, the setof elementary Barner PLS-spheres with any fixed number of inflexion faces(greater than 3) is non-empty and disconnected.Sketch of the proof. Given an oriented PLS-sphere, we can speak of its convexand concave edges. We paint all the convex (respectively, concave) edges red(respectively, blue).Combinatorially, a PLS-sphere is a planar graph with additional equipment:its edges are colored and some of the angles (the reflex ones) are marked. (Somesimilar phenomena are studied in [3] and [4].)Definition. For a face f of a PLS-sphere, we algorithmically define its index i(f):(1) At the beginning, put i(f) := 0. Start going along the boundary of the face f .(2) Once we pass by a vertex at which the color changes, put i(f) := i(f) + 1.(3) Once we pass by a vertex, if the color does not change and the angle we arepassing by is not reflex, put i(f) := i(f)+2. (If the color does not change andthe angle is reflex, we keep i(f) unchanged.)Some counts based on a discrete version of Segre’s Theorem yield that one ofthe following cases holds:(1) Γ has at least two faces with i(f) = 0.(2) Γ has one face with i(f) = 0 and at least two faces with (f) = 2(3) Γ has no faces with i(f) = 0 and at least 4 faces with i(f) = 2.This yields statement (1) of the Theorem. The statements (2–4) follow from (1),whereas (5) is proven in [5, 7]. Motivations.• The proof of the Theorem is based on and generalizes the Segre’s Theorem: Leta closed smooth simple curve c ⊂ S2 have a non-empty intersection with anygreat semicircle. Then c has at least 4 inflexion points.• There exist embedded saddle tori on RP 3. V. I. Arnold [2] formulated someconjectures about them (and about their higher dimensional versions). Someof the conjectures proved to be wrong [6], but two of them still stand open forRP 3. In particular, he conjectured that the set of all smooth embedded saddletori on RP 3 is connected (compare with Theorem, (5)).• Smooth elementary Barner saddle spheres arose originally in a relationship (see[5, 7]) to the following uniqueness conjecture proven for analytic surfaces by A.D. Alexandrov in [1]:Let K ⊂ R3 be a smooth convex body. If for a constant C, at every point of∂K, we have R1 ≤ C ≤ R2, then K is a ball. (R1 and R2 stand for the principalcurvature radii of ∂K).104 Oberwolfach Report 02Here is the link: Let K be a counterexample to the conjecture. Denote byhK its support function and denote by hC the support function of the ball ofradius C. The graph γ of the difference hK − hC is a conical surface in R4with the apex at the origin O. Its intersection with S3 is an elementary Barnersaddle sphere.Vice versa, a smooth elementary Barner saddle sphere spans a cone in R4which can be interpreted as the graph of some positively homogeneous functionh. For a sufficiently large C, the sum h + hC is a convex function. Then it isa support function of some convex body K which is a counterexample to theconjecture.References[1] A. D. Alexandrov, On uniqueness theorems for closed surfaces. (Russian) Doklady Akad.Nauk SSSR 22 (1939) no. 3, 99–102.[2] V. Arnold, Ramified covering CP 2 → S4, hyperbolicity, and projective topology, SiberianMath. J. 29 (1988) no. 5, 36–47.[3] S. Gortler, C. Gotsman, D. Thurston, Discrete one-forms on meshes and applications to3D mesh parameterization, Comput. Aided Geom. Design 23 (2006) no. 2, 83–112.[4] R. Haas, D. Orden, G. Rote, F. Santos, B. Servatius, H. Servatius, D. Souvaine, I. Streinu,and W. Whiteley, Planar minimally rigid graphs and pseudotriangulations, J. Comput.Geom. 31 (2005) no. 1–2, 31–61.[5] Y. Martinez-Maure, Contre-exemple à une caractérisation conjecturée de la sphère, C. R.Acad. Sci. Paris Sér. I Math. 332 (2001) no. 1, 41–44.[6] S. Orevkov, A la recherche de la topologie projective. Du cote de chez Arnold. Talk on theInternational Conference “Arnold-70”, Moscow, August 20–25, 2007.[7] G. Panina, New counterexamples to A. D. Alexandrov’s uniqueness hypothesis, Advances inGeometry 5 (2005) no. 2, 301–317.Discrete conformal equivalence for triangle meshesBoris Springborn(joint work with A. I. Bobenko, U. Pinkall, P. Schröder)Two Riemannian metrics g and g̃ on a smooth 2-manifold M (possibly with bound-ary) are conformally equivalent if g̃ = e2ug for some u : M → R. This means thatinfinitesimal lengths are scaled by a conformal factor eu which does not dependon direction but only on position. In this talk, we explore a straightforward dis-cretization of this concept of conformal equivalence [1]. Instead of smooth surfaceswe consider surface triangulations, and instead of Riemannian metrics g we con-sider functions ℓ on the set of edges which assign to each edge ij a length ℓij insuch a way that the triangle inequalities are satisfied for all triangles of the trian-gulation. We consider two such discrete metrics ℓ and ℓ̃ as discretely conformallyequivalent if a conformal factor eui can be associated to each vertex i such thatfor all edges ijℓ̃ij = e(ui+uj)/2ℓij .Discrete Differential Geometry 105In terms of the logarithmic edge lengths λ = 2 log ℓ, this relation is linear:(1) λ̃ij := λij + ui + uj .Edge lengths ℓ̃ij determine triangle angles. Let α̃kij be the angle at vertex k intriangle ijk with edge lengths ℓ̃ij , ℓ̃jk, ℓ̃ki, and let Θ̃k =∑ijk∋k α̃kij be the sum ofangles around vertex k. We consider the following problem:Given (i) a triangulation with a discrete metric ℓ, and (ii) a desired anglesum Θ̂k for each vertex k, find a discretely conformally equivalent metric ℓ̃ thathas at each vertex the desired angle sum, that is, Θ̃k = Θ̂k for each vertex k. Inparticular, we are interested in the case where the desired angle sum Θ̂k equals2π for all interior vertices k. In this case, the problem asks for a discretely confor-mally equivalent flat metric with prescribed total angles at the boundary vertices(discrete conformal flatterning problem).Analytically, the above problem amounts to one non-linear equation per vertexin the variables ui, which are also associated to the vertices. It turns out thatthese equations are variational: The discrete metric ℓ̃ solves the problem if andonly if u is a critical point of the functionS(u) =∑ijk(2f(12 λ̃ij ,12 λ̃jk,12 λ̃ki)− π(ui + uj + uk))+∑iΘ̂iui,where the first sum is taken over all triangles ijk of the triangulation, the secondsum is taken over all vertices i, the λ̃ij are defined by equation (1) as functionsof u, and the function f(x1, x2, x3) is defined as follows: If x1, x2, x3 are realnumbers such that there exists a Euclidean triangle with sides ex1, ex2 , ex3 , thenlet α1, α2, α3 be the opposite angles, and definef(x1, x2, x3) = α1x1 + α2x2 + α3x3 +Ë(α1) +Ë(α2) +Ë(α3),where Ë(α) = − ∫ α0log|2 sin t| dt is Milnor’s Lobachevsky function [2]. It turnsout that ∂f∂xi = αi, and therefore∂S∂ui= Θ̂i − Θ̃i, which proves the claim about thecritical points of S(u).The above definition of f(x1, x2, x3) assumes that ex1 , ex2 , ex3 satisfy the tri-angle inequalities. Otherwise, a triangle with these sides does not exist and theangles α1, α2, α3 are not defined. However, the definition of f can be extended toall of R3 by setting the angles to 0, 0, π whenever a triangle inequality is violated,where the value π is assigned to the angle opposite the side that is too long. If f isextended in this way, it is still continuously differentiable and convex. (It is strictlyconvex in the original domain and linear outside.) In fact, the extended functionf is (up to a factor of π) the Ronkin function of the polynomial z1 + z2 + z3:1π f(x1, x2, x3) = (12πi)3∫∫∫S1(ex1 )×S1(ex2 )×S1(ex3 )log∣∣z1 + z2 + z3∣∣ dz1z1∧ dz2z2 ∧dz3z3.(This Ronkin function also played an important role in Rick Kenyon’s talk.)106 Oberwolfach Report 02With f extended in this way, the function S(u) is convex and defined onR(#vertices). This reduces the discrete conformal flattening problem to an uncon-strained convex optimization problem which is easy enough to solve numericallyto render this approach useful for practical applications [3].The fact that Milnor’s Lobachevsky function appears in the variational principleindicates that there should be some connection with hyperbolic geometry. Indeed,the discrete conformal flattening problem is equivalent with the following one [4]:Given a surface with complete hyperbolic metric with cusps, find a hyperbolicpolyhedron with vertices at infinity whose boundary is isometric to the givensurface. The same variational principle can also be used to solve this problem.References[1] Feng Luo. Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6:5 (2004),765–780.[2] John Milnor. Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc. (NS) 6:1(1982), 9–24.