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Bridges 2007, San Sebastian

Bridges 2007, San Sebastian. Carlo H. S é quin EECS Computer Science Division University of California, Berkeley. Symmetric Embedding of Locally Regular Hyperbolic Tilings. Goal of This Study. Make Escher-tilings on surfaces of higher genus.

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Bridges 2007, San Sebastian

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  1. Bridges 2007, San Sebastian Carlo H. Séquin EECS Computer Science Division University of California, Berkeley Symmetric Embedding ofLocally Regular Hyperbolic Tilings

  2. Goal of This Study Make Escher-tilings on surfaces of higher genus. in the plane on the sphere on the torus M.C. Escher Jane Yen, 1997 Young Shon, 2002

  3. How to Make an Escher Tiling • Start from a regular tiling • Distort all equivalent edges in the same way

  4. Hyperbolic Escher Tilings All tiles are “the same” . . . • truly identical  from the same mold • on curved surfaces  topologically identical Tilings should be “regular” . . . • locally regular: all p-gons, all vertex valences v • globally regular: full flag-transitive symmetry(flag = combination: vertex-edge-face)

  5. “168 Butterflies,” D. Dunham (2002) Locally regular {3,7} tiling on a genus-3 surfacemade from 56 isosceles triangles “snub-tetrahedron”

  6. E. Schulte and J. M. Wills • Also: 56 triangles, meeting in 24 valence-7 vertices. • But: Globally regular tiling with 168 automorphisms! (topological)

  7. Generator for {3,7} Tilings on Genus-3 • Twist arms by multiples of 90 degrees ...

  8. Dehn Twists • Make a closed cut around a tunnel (hole) or around a (torroidal) arm. • Twist the two adjoining “shores” against each other by 360 degrees; and reconnect. • Network connectivity stays the same;but embedding in 3-space has changed.

  9. Fractional Dehn Twists • If the network structure around an arm or around a hole has some periodicity P,then we can apply some fractional Dehn twistsin increments of 360° / P. • This will lead to new network topologies,but may maintain local regularity.

  10. Globally Regular {3,7} Tiling • From genus-3 generator (use 90° twist) • Equivalent to Schulte & Wills polyhedron

  11. Smoothed Triangulated Surface • 56 triangles • 24 vertices • genus 3 • globally regular • 168 automorph.

  12. Generalization of Generator • Turn straight frame edges into flexible tubes

  13. From 3-way to 4-way Junctions Tetrahedral hubs 6(12)-sided arms

  14. 6-way Junction + Three 8-sided Loops

  15. Construction of Junction Elements 3-way junction construction of 6-way junction

  16. Junction Elements Decorated with 6, 12, 24, Heptagons

  17. Assembly of Higher-Genus Surfaces Genus 5:8 Y-junctions Genus 7

  18. Genus-5 Surface (Cube Frame) • 112 triangles, 3 butterflies each  . . .

  19. 336 Butterflies

  20. Creating Smooth Surfaces 4-step process: • Triangle mesh • Subdivision surface • Refine until smooth • Texture-map tiling design

  21. Texture-Mapped Single-Color Tilings • subdivide also texture coordinates • maps pattern smoothly onto curved surface.

  22. What About Differently Colored Tiles ? • How many different tiles need to be designed ?

  23. 24 Newts on the Tetrus (2006) One of 12 tiles 3 different color combinations

  24. Use with Higher-Genus Surfaces • Lack freedom to assign colors at will !

  25. New Escher Tile Editor • Tiles need not be just simple n-gons. • Morph edges of one boundary . . .and let all other tiles change similarly!

  26. Escher Tile Editor (cont.) Key differences: • Tiling pattern is no longer just a texture! • Tiles have a well-defined boundary,which is tracked in subdivision process. • This outline can be flood filled with color.

  27. Escher Tile Editor (cont.) • Possible to add extra decorations onto tiles

  28. Prototile Extraction • Flood-fill can also be used to identify all geometry that belongs to a single tile.

  29. Extract Prototile Geometry for RP • Two prototiles extracted and thickened

  30. Generalizing the Generator to Quads • 4-way junctions built around cube hubs • 4-sided prismatic arms

  31. Genus 7 Surface with 60 Quads • No twist

  32. {5,4} Starfish Pattern on Genus-7 • Polyhedral representation of an octahedral frame • 108 quadrilaterals (some are half-tiles) • 60 identical quad tiles: • Use dual pattern: • 48 pentagonal starfish

  33. Only Two Geometrically Different Tiles • Inner and outer starfish prototiles extracted, • thickened by offsetting, • sent to FDM machine . . .

  34. Fresh from the FDM Machine

  35. Red Tile Set -- 1 of 6 Colors

  36. 2 Outer and 2 Inner Tiles

  37. A Whole Pile of Tiles . . .

  38. The Assembly of Tiles Begins . . . Outer tiles Inner tiles

  39. Assembly(cont.):8 Inner Tiles • Forming inner part of octa-frame edge

  40. 8 tiles Assembly (cont.) • 2 Hubs • + Octaframe edge inside view 12 tiles

  41. More Assembly Steps

  42. More Assembly Steps

  43. Assembly Gets More Difficult

  44. Almost Done ...

  45. The Finished Genus-7 Object • . . . I wish . . . • “work in progress . . .”

  46. What about Globally Regular Tilings ? So far: • Method and tool set to make complex, locally regular tilings on higher-genus surfaces.

  47. BRIDGES, London, 2006 “Eight-fold Way” by Helaman Ferguson

  48. Visualization of Klein’s Quartic in 3D 24 heptagons on a genus-3 surface; 24x7 automorphisms (= maximum possible)

  49. Another View ... 168 fish

  50. Why Is It Called: “Eight-fold Way” ? • Since it is a regular polyhedral structure, it has a set of Petrie Polygons. • These are “zig-zag” skew polygons that always hug a face for exactly 2 consecutive edges. • On a regular polyhedron you can start such a Petrie polygon from any vertex in any direction.(A good test for regularity !) • On the Klein Quartic, the length of these Petrie polygons is always eight edges.

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