1. BID Seminar, Nov. 23, 2010 Symmetric Embedding of Regular Maps Inspired Guesses followed by Tangible Visualizations Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

2. 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

3. Tilings on Surfaces of Higher Genus 24 tiles on genus 3 48 tiles on genus 7

4. Two Types of “Octiles” u Six differently colored sets of tiles were used

5. 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

6. 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.

7. All the Regular Maps of Genus Zero Platonic SolidsDi-hedra (=dual) Hosohedra {3,4} {3,5} {3,3} {4,3} {5,3}

8. 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

9. 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°.

10. 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.

11. 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 Klein’s Quartic [Klein 1888]

12. 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:  Schläfli symbol {p,q} “Eight-fold Way” zig-zag path closes after 8 moves

13. 2006: Marston Conder’s List u http://www.math.auckland.ac.nz/~conder/OrientableRegularMaps101.txt 6104 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”

14. 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

15. A Tangible Physical Model u 3D-Print, hand-painted to enhance colors R3.2_{3,8}_6

16. Genus 5 {3,7} 336 Butterflies Only locally regular !

17. Globally Regular Maps on Genus 5

18. Emergence of a Productive Approach u Depict map domain on the Poincaré 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.

19. Depiction on Poincare Disk u Use Schläfli symbol  create Poincaré disk. {5,4}

20. 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

21. Complete Connectivity Information u Triangles of the same color represent the same face. u Introduce unique labels for all edges.

22. 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.

23. Numerology, Intuition, … u Example: R5.10_{6,6}_4 First try: oriented cube symmetry Second try: tetrahedral symmetry

24. A Valid Solution for R5.10_{6,6}_4 Virtual model Paper model (oriented tetrahedron) (easier to trace a Petrie polygon)

25. The Design Problem u Not “wicked” – just very difficult !

26. R5.12 and R5.13 From Conder’s 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 ]

27. R5.12 and R5.13 The two different Poincaré disks

28. Solutions for R5.12 C2 solution by Jack vanWijk My D2-symmetrical solution different

29. A First Genus-5 “Canvas” u A disk with 5 holes. u Paste on the vertex neighborhoods from the Poincaré disk. u Try to connect edge stubs with same labels: - without edge crossings - without holes in faces.

30. A Torus with 4 Handles u I glued the vertex neighborhoods onto the main torus and then tried to wire up corresponding edge stubs.

31. Connectivity of an Octagonal Facet u Would fit onto a genus-2 handle body

32. Connectivity of an Octagonal Facet A customized octagon and its curled-up state.

33. Two Connected Octagons (four edges shared between them) R5.12: Back-to-back R5.13: Twisted connections

34. R5.12: Toroidal Model A nice D2-symmetrical solution on a toroidal ring with 4 holes Template 2.5D paper model

35. Attempts to Establish Connectivity u Using the R5.12 solution as an inspiration… Placement of the four vertices: between the holes

36. Extracting the Fundamental Net Poincaré disk Symmetrical set of faces

37. Deforming the Fundamental Net Symmetrical set of faces

38. u Rolled-up into a torus Closing-up the Fundamental Net

39. A Cleaner, More Flexible Model The same basic structure with a cleaner template

40. From Paper Model to Virtual Model Mapping texture onto torus: Optimizing twist and azimuth

41. 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.

42. 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 A’s, B’s, move “bridge” to outside

43. 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)

44. Half-bridge Templates T-shaped pieces for the top and bottom of each half-bridge with tunnel.

45. Assembling the T-Shapes NOT one toroidal loop, but TWO smaller loops! Bridge with central tunnel Use 2 times

46. Capturing the Essence of the Solution Basic structure mapped onto strip geometry, maintaining D2- C2-symmetry

47. Model Refinement Equal-size holes, Match style of R5.12 solution

48. Model with D2-Symmetry Front and back of disk model: where no black edges, face wraps around. Face centers

49. Tubular Model (initially sought) Front and back view

50. Reflection on Design Process Successful solution path ?

51. Reusing What I Learned Embedding of R5.6: Disk- and paper-strip- models

52. Crucial Solution Step for R5.6 Fold-up of fundamental net Resulting paper-strip template

53. Design Process  CAD Tools ? u A variety of models ! u Interfaces ?

54. Epilog u “Doing math” is not just writing formulas! u It may involve paper, wires, styrofoam, glue… u Sometimes, tangible beauty may result !

55. More … Questions ?