[3] Boris Springborn, Peter Schröder, and Ulrich Pinkall. Conformal equivalence of trianglemeshes. ACM Transactions on Graphics 27:3. [Proceedings of ACM SIGGRAPH ’08][4] Alexander I. Bobenko, Ulrich Pinkall, and Boris Springborn. Discrete conformal equivalenceand ideal hyperbolic polyhedra. In preparation.Open problems in Discrete Differential GeometryCollected by Günter RoteProblem 1 (Günter M. Ziegler). What is the smallest possible maximum ver-tex degree f(d) for a centrally symmetric triangulation of the d-sphere? (Or asimplicial (d + 1)-polytope.) The known bounds are f(1) = 2, f(2) = 4, andd + 1 < f(d) ≤ 2d. The upper bound comes from the cross-polytope. If one couldshow f(d) < 3d/2 for some large d that would have interesting consequences.Problem 2 (Richard Kenyon). Let M be a closed polyhedral surface homeomor-phic to S2 which is entirely composed of equal regular pentagons. If M is immersedin 3-space, is it necessarily the boundary of a union of solid dodecahedra that areglued together at common facets? The pentagonal faces may intersect each other(and the “union of solid dodecahedra” must be defined appropriately) but twodifferent faces are not allowed to coincide. (The corresponding question for equalsquares has an affirmative answer.)Note (Ulrich Brehm): The great dodecahedron (Kepler-Poinsot polyhedron) isan interesting example, but it has genus 4. Moreover, the vertex-figures are self-crossing pentagrams, and therefore the surface is not immersed. The questionwould be interesting even for immersions of arbitrary closed surfaces, and for ori-entable closed surfaces of genus smaller than four where arbitrary selfintersecctionsare allowed.Discrete Differential Geometry 107Problem 3 (Wolfgang Kühnel). Let Md be a triangulated d-manifold (simplyconnected, closed) with a discrete metric such that the discrete curvature (angledefect) along any (d − 2)-simplex is positive. Give a discrete proof that Md ishomeomorphic to Sd. (Or give a counterexample.)For d = 3, recent unpublished work of Matveev/Shevshichin verifies this byexplicitly smoothing the metric; presumably the smoothing could also be doneusing Ricci flow. See also earlier work of Cheeger for general dimensions.Note: Nonnegative curvature is not sufficient, as demonstrated by a flat torus.CP 2 has a metric of positive sectional curvature, but the polyhedral conditionshould correspond to the stronger condition of positive curvature operator.Problem 4 (Joseph O’Rourke). Can a finite number of disjoint (closed) line seg-ments in the plane, acting as 2-sided mirrors, and a point light source be arrangedso that no light ray escapes to infinity? It seems most natural to treat the mirrorsas open segments, but they should be disjoint when closed. The conjecture is thatthis is impossible.Problem 5 (Günter Rote). Take the complete graph K4 embedded in the planein general position, with vertices p1, p2, p3, p4. Pick two arbitrary points a and band define two functions ωij and fij on the six edges of this graph:(1) ωij :=1[pipjpk][pipjpl], fij := [apipj ][bpjpj],where k and l are the two vertices different from i and j, and [q1q2 . . . qn] denotessigned area of the polygon q1q2 . . . qn. (The function ωij is a self-stress: theequilibrium condition∑j ωij(pj − pi) = 0 holds for every vertex i, where thesummation is over all edges ij incident to i.) Then we have the following identity:(2)∑1≤i 0.Definition (Discrete gradient vector field). A gradient vector field is a discretevector field V that does not admit any nontrivial closed V-paths.Definition (Critical cell). A cell σ is a critical cell with respect to a discretegradient vector field V if σ is not contained in any pair of V . A cell that is notcritical is a regular cell.The main technique for reducing the number of critical points is that of reversinga gradient vector field V along a V -path between two critical cells τ and σ:Theorem ([2], Theorem 11.1). Let σ and τ be critical cells of a gradient vectorfield V with a unique V -path Γ from ∂τ to σ. Then there is a gradient vectorfield Ṽ obtained by reversing V along the path Γ. The critial cells of Ṽ are exactlythe critical cells of V other than {σ, τ}. In particular, V = Ṽ except along thepath Γ.As in smooth Morse theory, a discrete gradient vector field can be understoodas the gradient of some function in the following sense:Definition ((Pseudo-)Morse function). A discrete Morse function is a functionf : K → R on the cells of a regular CW complex K if there is a gradient vectorfield V such that for all pairs of cells we haveσ is a facet of τ ⇒{f(σ) < f(τ) if (σ, τ) 6∈ V,f(σ) ≥ f(τ) if (σ, τ) ∈ V.For a discrete pseudo-Morse function, the strict inequality is replaced by a weakone, i.e., f(σ) ≤ f(τ) if (σ, τ) 6∈ V . In either case, we call V consistent with f .Definition (Induced partial order). The partial order ≺V induced by a discretegradient vector field V is the transitive relation generated byσ is a facet of τ ⇒{σ ≺V τ if (σ, τ) 6∈ V,σ ≻V τ if (σ, τ) ∈ V.For any pseudo-Morse function g consistent with V and any pair of cells (φ, ρ),φ ≺V ρ implies g(φ) ≤ g(ρ).3. Persistent Morse homologyHomological persistence [1] is used to investigate the change of the homologygroups in a sequence of nested topological spaces. We study nested subcomplexesof a given CW complex.Definition (Level subcomplex). Let f be a discrete Morse function on a regularCW complex K. For a cell σ ∈ K, the level subcomplex is the subcomplex of Kconsisting of all cells ρ with f(ρ) ≤ f(σ) together with their faces:K(σ) :=⋃ρ∈Kf(ρ)≤f(σ)⋃φ∈Kφ≤ρφ .118 Oberwolfach Report 02For K(φ) ⊂ K(ρ), let iφ, ρ∗ : H∗(K(φ)) → H∗(K(ρ)) denote the homomorphisminduced by inclusion. Let σ and τ be critical cells of dimension d and (d + 1),respectively, such that f(σ) < f(τ). The predecessor of σ is the cell σ− with thelargest f -value such that f(σ−) < f(σ), and similarly for τ−. Now consider thesequenceHd(K(σ−)) → Hd(K(σ)) → Hd(K(τ−)) → Hd(K(τ))induced by inclusion.Definition (Birth, death, persistence pair). Let f be an injective Morse functionon a regular CW complex. We say that a class h ∈ H∗(K(σ)) is born at (or createdby) σ ifh 6∈ im(iσ−, σ∗ ).Moreover, we say that a class h ∈ H∗(K(σ)) that is born at σ dies entering (orgets merged by) τ ifiσ, τd (h) ∈ im(iσ−, τ∗ ) but iσ, τ−d (h) 6∈ im(iσ−, τ−∗ ).If there exists a class h that is born at σ and dies entering τ , then (σ, τ) is apersistence pair. The difference f(τ) − f(σ) is called the persistence of (σ, τ).To uniquely define persistence pairs for a pseudo-Morse function f consistentwith some gradient vector field V , we require a total order on the cells. This canbe achieved by extending the partial order ≺V to a total order, which allows us tospeak about persistence pairs of (f, V ).4. Topological simplification of functionsFrom now on, let f be a pseudo-Morse function consistent with a discrete gra-dient vector field V on a combinatorial surface K. From the stability theorem forpersistence diagrams [4], we can deduce the following lower bound on the numberof persistence pairs, and therefore on the number of critical points:Lemma. For a pseudo-Morse function fδ with ‖fδ −f‖∞ < δ and consistent witha gradient vector field Vδ, the number of persistence pairs of (fδ, Vδ) is boundedfrom below by the number of persistence pairs of f with persistence ≥ 2δ.We are interested in functions that achieve this lower bound:Definition (Perfect δ-simplification). A perfect δ-simplification of (f, V ) is apseudo-Morse function fδ consistent with a gradient vector field Vδ, such that‖fδ − f‖∞ < δ and the number of persistence pairs of (fδ, Vδ) is equal to thenumber of persistence pairs of f with persistence ≥ 2δ.Our main result states that a perfect δ-simplification always exists for a discretepseudo-Morse function on a combinatorial surface.Theorem. Let f be a discrete pseudo-Morse function on a combinatorial surface.Then there exists a perfect δ-simplification of f .Discrete Differential Geometry 119The proof of this theorem is constructive. An analogous statement is not truein higher dimensions or for non-manifold complexes.References[1] H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification.Discrete Comput. Geom., 28(4):511–533, 2002.[2] R. Forman. Morse theory for cell complexes. Adv. Math., 134(1):90–145, 1998.[3] H. Edelsbrunner, D. Morozov, and V. Pascucci. Persistence-sensitive simplification of func-tions on 2-manifolds. In SCG ’06: Proceedings of the 22nd ACM Symposium on Computa-tional Geometry, pages 127–134. ACM, 2006.[4] D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. DiscreteComput. Geom., 37(1):103–120, 2007.Approximation of conformal mappings by circle patternsUlrike BückingConformal mappings constitute an important class in the field of complex analysis.They may be characterized by the fact that infinitesimal circles are mapped toinfinitesimal circles. Thurston first introduced in his talk [10] the idea to usefinite circles, in particular circle packings, to define a discrete conformal mapping.Remember that an embedded planar circle packing is a configuration of closed diskswith disjoint interiors in the plane C. Various connections between circle packingsand classical complex analysis have already been studied. A beautiful introductionand surway is presented by Stephenson in [9].The class of circle patterns generalizes circle packings as for each circle packingthere is an associated orthogonal circle pattern. Simply add a circle for eachtriangular face which passes through the three touching points. To define a circlepattern we use a planar graph as combinatorial data. The circles correspond tovertices and the edges specify which circles should intersect. The intersectionangles are given using a labelling on the edges. Thus an edge corresponds to akite of two intersecting circles as in Figure 1. A face of the graph corresponds toa point where all the circles corresponding to the incident vertices intersect. Suchintersection points are colored black in Figure 1. Moreover, for interior vertices thekites corresponding to the incident edges have disjoint interiors and their union ishomeomorphic to a closed disk. Thus to a circle pattern we also associate a kitepattern.