BID Seminar, Nov. 23, 2010 Symmetric Embedding of Regular Maps Inspired Guesses followed by Tangible Visualizations Carlo H. Squin EECS Computer Science Division University of California, Berkeley Background: Geometrical Tiling Escher-tilings on surfaces with different genus in the plane on the sphere on the torus M.C. Escher Jane Yen, 1997 Young Shon, 2002 Tilings on Surfaces of Higher Genus 24 tiles on genus 3 48 tiles on genus 7 Two Types of Octiles u Six differently colored sets of tiles were used From Regular Tilings to Regular Maps When are tiles the same ? u on sphere: truly identical from the same mold u on hyperbolic surfaces topologically identical (smaller on the inner side of a torus) Tilings should be regular... u locally regular: all p-gons, all vertex valences q u globally regular: full flag-transitive symmetry (flag = combination: vertex-edge-face) Regular Map The Symmetry of a Regular Map u After an arbitrary edge-to-edge move, every edge can find a matching edge; the whole network coincides with itself. All the Regular Maps of Genus Zero Platonic SolidsDi-hedra (=dual) Hosohedra {3,4} {3,5} {3,3} {4,3} {5,3} On Higher-Genus Surfaces: only Topological Symmetries Regular map on torus (genus = 1) NOT a regular map: different-length edge loops Edges must be able to stretch and compress 90-degree rotation not possible NOT a Regular Map u Torus with 9 x 5 quad tiles is only locally regular. u Lack of global symmetry: Cannot turn the tile-grid by 90. This IS a Regular Map u Torus with 8 x 8 quad tiles. Same number of tiles in both directions! u On higher-genus surfaces such constraints apply to every handle and tunnel. Thus the number of regular maps is limited. How Many Regular Maps on Higher-Genus Surfaces ? Two classical examples: R2.1_{3,8} _12 16 triangles Quaternion Group [Burnside 1911] R3.1d_{7,3} _8 24 heptagons Kleins Quartic [Klein 1888] Nomenclature R3.1d_{7,3}_8R3.1d_{7,3}_8 Regular map genus = 3 # in that genus-group the dual configuration heptagonal faces valence-3 vertices length of Petrie polygon: Schlfli symbol {p,q} Eight-fold Way zig-zag path closes after 8 moves 2006: Marston Conders List u Orientable 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 R2.2_{4,6}_12 R3.6_{4,8}_8 Low-Hanging Fruit Some early successes... R4.4_{4,10}_20 and R5.7_{4,12}_12 A Tangible Physical Model u 3D-Print, hand-painted to enhance colors R3.2_{3,8}_6 Genus 5 {3,7} 336 Butterflies Only locally regular ! Globally Regular Maps on Genus 5 Emergence of a Productive Approach u Depict map domain on the Poincar disk; establish complete, explicit connectivity graph. u Look for likely symmetries and pick a compatible handle-body. u Place vertex stars in symmetrical locations. u Try to complete all edge-interconnections without intersections, creating genus-0 faces. u Clean-up and beautify the model. Depiction on Poincare Disk u Use Schlfli symbol create Poincar disk. {5,4} Relators Identify Repeated Locations Operations: R = 1-click ccw-rotation around face center; r = cw-rotation. S = 1-click ccw-rotation around a vertex; s = cw-rotation. R3.4_{4,6}_6 Relator: R s s R s s Complete Connectivity Information u Triangles of the same color represent the same face. u Introduce unique labels for all edges. Low-Genus Handle-Bodies u There is no shortage of nice symmetrical handle-bodies of low genus. u This is a collage I did many years ago for an art exhibit. Numerology, Intuition, u Example: R5.10_{6,6}_4 First try: oriented cube symmetry Second try: tetrahedral symmetry A Valid Solution for R5.10_{6,6}_4 Virtual model Paper model (oriented tetrahedron) (easier to trace a Petrie polygon) The Design Problem u Not wicked just very difficult ! R5.12 and R5.13 From Conders List: u R5.12 : Type {8,8}_4 Order 64 mV = 4 mF = 4 Self-dual [ TT, RSRS, RsRs, RTRT, STST, R^8, sRRRRsss ] u R5.13 : Type {8,8}_4 Order 64 mV = 4 mF = 4 Self-dual [ TT, RSRS, RTRT, STST, R^8, SRRRSr, SRsRSS ] R5.12 and R5.13 The two different Poincar disks Solutions for R5.12 C2 solution by Jack vanWijk My D2-symmetrical solution different A First Genus-5 Canvas u A disk with 5 holes. u Paste on the vertex neighborhoods from the Poincar disk. u Try to connect edge stubs with same labels: - without edge crossings - without holes in faces. A Torus with 4 Handles u I glued the vertex neighborhoods onto the main torus and then tried to wire up corresponding edge stubs. Connectivity of an Octagonal Facet u Would fit onto a genus-2 handle body Connectivity of an Octagonal Facet A customized octagon and its curled-up state. Two Connected Octagons (four edges shared between them) R5.12: Back-to-back R5.13: Twisted connections R5.12: Toroidal Model A nice D2-symmetrical solution on a toroidal ring with 4 holes Template 2.5D paper model Attempts to Establish Connectivity u Using the R5.12 solution as an inspiration Placement of the four vertices: between the holes Extracting the Fundamental Net Poincar disk Symmetrical set of faces Deforming the Fundamental Net Symmetrical set of faces u Rolled-up into a torus Closing-up the Fundamental Net A Cleaner, More Flexible Model The same basic structure with a cleaner template From Paper Model to Virtual Model Mapping texture onto torus: Optimizing twist and azimuth Back to Paper Model Adding two handles... to route the green/yellow edges to the proper location, so that the four yellow face centers can be merged. Two Octagons again Glue these faces together at edges of bridge region to form a slim tunnel. Step 2: Join hammer-heads Step 3: Merge As, Bs, move bridge to outside The Crucial Breakthrough u Bridge with Moebius loop to connect A and B: u Replace ribbon that carries edges 5 and 7 with a tunnel in bridge. (Reconstructed model) Half-bridge Templates T-shaped pieces for the top and bottom of each half-bridge with tunnel. Assembling the T-Shapes NOT one toroidal loop, but TWO smaller loops! Bridge with central tunnel Use 2 times Capturing the Essence of the Solution Basic structure mapped onto strip geometry, maintaining D2- C2-symmetry Model Refinement Equal-size holes, Match style of R5.12 solution Model with D2-Symmetry Front and back of disk model: where no black edges, face wraps around. Face centers Tubular Model (initially sought) Front and back view Reflection on Design Process Successful solution path ? Reusing What I Learned Embedding of R5.6: Disk- and paper-strip- models Crucial Solution Step for R5.6 Fold-up of fundamental net Resulting paper-strip template Design Process CAD Tools ? u A variety of models ! u Interfaces ? Epilog u Doing math is not just writing formulas! u It may involve paper, wires, styrofoam, glue u Sometimes, tangible beauty may result ! More Questions ?