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Bridges, Pécs, 2010Math ArtRegular Maps of Genus ZeroThe Symmetry of a Regular MapOn Higher-Genus Surfaces: only “Topological” SymmetriesHow Many Regular Maps on Higher-Genus Surfaces ?Nomenclature2006: Marston Conder’s ListMacbeath SurfaceR7.1_{3,7}_18 Paper ModelsStyrofoam Model for R7.1_{3,7}_18Globally Regular Tiling of Genus 4A Solution for R4.2_{4,5}_6Nice Color Pattern for R4.2_{4,5}_6A Graph-Embedding ProblemAn Intuitive ApproachA Tangible Physical ModelA Virtual Genus-3 Tiffany LampLight Cast by Genus-3 “Tiffany Lamp”“Low-Hanging Fruit”Genus 5Globally Regular Maps on Genus 5Emergence of a Productive ApproachDepiction on Poincare DiskRelators Identify Repeated LocationsComplete Connectivity InformationLow-Genus Handle-BodiesNumerology, Intuition, …An Valid Solution for R5.10_{6,6}_42 Methods to Find EmbeddingsThe General Text-book Method (1)The General Text-book Method (2)The General Text-book Method (3)The General Text-book Method (4)Jack J. van Wijk’s Method (1)Jack J. van Wijk’s Method (3)Jack J. van Wijk’s Method (2)J. van Wijk’s Method (5)Jack J. van Wijk’s Method (5)Jack J. van Wijk’s Method (6)Jack J. van Wijk’s Method (7)“Vertex Flowers” for Any GenusPaper Models for “Vertex Flowers”Anatomy of a Paper-Strip ModelThe Regular Map R3.3_{3,12}_8Deforming & Folding the Map DomainEvolution of the Topology ModelModels for R3.3_{3,12}_8Models for R3.3_{3,12}_8Good Solutions Can Be Re-used !Re-use of R3.3 Topology for R3.5Visualizing R4.2_{4,5}_6R4.2_{4,5}_6 LatticeR4.2_{4,5}_6 Lattice (wide angle)More …ConclusionsBridges, Pécs, 2010Bridges, Pécs, 2010My Search forSymmetrical Embeddingsof Regular MapsEECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, BerkeleyCarlo H. SéquinMath Math Art ArtThis is a “math-first” talk !“Art” comes into it in secondary ways:The way I find my solutions is more an “art”than a science or a formal math procedure;How to make the results visible is also an “art”;Some of the resulting models can be enhancedso that they become “art-objects” on their own.Regular Maps of Genus ZeroRegular Maps of Genus ZeroPlatonic SolidsDi-hedraHosohedraThe Symmetry of a Regular MapThe Symmetry of a Regular MapAfter an arbitrary edge-to-edge move, every edge can find a matching edge;the whole network coincides with itself.On Higher-Genus Surfaces:On Higher-Genus Surfaces:only “Topological” Symmetriesonly “Topological” SymmetriesRegular map on torus (genus = 1) NOT a regular map: different-length edge loopsEdges must be able to stretch and compress90-degree rotation not possibleHow Many Regular Maps How Many Regular Maps on Higher-Genus Surfaces ?on Higher-Genus Surfaces ?Two classical examples:R2.1_{3,8} _1216 trianglesQuaternion Group [Burnside 1911]R3.1d_{7,3} _824 heptagonsKlein’s Quartic [Klein 1888]NomenclatureNomenclatureR3.1d_{7,3}_8Regular mapgenus = 3# in that genus-groupthe dual configurationheptagonal facesvalence-3 verticeslength of Petrie polygon: Schläfli symbol“Eight-fold Way” zig-zag path closes after 8 moves2006: Marston Conder’s List2006: Marston Conder’s Listhttp://www.math.auckland.ac.nz/~conder/OrientableRegularMaps101.txtOrientable regular maps of genus 2 to 101:R2.1 : Type {3,8}_12 Order 96 mV = 2 mF = 1 Defining relations for automorphism group: [ T^2, R^-3, (R * S)^2, (R * T)^2, (S * T)^2, (R * S^-3)^2 ] R2.2 : Type {4,6}_12 Order 48 mV = 3 mF = 2 Defining relations for automorphism group: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^6 ] R2.3 : Type {4,8}_8 Order 32 mV = 8 mF = 2 Defining relations for automorphism group: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^-2 * R^2 * S^-2 ] R2.4 : Type {5,10}_2 Order 20 mV = 10 mF = 5 Defining relations for automorphism group: [ T^2, S * R^2 * S, (R, S), (R * T)^2, (S * T)^2, R^-5 ] = “Relators”Macbeath SurfaceMacbeath SurfaceEntry for the Macbeath surface of genus 7:R7.1 : Type {3,7}_18 Order 1008 mV = 1 mF = 1 Defining relations for automorphism group: [ T^2, R^-3, (R * S)^2, (R * T)^2, (S * T)^2, S^-7, S^-2 * R * S^-3 * R * S^-2 * R^-1 * S^2 * R^-1 * S^2 * R^-1 * S^-2 * R * S^-1 ]This is the 2nd-simplest surface for which the Hurwitz-limit of 84*(genus-1) can be achieved.For my 2006 Bridges talk I wanted to make a nice sculptural model of this surface …I simply could not find a solution !Even though I tried really hard …R7.1_{3,7}_18R7.1_{3,7}_18Paper ModelsPaper ModelsStyrofoam Model for R7.1_{3,7}_18Styrofoam Model for R7.1_{3,7}_18Globally Regular Tiling of Genus 4Globally Regular Tiling of Genus 4But this is not an embedding! Faces intersect heavily!Actual cardboard model(Thanks to David Richter)Conder: R4.2d_{5,4}_6A Solution for R4.2_{4,5}_6A Solution for R4.2_{4,5}_6 Inspiration R3.1: Petrie polygons zig-zag around arms. R4.2: Let Petrie polygons zig-zag around tunnel walls.It works !!!A look into a tunnelNice Color Pattern for R4.2_{4,5}_6Nice Color Pattern for R4.2_{4,5}_6Use 5 colorsEvery color is at every vertexEvery quad is surrounded by the other 4 colorsA Graph-Embedding ProblemA Graph-Embedding ProblemDyck’s graph = K4,4,4Tripartite graphNodes of the same color are not connected.Find surface of lowest genus in which Dyck’s graph can be drawn crossing-freeAn Intuitive ApproachStart with highest-symmetry genus-3 surface: “Tetrus”Place 12 points so that the missing edges do not break symmetry: Inside and outside on each tetra-arm.Do not connect the nodes that lie on thesame symmetry axis(same color)(or this one).A Tangible Physical ModelA Tangible Physical Model3D-Print, hand-painted to enhance colorsR3.2_{3,8}_6A Virtual Genus-3 Tiffany LampA Virtual Genus-3 Tiffany LampLight Cast by Genus-3 “Tiffany Lamp”Light Cast by Genus-3 “Tiffany Lamp”Rendered with “Radiance” Ray-Tracer (12 hours)R2.2_{4,6}_12 R3.6_{4,8}_8““Low-Hanging Fruit”Low-Hanging Fruit”Some early successes . . .R4.4_{4,10}_20 and R5.7_{4,12}_12Genus Genus 55336ButterfliesOnly locallyregular !Globally Regular Maps on Genus 5Globally Regular Maps on Genus 5Emergence of a Productive ApproachEmergence of a Productive ApproachDepict map domain on the Poincaré disk; establish complete, explicit connectivity graph.Look for likely symmetries and pick a