αFigure 1. The exterior intersection angle α of two intersectingcircles and the associated kite built from centers and intersectionpoints.120 Oberwolfach Report 02Given two circle patterns C1 and C2 with the same combinatorics and intersec-tion angles, define a mapping gC : C1 → C2. Namely, take gC to map the centersof circles and the intersection points of C1 corresponding to vertices and faces ofG to the corresponding centers of circles and intersection points of C2 and extendit to an affine map on each kite.For a given conformal map g we use an analytic approach and specify suitableboundary values for the radius function according to |g′| in order to define the(approximating) mappings gC . Generalizing ideas of Schramm’s convergence proofin [8] we obtain convergence in C1 on compact sets if we take for C1 a sequence ofisoradial circle patterns (i.e., all radii are equal) with decreasing radii ε → 0 whichapproximate the domain of g. Note in particular that the combinatorics of thecircle patterns C1 may be irregular or change within the sequence. More precisely,we obtain for example the following result.Theorem. Let W ⊂ C be open and let g : W → C be a locally injective holomor-phic function. Let K ⊂ C be a compact set whose interior is simply connected andwhich is contained in W .Consider a sequence C(n)1 of isoradial circle patterns with decreasing radii εn →0. Assume that the region covered by the closed disks filling the circles of the circlepattern C(n)1 is contained in W and covers K for each n ∈ N. Furthermore, thecenters of circles not contained in K have a distance of at most 2εn to K. As-sume further that the intersection angles of the circle patterns C(n)1 are uniformlybounded away from 0 and π.Then a sequence of image circle patterns C(n)2 can be defined by setting rn(v) =εn|g′(v)| for the radius rn(v) of the boundary centers of circles, where v is thecorresponding center of circle of C(n)1 . Furthermore, we adjust the translationaland rotational freedom of C(n)2 according to the image of g at one vertex (perconnected component of the circle pattern). Then we have for all centers of circlesw of C(n)1g(w) = gC (n)(w) + O(εn)(1)|g′(w)| = rn(w)/εn + O(εn).(2)The implicit constants in O(. . . ) of these estimations depend on g and W , K,and the uniform bound on the intersection angles, but not on the combinatorics ofC(n)1 .The main idea of the proof is to consider a “nonlinear discrete Laplace equation”for the radius function rn. This equation turns out to be a (good) approximationof a known linear Laplace equation and can be used in the case of isoradial cir-cle patterns C(n)1 to compare discrete and smooth solutions of the correspondingelliptic problems. In particular, we obtain (2) if εn is small enough.Isoradial circle patterns are closely related to rhombic embeddings. Simplybicolor the vertices of such an embedding and add circles centered at all white (orall black) vertices with radius equal to the edge length. For rhombic embeddings,Discrete Differential Geometry 121there is an asymptotic development given by Kenyon in [7] of a discrete Green’sfunction which is slightly generalized in [2] and in [4]. Using similar ideas asDuffin in [5], we can generalize theorems of discrete potential theory concerningthe regularity of solutions of a discrete Laplace equation, see [2]. This regularitylemma, estimation (2), and a Taylor expansion of the analytic closing condition forthe radii form the basis of our proof of C1-convergence claimed in the theorem.Theproof generalizes a method used by He and Schramm in [6].Using a repeated application of the regularity lemma, we also obtain C∞-convergence on compact sets for a class of isoradial circle patterns. In order todefine higher derivatives, we consider the rhombic embeddings as combinatorialsurfaces in Zdn for appropriate dn as in [1]. Using flips of 3-dimensional cubes,this original surfaces can be changed without changing their boundary curve. Thisleads to some region reached by flips in Zdn . Here we still have the same estima-tions. For a fixed copamct subset K we now additionally assume that the com-binatorial distance in at least two directions of this region is comparable to thecombinatorial distance to the boundary, independently of n.An extended and more detailed version of partially weaker results can be foundin [3, 2].References[1] Alexander I. Bobenko, Christian Mercat, and Yuri B. Suris, Linear and nonlinear theoriesof discrete analytic functions. Integrable structure and isomonodromic Green’s function,J. reine angew. Math. 583 (2005), pp. 117–161.[2] Ulrike Bücking, Approximation of conformal mappings by circle patterns. Geom. Dedicata137, 163–197 (2008)[3] Ulrike Bücking, Approximation of conformal mappings by circle patterns and discreteminimal surfaces. Ph.D. thesis, Technische Universität Berlin (2007). Published online athttp://opus.kobv.de/tuberlin/volltexte/2008/1764/[4] Dmitry Chelkak and Stanislav Smirnov, Discrete complex analysis on isoradial graphs.arXiv:0810.2188v1 [math.CV] (2008)[5] Richard J. Duffin, Discrete potential theory. Duke Math. J. 20, 233–251 (1953)[6] Zheng-Xu He and Oded Schramm, The C∞-convergence of hexagonal disk packings to theRiemann map. Acta Math. 180, 219–245 (1998)[7] Richard Kenyon, The Laplacian and Dirac operators on critical planar graphs. Invent. Math.150, 409–439 (2002)[8] Oded Schramm, Circle patterns with the combinatorics of the square grid. Duke Math. J.86, 347–389 (1997)[9] Kenneth Stephenson, Introduction to circle packing: the theory of discrete analytic func-tions. Cambridge University Press, New York (2005)[10] William Thurston, The finite Riemann mapping theorem (1985). Invited address at theInternational Symposioum in Celebration of the proof of the Bieberbach Conjecture, PurdueUniversity.122 Oberwolfach Report 02Stability of the foldHerbert Edelsbrunner(joint work with Dmitriy Morozov and Amit Patel)The fold of a smooth mapping f : M → Rk from a compact n-manifold to Eu-clidean space of dimension k ≤ n is the image of the points at which the gradientsof the k component functions are linearly dependent. The fold decomposes Rkinto chambers which we glue along shared faces to form immersed k-manifolds.Measuring difference with the erosion distance (the Hausdorff distance for thecomplements adapted to immersed k-manifolds), we prove that the fold is stable.Specifically, for a second smooth mapping g : M → Rk, we can form correspondingimmersed k-manifolds such that the erosion distance between corresponding pairsis bounded from above by the maximum Euclidean distance between points f(x)and g(x).References[1] V. I. Arnol’d. Catastrophe Theory. Third edition, Springer-Verlag, Berlin, Germany, 1992.[2] D. Cohen-Steiner, H. Edelsbrunner and J. Harer. Stability of persistence diagrams.Discrete Comput. Geom. 37 (2007), 103–120.[3] H. Edelsbrunner, D. Morozov and A. Patel. Stability of the fold. Manuscript, Dept.Comput. Sci., Duke Univ., Durham, North Carolina, 2009.[4] M. Golubitsky and V. Guillemin. Stable Mappings and Their Singularities. Springer-Verlag, New York, 1973.[5] J. J. Koenderink. What does the occluding contour tell us about solid shape? Perception13 (1984), 321–330.[6] H. Whitney. On singularities of mappings of Euclidean space. I. Mappings of the plane tothe plane. Ann. Math. 62 (1955), 374–410.The Angle Defect and Its GeneralizationsEthan D. BlochThe talk started with a review of what is most likely the first theorem in DiscreteDifferential Geometry, namely Descartes’ Theorem concerning the angle defect forconvex polyhedra in R3. This theorem was formulated before there was a subjectcalled differential geometry. For a triangulated polyhedral surface M2 in Euclideanspace, the usual notion of curvature at a vertex v is the classical angle defect dv =2π−∑αi, where the αi are the angles of the triangles containing v. The classicalangle defect satisfies some standard properties one would expect: it is invariantunder polyhedral local isometries; it is locally defined; it is zero at a vertex thathas a flat star; it is invariant under subdivision; and it satisfies the polyhedralGauss-Bonnet Theorem, which says∑v dv = 2πχ(M2), where the summation isover all the vertices of M2, and χ(M2) is the Euler characteristic of M2. Descartes’Theorem, which predates Euler, is the special case of the polyhedral Gauss-BonnetTheorem when M2 is a convex polyhedron in R3. See [8] for the text of Descartes’work on polyhedra, though Descartes did not give a proof of his theorem; see [12,Discrete Differential Geometry 123Chapter 25] for two proofs, the second of which is plausibly the one Descartes hadin mind.The bulk of the talk was a description of three methods for generalizing theclassical angle defect to arbitrary finite simplicial complexes of all dimensionsimmersed in Euclidean space. One method, which I refer to as standard curvature,has been studied from a differential geometric point of view, for example in [1], [6]and [14]. This approach to generalizing the angle defect, which is based on exteriorangles, is simple to define (though it does not directly resemble the classical angledefect), and its convergence properties have been well studied. On the other hand,the definition of standard curvature concentrates all the curvature at the vertices ofsimplicial comlexes, which does not necessarily correspond to our intuition aboutcurvature in dimension greater than 2.A different approach to generalizing the classical angle defect, known simplyas the angle defect (or angle deficiency), has been studied in the case of convexpolytopes by a number of combinatorialists, for example [13] and [9]. In [10] aGauss-Bonnet theorem is proved for the angle defect in polytopes with underlyingspaces that are manifolds. In contrast to standard curvature, the angle defect forconvex polytopes is found at each cell of codimension at least 2, and the definitioncompletely resembles the classical case.The talk then discussed the possibility of extending the angle defect, as definedfor polyhedral manifolds, to arbitrary finite immersed simplicial complexes. Thedefinition of angle defect for polyhedral manifolds cannot be used without mod-ification for arbitrary simplicial complexes, because it does not satisfy a Gauss-Bonnet theorem. A method of extending the angle defect to all finite immersedsimplicial complexes, called the generalized angle defect, was then discussed, fol-lowing [3]. This approach is based upon a simple topological decomposition ofeach simplicial complex, which allows for a suitable replacement for “2π” in thedefinition of the classical angle defect, where the replacement for a given simplexdepends upon the topological nature of the neighborhood of the simplex. Bothstandard curvature and the generalized angle defect satisfy the expected proper-ties, such as being locally defined, invariant under local isometries, and satisfyinga Gauss-Bonnet theorem, though the Gauss-Bonnet Theorem for the generalizedangle defect uses a modified Euler characteristic rather than the standard Eulercharacteristic; this modified Euler characteristic reduces to the standard Eulercharacteristic in the case of pseudomanifolds.Although in many respects the generalized angle defect behaves as nicely asstandard curvature, there is one exception. In [1, Section 5] it is stated that foran odd-dimensional polyhedral manifold, the standard curvature is zero at everyvertex. It then follows from the Gauss-Bonnet theorem for standard curvature thatevery odd-dimensional polyhedral manifold has Euler characteristic zero (a well-known fact, but the method of [1, Section 5] yields a completely elementary proof).By contrast, the generalized angle defect is never zero for a convex polytope ofany dimension, as seen in [13]. The talk then discussed the method of [4], where avariant of the generalized angle defect is defined that satisfies the nice properties124 Oberwolfach Report 02of the generalized angle defect, and yet also has the additional property that itis identically zero for any odd-dimensional simplicial complex K (of dimensionat least 3) such that χ(link(ηi, K)) = 2 for all i-simplices ηi of K, where i isan even integer such that 0 ≤ i ≤ n − 1. As a corollary, we deduce that ifK is an odd-dimensional simplicial complex (of dimension at least 3) such thatχ(link(ηi, K)) = 2 for all i-simplices ηi of K, where i is an even integer such that0 ≤ i ≤ n−1, then K has Euler characteristic zero. This last result was proved forEuler spaces by different methods in [2], [7] and [11]; the result for Euler spacesturn out to be equivalent to the above corollary. The definition of this variant ofthe generalized angle defect is based, somewhat surprisingly, on a sequence thatmakes use of the Bernoulli numbers.Given that there is more than one way to generalize the classical angle defect,it would be helpful to understand the relations between these generalizations, todetermine which approach is useful in which situations. One method of comparingthe different approaches would be to find an axiomatic characterization of each.Although the classical angle defect has been widely studied, there does not appearto be in the literature an axiomatic characterization of it. The final part of thetalk discussed such a characterization in dimension 2, as found in [5], where it isproved that the classical angle defect for embedded simplicial surfaces is character-ized by being invariant under simplicial isometries of stars and under subdivision,continuous, and satisfying the Gauss-Bonnet theorem with respect to the Eulercharacteristic. The same characterization also works for both standard curvatureand the generalized angle defect for arbitrary immersed 2-dimensional simplicialcomplexes, except that for the generalized angle defect it is necessary to use themodified Euler characteristic.References[1] Thomas Banchoff, Critical points and curvature for embedded polyhedra, J. DifferentialGeom., 1, (1967), 245–256.[2] Thomas Banchoff, Critical points and curvature for embedded polyhedra, II, Progress inMath., 32, (1983), 34–55.[3] Ethan D. Bloch, The angle defect for arbitrary polyhedra, Beiträge Algebra Geom., 39,(1998), 379–393.[4] Ethan D. Bloch, The angle defect for odd-dimensional simplicial manifolds, Discrete Com-put. Geom., 35, (2006), 311–328.[5] Ethan D. Bloch, A characterization of the angle defect and the Euler characteristic indimension 2, to appear in Discrete Comput. Geom.[6] Jeff Cheeger, Werner Müller, and Robert Schrader, On the curvature of piecewise flat spaces,Commun. Math. Phys., 92, (1984), 405–454.[7] Beifang Chen and Min Yan, Eulerian 2-strata spaces, J. Combin. Theory Ser. A, 85, (1999),1–28.[8] Pasquale J. Federico, Descartes on Polyhedra, Springer-Verlag, New York, 1982.[9] Branko Grünbaum, Grassmann angles of convex polytopes, Acta Math., 121, (1968), 293–302.[10] Branko Grünbaum and G. C. Shephard, Descartes’ theorem in n dimensions, Enseign.Math. (2), 37, (1991), 11–15.[11] Colin MacLaurin and Guyan Robertson, Euler Characteristic in Odd Dimensions, Austral.Math. Soc. Gaz., 30, (2003), 195–199.Discrete Differential Geometry 125[12] Igor Pak, Lectures on Discrete and Polyhedral Geometry, http://www.math.umn.edu/∼pak/book.htm.[13] G. C. Shephard, Angle deficiencies of convex polytopes, J. London Math. Soc., 43, (1968),325–336.[14] M. Zähle, Approximation and characterization of generalized Lipschitz-Killing curvatures,Ann. Global Anal. Geom., 8, (1990), 249–260.Square pegs and beyondBenjamin MatschkeThis talk gave a collection of new theorems and re-opened conjectures related tothe following famous square peg problem.Conjecture (Square Peg Problem, Toeplitz [9]). Does every continuously embed-ded circle in the plane contain four points spanning a square?This conjecture has been proved in the special case whenthe curves are smooth enough [7], [8], but it is still open forcontinuous curves. In particular, certain local spirals invali-date all known approaches for continuous curves, all of whichuse the fact that “generic” curves circumscribe an odd numberof squares.Equilateral Triangles on CurvesTheorem ([5, Thm. III.3.2]). Let d : S1 × S1 −→ R be a contin-uous symmetric map (a generalised metric). Then there are threepoints x, y, z ∈ S1, not all of them equal, such thatd(x, y) = d(x, z) = d(y, z).Polygons on CurvesLemma ([5, Lem. III.4.7]). Let γ : S1 −→ M be a C∞-embedded curve in aRiemannian manifold M . Let n be a prime power ≥ 3, ε > 0 and let P ⊂ (S1)nbe the set of polygons whose vertices lie counter-clockwise on γ. Then there is aclosed one-parameter family S1 −→ P of such polygons such that(1) each of the polygons are up to ε edge-regular, i. e. the edge ratios lie in theinterval [1 − ε, 1 + ε], and(2) this one-parameter family is invariant under cyclic permutation of the vertices.It can be proved using the fact that an isotopy of γ changesthe solution set of all edge-regular n-gons on γ by a bordism.It turned out that Makeev already gave a very similar re-sult [4, Thm. 3]. Here are two known and one new directcorollaries:Corollary (Smooth Square Peg Problem, [7], [8]). Each C∞-embedded circle inthe plane contains four points spanning a square.126 Oberwolfach Report 02Corollary (A Conjecture of Hadwiger, [4, Thm. 4], [10, Thm. 11]). Each knot,that is, a C∞-embedded circle in R3, contains four points spanning a planar rhom-bus.Corollary (Blagojević–M., see [5, Thm. III.6.1]). Let d1 and d2 be two symmetricdistance functions on S1, where d1 is given by a C∞-embedding of S1 into aRiemannian manifold. Then there are three pairwise distinct points on S1 formingan equilateral triangle with respect to d1 and an isosceles triangle with respect tod2.Rectangles on CurvesConjecture. Does every C∞-embedded circle in R2 contain 4 points spanning arectangle with prescribed edge-ratios?Griffiths [1] gave a proof of this conjecture, however thereare slight errors in his calculation of orientations, such thatan intersection number is zero instead of 16. Hence a desiredintersection was not shown (that is, I do not know how to fixit [5, III.7]).An ansatz towards solving this conjecture is the following: If the conjecturewere wrong, then there would be a ratio r > 0 and a curve γ : S1 → R2 notcontaining rectangles of ratio r such that for all ε > 0 there is a Z4-invariantone-parameter family of up-to-ε-rectangles S1 → (γ(S1))4, such that the verticesof the rectangles lie counter-clockwise on γ [6]. It seems that this is essentiallywhat topology can give us, but now more geometic ideas are needed to deduce acontradiction.Tetrahedra on Closed SurfacesTheorem ([5, Thm. III.8.5]). Every smooth closed surfacein R3 contains four points spanning a tetrahedron that issimilar to a given one.One can even prescribe an arbitrary vertex.Regular Octahedra on Embedded SpheresConjecture. Does every C∞-embedded S2 in R3 containsix points spanning a regular octahedron?Guggenheimer [2] gave a proof of this conjecture, howeverthere is an error in his main lemma, which he used to provethe smooth square peg problem, and which he claimed tobe generalisable to prove the above conjecture.Discrete Differential Geometry 127This conjecture is probably very difficult and a solution would involve deepergeometric reasoning, since there is the following “topological counter-example”.Let G be the symmetry group of the regular octahedron and Gor ⊂ G the sub-group of orientation preserving symmetries. G acts on (S2)6 by permuting thecoordinates in the same way as it permutes the vertices of the regular octahedron.Let G act on R12 by permuting the coordinates in the same way as it permutes theedges of the regular octahedron. The subrepresentation (R ·1)⊥ ⊂ R12 is denotedby Y . LetX := {(x1 . . . x6) ∈ (S2)6 | xi are pairwise distinct}.Then any embedding Γ : S2 → R3 gives us a test mapt : (S2)6 −→G Y,which measures the edges of the parametrised octahedra. The solution set S ofregular octahedra on Γ is S := t−1(0) ∩ X . The subset of S of positively orientedoctahedra gives us a well-defined element in the equivariant normal bordism group(see [3])ΩGor1 (X, const, X × Y − TX) = Ω1(X/G, const, X ×G Y − T (X/G)),since isotopies of Γ change S only by a normal bordism. This element can beshown to be zero [6], which I call a topological counter-example. In particular, thetest map t can be deformed G-equivariantly rel ∂X = (S2)6 \X to a map t′, suchthat t′−1(0) ∩ X = ∅.References[1] H. B. Griffiths, The topology of square pegs in round holes, Proc. London Math. Soc. 62(1991), 647–672.[2] H. Guggenheimer, Finite sets on curves and surfaces, Israel J. Math. 3 (1965), 104–112.[3] U. Koschorke: Vector Fields and Other Vector Bundle Morphisms – A Singularity Approach,Lecture Notes in Mathematics, Vol. 847, Springer-Verlag, 1981.[4] V. V. Makeev, Quadrangles inscribed in a closed curve and the vertices of a curve, J. Math.Sci., Vol 131, No. 1 (2005), 5395–5400.[5] B. Matschke, Equivariant Topology and Applications, diploma thesis, TU Berlin, 2008.[6] B. Matschke, in preparation.[7] L. G. Shnirel’man, On some geometric properties of closed curves (in Russian), Usp. Mat.Nauk 10 (1944), 34–44.[8] W. Stromquist, Inscribed squares and square-like quadrilaterals in closed curves, Mathe-matika 36 (1989), 187–197.[9] O. Toeplitz, Ueber einige Aufgaben der Analysis situs, Verhandlungen der SchweizerischenNaturforschenden Gesellschaft in Solothurn 4 (1911), p. 197.[10] S. T. Vrećica, Živaljević, Fulton-MacPherson compactification, cyclohedra, and the polygonalpegs problem, arXiv:0810.1439, 2008.128 Oberwolfach Report 02Optimizing circle arrangements: questions and commentsRobert ConnellyConsider a Boolean expression for an arrangement of overlapping circles in theplane. A simple example is (B1 ∪B2)∩B3, where each Bi is a circular disk with afixed radius. A sample problem asks when is the area of (B1∪B2)∩B3 maximized.When the two disks B1 and B2 can be packed inside B3 any such packing achievesthe maximum area. When one of the two disks B1 or B2 has a larger radius thanB3, it can cover B3, and the area of B3 is the maximum area of the Booleanexpression. For the intermediate case, a configuration such as the one in the figurewill achieve the maximum area.xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx321BBBSimilarly for the expression (B1∪B2 · · ·∪Bn)∩Bn+1 when the first disks cannotbe packed into or cover the last disk Bn+1, there will be some particular optimumconfiguration. But what can be said about such configurations?In general it seems that it is quite complicated to determine exact solutions tothis type of maximum problem. It is possible, though, to make some observationsand to get more information about particular cases. For example, a result of B.Csikós in [2] together with connections to tensegrity structures gives a criterionfor the area function to be critical in the space of configurations of disks with fixedradii. This criterion can also be used to verify when a circle with a fixed radiusmaximizes its intersection with a fixed acute triangle. See the result of B. M.Stewart in [5]. This is also related to the problem of “How must n equal circles(spherical caps) of given angular radius r be arranged on the surface of a sphere sothat the area covered by the circles will be as large as possible,” which is discussedin the paper [4] by P. W. Fowler and T. Tarnai. Another example is the problemdiscussed by G. Fejes Tóth in [3]. He shows that “the density of the part of theplane covered by a system of congruent circles of density d is at most df(1/d),where the function f(x) is defined as the maximum of the area of the intersectionof a circle of unit area and a hexagon of area x.”The critical formula in [2] can be used in applications to protein folding, where,instead of using energy functions, one can maximize various weighted lattice poly-nomials of Boolean area/volumes functions of disks. This follows an idea withK. Bezdek in [1]. This has the potential of taking into account not only thepairwise interactions of the atoms, but also their hydrophobic-hydrophilic na-ture. In addition, using Csikós’s theory, the Voronoi construction in [2] shouldallow a computationally feasible algorithm to compute the gradient of the Booleanarea/volume.Discrete Differential Geometry 129References[1] Károly Bezdek and Robert Connelly, On the weighted Kneser-Poulsen conjecture, (submit-ted).[2] Balázs Csikós, On the volume of flowers in space forms. Geom. Dedicata 86 (2001), no. 1-3,59–79.[3] Gábor Fejes Tóth, Best partial covering of a convex domain by congruent circles of a giventotal area. Discrete Comput. Geom. 38 (2007), no. 2, 259–271.[4] Patrick W. Fowler and Tibor Tarnai, Transition from spherical circle packing to covering:geometrical analogues of chemical isomerization. Proc. Roy. Soc. London Ser. A 452 (1996),no. 1952, 2043–2064.[5] B. M. Stewart, The two-area covering problem. Amer. Math. Monthly 58, (1951). 394–403.Unfolding convex polyhedra via quasigeodesic source & star unfoldingsJoseph O’Rourke(joint work with Jin-ichi Itoh and Costin Vı̂lcu)1. IntroductionThere were two general methods known to unfold the surface P of any convexpolyhedron to a simple polygon in the plane: the source unfolding and the starunfolding. Both unfoldings are with respect to a point x ∈ P . The source unfoldingcuts the cut locus of x on P , and the star unfolding cuts the shortest paths from xto every vertex of P . Our extension replaces x by a simple closed polygonalcurve Q. Although we do not yet know the widest classes of curves for whichthe unfoldings avoid overlap, in both cases the classes include quasigeodesic loops.Again the source unfolding cuts (a portion of) the cut locus, and the star unfoldingcuts shortest paths from vertices to Q. And both cut all but a segment of Q.Quasigeodesics extend the notion of geodesics to nondifferentiable, and in par-ticular, to polyhedral surfaces. Let Γ be any directed curve on a convex surface P ,and p ∈ Γ be any point in the relative interior of Γ. Let L(p) be the total faceangle incident to the left side of p, and R(p) the angle to the right side. If Γis a geodesic, then L(p)=R(p) = π. A quasigeodesic Γ loosens this condition toL(p) ≤ π and R(p) ≤ π, again for all p interior to Γ [2, p. 16]. A simple, closedquasigeodesic is a simple closed curve on P that is quasigeodesic throughout itslength. As all curves we consider must be simple, we will henceforth drop thatprefix. Although Pogorelov showed that any convex polyhedron P has at leastthree closed quasigeodesics, there is no polynomial-time algorithm known to findthem, so we consider instead a wider class. A geodesic loop is a closed geodesicwith one exceptional loop point x at which the condition L(x)=R(x) = π may beviolated.2. Quasigeodesic Star UnfoldingP \ Q separates P into two “halves” P1 and P2. If we view the star unfoldingas an algorithm, it consists of three main steps:130 Oberwolfach Report 02(1) Select shortest paths sp(vi) from each vi ∈ Pk to Q.(2) Cut along sp(vi) and flatten each half.(3) Cut along Q, joining the two halves at an uncut segment s ⊂ Q.After cutting along sp(vi), we conceptually insert an isosceles triangle withapex angle equal to the curvature ω(vi) at each vi, which flattens each half, atechnique used by Alexandrov [1]. The procedure is illustrated in Figure 1, andfully described in [3].Figure 1. (a) Geodesic loop Q on cube. Shortest paths sp(vi) areshown. Faces are labeled {F, T, L, R, Bt, Bk}. (b) Star unfoldingwith respect to Q, joined at s = v′0v′7.3. Quasigeodesic Source UnfoldingThe point source unfolding cuts the cut locus of the point x: the closure of setof all those points y to which there is more than one shortest path on P fromx. Our method also relies on the cut locus, but now the cut locus CPk on thehalf-surface Pk with respect to its boundary ∂P=Q. We cut only the edges of CPnot incident to Q, plus one further cut. An example of the cut loci for a simpleclosed quasigeodesic Q are shown in Figure 2.The main steps of the source unfolding argument for closed quasigeodesics areas follows:(1) Develop Q to the plane. It is known that a closed convex curve developswithout self-intersection [4]. Closed quasigeodesics are convex curves to bothsides, and quasigeodesic loops are convex to one side.(2) From a neighborhood of Q in Pk, determine a doubly covered planar cone P∗kwhose boundary is Q. This cone in some sense envelops Pk.(3) Show that the “peels” of the cut locus of Pk nest inside the peels of the cutlocus of P ∗k . This implies that those peels will develop in the plane withoutoverlap by opening P ∗k along a generator of the cone.(4) The nested embedding is achieved by cutting the cut loci CP1 and CP2 notincident to Q.Discrete Differential Geometry 131Figure 2. (a) Truncated cube and quasigeodesic Q =(v0, v1, v7, v10). (b) View of CP1 from “front” side P1 (c) Viewof CP2 from “back” side P2.(5) Finally, we cut along Q, joining the two halves at an uncut segment s ⊂ Q.To capture quasigeodesic loops, and in fact all convex curves, the final step aboveis altered.Theorem. Both the source and the star unfolding of a convex polyhedron P withrespect to a quasigeodesic loop Q unfold P to a simple planar polygon.References[1] Aleksandr D. Alexandrov. Intrinsic Geometry of Convex Surfaces. Chapman & Hall/CRC,Boca Raton, FL, 2006. A. D. Alexandrov Selected Works. Edited by S. S. Kutateladze.Translated from the Russian by S. Vakhrameyev.[2] Aleksandr D. Alexandrov and Victor A. Zalgaller. Intrinsic Geometry of Surfaces. AmericanMathematical Society, Providence, RI, 1967.[3] Jin-ichi Itoh, Joseph O’Rourke, and Costin Vı̂lcu. Unfolding convex polyhedra viaquasigeodesic star unfoldings. Technical Report 091, Smith College, December 2008.arXiv:0821.2257v1 [cs.CG].[4] Joseph O’Rourke and Catherine Schevon. On the development of closed convex curves on3-polytopes. J. Geom., 13:152–157, 1989.Geometric aspects of discrete elastic rodsMax Wardetzky(joint work with M. Bergou, S. Robinson, B. Audoly and E. Grinspun)What this is aboutElastic rods are curve-like elastic bodies that have one dimension (length) muchlarger than the others (cross-section). Their elastic energy breaks down into threecontributions: stretching, bending, and twisting. Stretching and bending are cap-tured by the deformation of a space curve called the centerline, while twistingis captured by the rotation of a material frame associated to each point on the132 Oberwolfach Report 02centerline. Building on the notions of framed curves, parallel transport, and ho-lonomy, we present a smooth and a corresponding discrete theory that establishesan efficient model for simulating thin flexible rods with arbitrary cross sectionand undeformed configuration. To large parts, the material herein is an excerptfrom [1].Elastic energyWe describe the configuration of a smooth elastic rod by an adapted framedcurve Γ = {γ ; t,m1,m2}. Here γ(s) is an arc length parameterized space curvedescribing the rod’s centerline; the assignment of an orthonormal material frame{t(s),m1(s),m2(s)} to each point on the centerline contains the requisite infor-mation for measuring twist. We require the material frame to be adapted to thecenterline, i.e., to satisfy t(s) = γ ′(s). As usual, we refer to κ = t′ as the center-line’s curvature (normal) vector and to τ = m′1· m2 as the material frames twistmeasuring the rotation of the material around its centerline. The Kirchhoff modelof elastic energy of inextensible (no stretching of the centerline) and isotropic (nopreferred bending direction) elastic rods is given by(1) E =12∫γακ2 + βτ2 ds ,where α and β are constants encoding bending and twisting stiffness, respectively.Curve-angle representation & the Bishop frameWhile (1) completely describes an energy model for inextensible isotropic rods,there is a more convenient description when turning to simulations—one thatrenders the formulation of the material frame more explicit. The requisite toolis provided by the Bishop (or parallel) frame, an adapted orthonormal frame{t(s),u(s),v(s)} that has zero twist uniformly, i.e., u′ · v = −v′ · u = 0. Theassignment of an adapted frame to one point on the curve uniquely pins down theBishop frame throughout the entire curve. Every smoothly parameterized spacecurve with nowhere vanishing derivative carries a Bishop frame—one of severalproperties that sets the Bishop frame apart from the Frenet frame (which is nottwist-free).Denoting by θ the angle between the Bishop and the material frame in the crosssection orthogonal to the centerline’s tangent, i.e., θ = ∠(u,m1) = ∠(v,m2), onereadily checks that the material frame’s twist satisfies τ = θ′. Therefore, we canrewrite elastic energy of inextensible isotropic rods asE =12∫γακ2 + β(θ′)2 ds .We refer to this formulation as the curve-angle representation, as it previouslyalso appeared in [3]. This representation reveals a fascinating analogy betweenthe potential energy of elastic rods and the kinetic energy of Lagrange spinningtops. Indeed, by identifying the axis of the top with the direction of the rod’s unittangent, t, and furthermore identifying the rod’s arc length with the top’s physicalDiscrete Differential Geometry 133time, we find that∫κ2 =∫(t′)2 and∫τ2 =∫(θ′)2 measure the kinetic energy ofthe motion of the top’s center of mass and rotation around its axis, respectively.Holonomy & Fuller’s formulaFor a frame to be parallel along a space curve has the following interpretation.Consider the centerline’s Gauss image, γ̃ , traced out on the unit 2-sphere, S2, bythe unit tangent, t. For {u,v} to be parallel (twist-free) along γ is then equivalentfor {u,v} to be parallel-transported along γ̃ in the usual sense of the Levi-Civitaconnection on S2.Assume γ is a closed curve, then γ̃ is closed as well. When parallel transporting{u,v} once around γ (or γ̃), the resulting final frame will usually differ from theinitial one by an angle called holonomy, Hol . This angle is related to the so-calledwrithe. More precisely, whenever γ is a non self-intersecting closed space curvewith (material) frame {m1,m2}, let Lk denote the (unique) linking number of thetwo curves {γ±(s)} = {γ(s) ± ǫm1(s)} for some small enough ǫ > 0. Then(2) Lk = Tw + Wr ,where Tw = (1/2π)∫γτds is the total twist of the material frame, while writhesatisfies Wr ≡ Hol/2π modulo 1. Equation (2) is sometimes referred to as theCǎlugǎreanu-White-Fuller formula, see, e.g., [2].Furthermore, the Gauss-Bonnet theorem implies that Hol ≡ A modulo 2π,where A is the signed area enclosed by γ̃ on S2.Centerline variationIn physical simulations, in order to compute forces, we are required to expresschanges of elastic energy due to variations of the position (shape) of the center-line. The corresponding change of bending energy,∫γκ2ds, is straightforward tocalculate, while computing the change of twisting energy,∫γ τ2ds, is slightly moreinvolved since it requires the computation of the change of holonomy (or writhe).If γ is a closed curve, then it follows from Gauss-Bonnet that the change in holo-nomy, δHol , with respect to varying the centerline’s tangent (the position of γ̃ onS2) by δt is given by(3) δHol = δA = −∫γδt · (t × t′) ds and hence δθ′ = δt · (κb) ,where κb = t × t′ is the centerline’s curvature binormal vector. For closedcuves, (3) is the infinitesimal version of Fuller’s calculation for the differencebetween the writhe of two space curves, see [2]. From (3) we obtain that theL2-gradient of Hol with respect to variations of positions (not tangents) is (κb)′.The discrete pictureWe represent a discrete rod’s centerline as a piecewise straight polygonal spacecurve, and we associate discrete adapted orthonormal frames with edges of thiscurve. Along each edge we assume these frames to be constant. For each pair134 Oberwolfach Report 02(ei−1, ei) of consecutive edges, discrete parallel transport from one edge to the nextis given by rotating by the angle ∠(ei−1, ei) about the normal to the plane spannedby ei−1 and ei. This gives rise to discrete Bishop (parallel) frames. Accordingly,we obtain a discrete notion of holonomy (or writhe) for closed polygonal curves.We require elastic energy and hence a discrete notion of curvature and twist.As in the smooth case, twist is nothing but the change of the angle between theBishop and the material frame at each edge of the polygonal curve, γ .Consider once more the Gauss image, γ̃ of γ , on S2. The vertices of γ̃ correspondto unit tangents, ti, along the edges of γ , while the edges of γ̃ are arcs of greatcircles. As in the smooth case before, Gauss-Bonnet tells us that discrete holonomyis related to the signed area enclosed by γ̃ . To obtain forces, it therefore suffices tostudy variations of this area with respect to variations of the vertices of γ̃ . To thisend, consider an arc of a great circle of length φi = ∠(ti−1, ti) < π between ti−1and ti and consider respective variations by δti−1 and δti on S2. Consider furtherthe area swept out by the geodesics that connect the two varying endpoints. Itfollows (for example by considering Jacobi fields) that this area (and thereforediscrete holonomy) satisfiesδHol = δA = −δti−1 + δti2· (2 tanφi2bi) with bi =ti−1 × ti|ti−1×ti|,which is the discrete analogue of (3). By postulating in the discrete case relation (3)between the gradient of holonomy and curvature, we may define discrete curvaturesat the vertices of γ by κi = 2 tan(φi/2), where φi is the angle between the edgesincident to a particular vertex.For additional material, including anisotropic rods and simulation results, see [1].References[1] M. Bergou, M. Wardetzky, S. Robinson, B. Audoly, and E. Grinspun. Discrete elastic rods.In ACM Transactions on Graphics, 27(3), 2008. [Proceedings of ACM SIGGRAPH ’08][2] F. B. Fuller. Decomposition of the linking number of a closed ribbon: A problem frommolecular biology. PNAS, 75(8):3557–3561, 1978.[3] J. Langer and D. A. Singer. Lagrangian Aspects of the Kirchhoff Elastic Rod. SIAM Review,38(4):605–618, 1996.Complex barycentric coordinates for shape deformationMirela Ben-Chen(joint work with Ofir Weber, Craig Gotsman)Barycentric coordinates are a very useful mathematical tool for computer graphicsapplications. Since they allow inferring continuous data over a domain from dis-crete or continuous values on the boundary of the domain, barycentric coordinatesare used in a wide range of applications.Traditionally, barycentric coordinates in Rn are defined as the real coefficientsof an affine combination of vectors in Rn, see for example [FHK06, Flo03]. Assuch, they operate identically on each coordinate. When working in the plane,Discrete Differential Geometry 135barycentric coordinates in Rn can also be considered as an affine combination ofcomplex numbers with real coefficients, thus it is natural to consider also the casewhere the coefficients are themselves allowed to be complex. This new point ofview has a few advantages: First, it allows the definition of complex barycentriccoordinates, permitting a different linear operation for each of the two coordinates,through which new effects can be achieved. Second, it unleashes the rich theoryof complex analysis, simplifying the underlying theory considerably.Complex barycentric coordinates are especially useful for 2D shape and imagedeformation. In a typical application scenario, the user defines a source contour,usually a polygon, and deforms it to a target contour by moving its vertices. Thisindicates to the application that the region within the source contour should bedeformed in some natural way to the region within the target contour such that theper-edge correspondence is respected. For such applications it is important thatthe mapping has certain properties - for example, if the target polygon is identicalto the source polygon, then the mapping should be the identity map. In addition,for shape deformation applications, conformal mappings are preferred, since theypreserve angles and hence preserve details better than arbitrary mappings.Complex barycentric coordinates are defined as follows. Let S = {v1, v2, . . . , vn}⊂ R2 be the vertices of a simply connected planar polygon, oriented in the counterclockwise direction, vj = (xj , yj). Let zj = xj + iyj be the representation of thevertices as complex numbers, zj ∈ C. Denote by Ω the interior of S. Given apoint v = (x, y) ∈ Ω, define z = x + iy and consider the following complex linearcombination∑nj=1 kj(z)zj , where kj : Ω → C. We say that the functions kj(z) arecomplex barycentric coordinates with respect to S if the following two propertieshold for all z ∈ Ω:Constant precision:n∑j=1kj(z) = 1, Linear precision:n∑j=1kj(z)zj = zGiven complex barycentric coordinates kj(z) for S, we may consider the com-plex function gS,F (z) which results from applying the complex barycentric coor-dinates to the vertices of a target polygon F = {f1, f2, . . . , fn} ⊂ C:gS,F (z) =n∑j=1kj(z)fjComplex barycentric coordinates can easily be generalized to continuous con-tours in the following way. Let Ω be a simply connected open planar region witha smooth boundary S. Given z ∈ Ω and w ∈ S, consider the complex functionk(w, z) : S × Ω → C. As in the discrete case, we say that k(w, z) is a barycentriccoordinate function if it satisfies the following properties for all z ∈ Ω:Constant precision:∫Sk(w, z) dw = 1, Linear precision:∫Sk(w, z)w dw = zThe function k(w, z) is sometimes called a kernel function. The integral over Sis a complex integral, where dw = T (w)ds, T (w) is the unit-length tangent vector136 Oberwolfach Report 02to S at w, and ds is the usual arc-length differential element. Given a continuouscomplex function f : S → C, we can define a planar mapping gS,f(Ω) as follows:gS,f(z) =∫Sk(w, z)f(w) dwAs in the case of real barycentric coordinates, the main challenge is to findkernels k(w, z), or, in the discrete case, coordinate functions kj(z), which sat-isfy the required properties. As it turns out, simply choosing the Cauchy kernelC(w, z) = 12πi1w−z provides us with the required properties. Due to the Cauchyintegral formula, we have that for z ∈ Ω12πi∫S1w − zdw = 1,12πi∫Sww − zdw = z,which are the two properties we have required - constant precision and linearprecision. We call the resulting coordinates Cauchy coordinates.Applying the Cauchy coordinates to a target contour f(S) defines the followingmapping:gS,f(z) =12πi∫Sf(w)w − zdwThis mapping is called the Cauchy transform of f [Bel92]. It has various in-teresting properties, one of which being that if f is continuous on S, then g isalways holomorphic on Ω. Hence, if we apply these coordinates in the context ofplanar shape deformation, the deformation is guaranteed to be conformal (if thederivatives do not vanish). In addition, since holomorphic functions are infinitelydifferentiable, the mapping will be smooth.In a practical shape deformation scenario, the con-tour S is usually a polygon (sometimes called “cage”)which the user deforms to a new polygon F with ver-tices {f1, f2, . . . , fn}. By integrating over the edges ofthe source polygon and taking into consideration thatthe target contour is also a polygon, we get:zjzj+1zBjzj-1Bj+1 Bj-1AjAj+1gS,f(z) =n∑j=1Cj(z)fjCj(z) =12πi(Bj+1(z)Aj+1log(Bj+1(z)Bj(z))−Bj−1(z)Ajlog( Bj(z)Bj−1(z)))As it turns out, these complex coordinates are equivalent to the recently intro-duced Green coordinates [LLCO08] which generalized the concept of barycentriccoordinates to be a linear combination of the coordinates of the vertices of thepolygon plus a linear combination of the normals to the edges of the polygon.In addition to the Cauchy-Green barycentric coordinates, we showed how to de-fine coordinates which minimize a given energy functional. We proposed two suchfunctionals, one which improves the fit between the target polygon and the bound-ary of the resulting deformation (inspired by the continuous Szegö transform), andDiscrete Differential Geometry 137one which allows the user to manipulate a small set of positional constraints, in-stead of manipulating the vertices of the target polygon.To conclude, we have generalized the concept of barycentric coordinates fromreal numbers to complex numbers, and provided a few examples of known and newcoordinates which can be expressed quite simply in this framework. We believethere is still much research to be done on the theory and applications of complexbarycentric coordinates. One challenge is to find non-conformal complex coordi-nates which will generate the more useful “as rigid as possible”-type deformations.Another interesting theoretical issue is the connection between complex barycen-tric coordinates and the so-called “primal/dual ratio”. As Mercat [Mer08] pointedout, complex primal/dual ratios will arise when the primal and dual edges are notorthogonal. We believe more insight into complex barycentric coordinates can begained by studying these concepts.References[Bel92] Bell S.-R.: The Cauchy Transform, Potential Theory and Conformal Mapping.CRC Press, 1992.[FHK06] Floater M. S., Hormann K., Kòs G.: A general construction of barycentric coor-dinates over convex polygons. Adv. Comp. Math. 24, 1–4 (2006), 311–331.[Flo03] Floater M. S.: Mean-value coordinates. Comp. Aided Geom. Design 20, 1 (2003),19–27.[LLCO08] Lipman Y., Levin D., Cohen-Or D.: Green coordinates. ACM Trans. Graph. 27, 3(2008). [Proceedings of ACM SIGGRAPH ’08][Mer08] Mercat C.: Discrete complex structure on surfel surfaces. Discrete Geometry forComputer Imagery, LNCS 4992 (2008), 153–164.Circle packing samplerKen StephensonAfter a brief overview of “circle packing”, we discuss three topics of current inter-est: (1) The “spontaneous geometry” circle packings bring with random triangu-lations, (2) the “fast geometry” available with a new circle packing algorithm, and(3) the “warped geometry” one sees in affine circle packings.Circle packings are configurations P of circles having prescribed patterns Kof tangency. K can be essentially any triangulation of a topological surface, andthe heart of the topic is that P imposes a geometry on K — a geometry that is“conformal” in nature. For general backgound, see [1].(1) Random Geometry: Circle packing got its start (in analysis) with a con-jecture of Thurston regarding the approximation of classical conformal maps bydiscrete conformal maps — maps between circle packings sharing the same combi-natoric pattern K. In the adjustment of circle sizes from one packing to another,conformal structure is (roughly) perserved. His conjecture was proven by Rodinand Sullivan, see [3].However, fairly extensive experiments suggest that much more may be true:conformal structure seems to be an “emergent” property. A circle packing of a138 Oberwolfach Report 02random triangulation of a region Ω imposes a geometry on Ω. As the complexityof such random triangulations grows, the imposed geometries appear to convergein probability to the conformal geometry Ω inherits from the plane. The followingfigure illustrates estimates of two companions of conformal structure, harmonicmeasure and extremal length. On the left is the traditional circle packing approach,which starts with a circle packing of Ω, then repacks it for various purposes. On theright is our alternate approach, which differs only in that the initial triangulationof Ω is random.gfgfThe robustness of convergence of such conformal quantities is suggested by ex-periments with Ω a square. On the left of the following figure, a random Delaunaytriangulation of Ω is packed as a rectangle, with corner circles associated withvertices of K closest to the corners of Ω.−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.101020304050607080K PΩThe logarithm of the aspect ratio (width/height) of the resultant rectangle is arandom variable which by symmetry will have mean zero. Trial runs are illustratedon the right: each plot gives the distribution of outcomes for 5000 random trialsfor, respectively, N = 200, 400, 1600, 3200, 6400, or 12800 random points in Ω.Observe that doubling the number of points almost precisely halves the variance.Emergent conformality could have broad implications. By varying the notionof random triangulations, for instance, one might be able to approximate solutionsof the Beltrami equation. In applications, there may be a “central limit” theoremjustifying circle packing as a mechanism for imposing conformal structures.(2) Fast Geometry: As combinatorial complexity has grown, circle packing com-putations have become progressively more unweildy. Current methods computeDiscrete Differential Geometry 139local data — radii R — and only then lay out the global configuration, one circle ata time, to get the centers Z. With large packings, even highly accurate radii leadto cumulative layout errors that prevent a coherent global packing. My student,Gerald Orick, has developed a novel new algorithm, applicable now to maximalpackings of complexes K triangulating the disc or sphere.Orick’s approach is iterative, but alternates between adjustments to R and toZ via Tutte-style embeddings. Assuming that K triangulates a disc, assign anarbitrary initial radius label R. In each iteration cycle, the current R determinescenters for boundary circles as well as geometric weights for a Laplace operatoron K. Solving the Dirichlet problem (efficiently done via sparse matrices) givesan embedding of centers Z. The local geometry of the embedding leads to a newlabel R, and the process repeats. This is extremely fast. For instance, a typicalbrain flattening application involving 250,000 circles would take half a day on alaptop and might still suffer layout problems. The new algorithm will computethe packing in 3 minutes with no layout problems — in fact, reasonably accurateintermediate layouts will become available within the first few seconds.The key bit of local geometry is reflected in this image of thestar of v: were the embedding associated with a circle packing,all the face sectors at v would have the same radius. When theydon’t, a new label r is set for v so that 2πr2 equals the sum ofsector areas. Once new labels are set for all vertices, Dubejko-style weights [5] can be computed for the edges, giving a Laplaceoperator for K. This leads to a new embedding, etc., etc.Fast computation can be a game-changer in applications ofcircle packing.(3) Warped Geometry: If K triangulates a compact oriented surface, then itimposes, via circle packing, a unique conformal structure on that surface. A newapproach to the fundamental packing procedures allows us to compute general“affine” structures. The figure below is based on a combinatorial torus, K. On theleft is the associated conformal torus T , determined by laying out a fundamentaldomain — edge shading indicates side-pairings. On the right is an affine torus forthis same K.The side-pairing maps for the affine torus are generated by z 7→ αz, z 7→ βz. Theconstruction begins by specifying the moduli |α| and |β| and applying a packing140 Oberwolfach Report 02procedure to determine labels [r1 : r2 : r3] (in homogeneous coordinates) for thefaces. These induce, for any vertex v, a locally consistent set of radii defininga flower for v. In general, however, there is no globally consistent set of radii.Arg(α) and Arg(β) are uniquely determined in the packing process, reflecting theanticipated rigidity associated with K.Circle “packings” consisting of one self-tangent circle have been used to studythe space of affine tori as fibered over the space of conformal tori, [2]. Now one canexperiment not only with more general combinatorial tori, but also with highergenus surfaces. These affine structures also suggest an approach to discrete holo-morphic differential forms.References[1] Ken Stephenson, “Introduction to Circle Packing: The Theory of Discrete Analytic Func-tions”, Camb. Univ. Press 2005, New York.[2] S. Kojima, S. Mizushima, and S. P. Tan, Circle packings on surfaces with projective struc-tures, J. Differential Geom., 63 (2003), 349–397.[3] B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping, J.Differential Geom., 26 (1987), 349–360.[4] William Thurston, The finite Riemann mapping theorem, unpublished talk at An Interna-tional Symposium at Purdue University in celebrations of de Branges’ proof of the Bieber-bach conjecture, 1985.[5] Tomasz Dubejko, Circle-packing connections with random walks and a finite volume method,Sémin. Théor. Spectr. Géom., 15 (1996–97), 153–161.Reporter: Boris SpringbornDiscrete Differential Geometry 141ParticipantsUlrich BauerMathematisches InstitutGeorg-August-UniversitätBunsenstr. 3-537073 GöttingenMirela Ben-ChenComputer Science DepartmentTECHNIONIsrael Institute of TechnologyHaifa 32000ISRAELBruno BenedettiInstitut für MathematikMA 6-2Technische Universität BerlinStraße des 17. Juni 13610623 BerlinProf. Dr. Ethan BlochDepartment of MathematicsBard CollegeAnnandale-on-Hudson , NY 12504USAProf. Dr. Alexander I. BobenkoInstitut für MathematikFakultät II - Sekr. MA 8 - 3Technische Universität BerlinStraße des 17. Juni 13610623 BerlinProf. Dr. Ulrich BrehmInstitut für GeometrieTU Dresden01062 DresdenDr. Ulrike BückingInstitut für MathematikFakultät II - Sekr. MA 8 - 3Technische Universität BerlinStraße des 17. Juni 13610623 BerlinProf. Dr. Robert ConnellyDepartment of MathematicsCornell UniversityMalott HallIthaca , NY 14853-4201USAProf. Dr. Herbert EdelsbrunnerDepartment of Computer ScienceDuke UniversityBox 90129Durham , NC 27708-0129USAFelix EffenbergerFachbereich MathematikInstitut für Geometrie u. TopologieUniversität StuttgartPfaffenwaldring 5770550 StuttgartDipl. Math. Martin von GagernZentrum für MathematikTU MünchenBoltzmannstr. 385748 Garching bei MünchenDr. Udo Hertrich-JerominDepartment of Mathematical SciencesUniversity of BathClaverton DownGB-Bath BA2 7AY142 Oberwolfach Report 02Prof. Dr. Tim HoffmannZentrum für MathematikTU MünchenBoltzmannstr. 385748 Garching bei MünchenEmanuel Huhnen-VenedeyInstitut für MathematikMA 8 - 3Technische Universität BerlinStraße des 17. Juni 13610623 BerlinDr. Ivan IzmestievInstitut für MathematikMA 8 - 3Technische Universität BerlinStraße des 17. Juni 13610623 BerlinProf. Dr. Michael JoswigFachbereich MathematikTU DarmstadtSchloßgartenstr. 764289 DarmstadtProf. Dr. Richard KenyonDepartment of MathematicsBrown UniversityBox 1917Providence , RI 02912USAProf. Dr. Wolfgang KühnelFachbereich MathematikInstitut für Geometrie u. TopologieUniversität StuttgartPfaffenwaldring 5770550 StuttgartDr. Carsten LangeInstitut für MathematikFreie Universität BerlinArnimallee 314195 BerlinInna LukyanenkoInstitut für MathematikMA 8 - 3Technische Universität BerlinStraße des 17. Juni 13610623 BerlinProf. Dr. Feng LuoDepartment of MathematicsRutgers UniversityHill Center, Busch Campus110 Frelinghuysen RoadPiscataway , NJ 08854-8019USADr. Frank H. LutzFakultät II-Institut f. MathematikTechnische Universität BerlinSekr. MA 3-2Straße des 17. Juni 13610623 BerlinDr. Benjamin MatschkeFachbereich Mathematik, Sekr.MA 8-5Technische Universität BerlinStraße des 17. Juni 13610623 BerlinProf. Dr. Christian MercatDepartement de MathematiquesUniversite Montpellier IIPlace Eugene BataillonF-34095 Montpellier Cedex 5Prof. Dr. Joseph O’RourkeDepartment of Computer ScienceSmith CollegeClark Science CenterNorthampton MA 01063USAProf. Dr. Yann Ollivier35, rue Rene LeynaudF-69001 LyonDiscrete Differential Geometry 143Prof. Dr. Igor PakSchool of MathematicsUniversity of Minnesota127 Vincent Hall206 Church Street S. E.Minneapolis MN 55455-0436USAProf. Dr. Gaiane PaninaInstitute for Informatics and AutomationRAS14, lin, V.O.39199171 St. PetersburgRussiaProf. Dr. Ulrich PinkallFakultät II-Institut f. MathematikTechnische Universität BerlinSekr. MA 3-2Straße des 17. Juni 13610623 BerlinProf. Dr. Konrad PolthierInstitut für MathematikFreie Universität BerlinArnimallee 314195 BerlinProf. Dr. Jürgen Richter-GebertZentrum für MathematikTU MünchenBoltzmannstr. 385748 Garching bei MünchenDr. Günter RoteInstitut für InformatikFreie Universität BerlinTakustr. 914195 BerlinWolfgang K. SchiefInstitut für MathematikMA 8 - 3Technische Universität BerlinStraße des 17. Juni 13610623 BerlinProf. Dr. Jean-Marc SchlenkerInstitut de Mathematiques de ToulouseUniversite Paul Sabatier118, route de NarbonneF-31062 Toulouse Cedex 9Michael SchmidZentrum für MathematikTechnische Universität MünchenBoltzmannstr. 385748 Garching bei MünchenBernd SchulzeDepartment of MathematicsYork University4700 Keele StreetNorth York , Ont. M3J 1P3CANADAStefan SechelmannInstitut für MathematikFakultät II - Sekr. MA 8 - 3Technische Universität BerlinStraße des 17. Juni 13610623 BerlinDr. Boris SpringbornInstitut für MathematikFakultät II - Sekr. MA 8 - 3Technische Universität BerlinStraße des 17. Juni 13610623 BerlinProf. Dr. Kenneth StephensonDepartment of MathematicsUniversity of Tennessee121 Ayres HallKnoxville , TN 37996-1300USAProf. Dr. Ileana StreinuDepartment of Computer ScienceSmith CollegeClark Science CenterNorthampton MA 01063USA144 Oberwolfach Report 02Prof. Dr. John M. SullivanFakultät II-Institut f. MathematikTechnische Universität BerlinSekr. MA 3-2Straße des 17. Juni 13610623 BerlinProf. Dr. Yuri B. SurisZentrum für MathematikTechnische Universität MünchenBoltzmannstr. 385748 Garching bei MünchenProf. Dr. Serge TabachnikovDepartment of MathematicsPennsylvania State UniversityUniversity Park , PA 16802USAProf. Dr. Johannes WallnerInstitut für GeometrieTU GrazKopernikusgasse 24A-8010 GrazProf. Dr. Max WardetzkyInstitut für Numerischeund Angewandte MathematikUniversität GöttingenLotzestr. 16-1837083 GöttingenProf. Dr. Günter M. ZieglerInstitut für MathematikMA 6-2Technische Universität BerlinStraße des 17. Juni 13610623 Berlin

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