Bridges 2008, Leeuwarden

1. Bridges 2008, Leeuwarden Carlo H. Séquin EECS Computer Science Division University of California, Berkeley Intricate Isohedral Tilings of 3D Euclidean Space

2. My Fascination with Escher Tilings in the plane on the sphere on the torus M.C. Escher Jane Yen, 1997 Young Shon, 2002

3. My Fascination with Escher Tilings • on higher-genus surfaces: London Bridges 2006 • What next ?

4. A fascinating intellectual excursion ! Celebrating the Spirit of M.C. Escher Try to do Escher-tilings in 3D …

5. A Very Large Domain ! • A very large domain • keep it somewhat limited

6. Monohedral vs. Isohedral monohedral tiling isohedral tiling In an isohedral tiling any tile can be transformed to any other tile location by mapping the whole tiling back onto itself.

7. Still a Large Domain!  Outline • Genus 0 • Modulated extrusions • Multi-layer tiles • Metamorphoses • 3D Shape Editing • Genus 1: “Toroids” • Tiles of Higher Genus • Interlinked Knot-Tiles

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

9. Genus 0: Simple Extrusions • Start from one of Escher’s 2D tilings … • Add 3rd dimension by extruding shape.

10. Extruded “2.5D” Fish-Tiles Isohedral Fish-Tiles Go beyond 2.5D !

11. Modulated Extrusions • Do something with top and bottom surfaces ! Shape height of surface before extrusion.

12. Tile from a Different Symmetry Group

13. Flat Extrusion of Quadfish

14. Modulating the Surface Height

15. Three tiles overlaid Manufactured Tiles (FDM)

16. Offset (Shifted) Overlay • Let Thick and thin areas complement each other: • RED = Thick areas;BLUE = THIN areas;

17. Shift Fish Outline to Desired Position • CAD tool calculates intersections with underlying height map of repeated fish tiles.

18. As QuickSlice sees the shape … 3D Shape is Saved in .STL Format

19. Fabricated Tiles …

20. Building Fish in Discrete Layers • How would these tiles fit together ? need to fill 2D plane in each layer ! • How to turn these shapes into isohedral tiles ? selectively glue together pieces on individual layers.

21. M. Goerner’s Tile • Glue elements of the two layers together.

22. Movie on YouTube ?

23. Escher Night and Day • Inspiration: Escher’s wonderful shape transformations (more by C. Kaplan…)

24. Escher Metamorphosis • Do the “morph”-transformation in the 3rd dim.

25. Bird into fish … and back

26. “FishBird”-Tile Fills 3D Space 1 red + 1 yellow  isohedral tile

27. True 3D Tiles • No preferential (special) editing direction. • Need a new CAD tool ! • Do in 3D what Escher did in 2D:modify the fundamental domain of a chosen tiling lattice

28. A 3D Escher Tile Editor • Start with truncated octahedron cell of the BCC lattice. • Each cell shares one face with 14 neighbors. • Allow arbitrary distortions and individual vertex moves.

29. Cell 1: Editing Result • A fish-like tile shape that tessellates 3D space

30. Another Fundamental Cell • Based on densest sphere packing. • Each cell has 12 neighbors. • Symmetrical form is the rhombic dodecahedron. • Add edge- and face-mid-points to yield 3x3 array of face vertices,making them quadratic Bézier patches.

31. Cell 2: Editing Result • Fish-like shapes … • Need more diting capabilities to add details …

32. Lessons Learned: • To make such a 3D editing tool is hard. • To use it to make good 3D tile designsis tedious and difficult. • Some vertices are shared by 4 cells, and thus show up 4 times on the cell-boundary; editing the front messes up back (and some sides!). • Can we let a program do the editing ?

33. Iterative Shape Approximation A closest matching shape is found among the 93 possible marked isohedral tilings; That cell is then adaptively distorted to match the desired goal shape as close as possible. • Try simulated annealing to find isohedral shape:“Escherization,” Kaplan and Salesin, SIGGRAPH 2000).

34. “Escherization” Resultsby Kaplan and Salesin, 2000 • Two different isohedral tilings.

35. Towards 3D Escherization • The basic cell – and the goal shape

36. Subdivided and partially annealed fish tile Simulated Annealing in Action • Basic cell and goal shape (wire frame)

37. The Final Result • made on a Fused Deposition Modeling Machine, • then hand painted.

38. More “Sim-Fish” • At different resolutions

39. Part II: Tiles of Genus > 0 • In 3D you can interlink tiles topologically !

40. Genus 1: Toroids • An assembly of 4-segment rings,based on the BCC lattice (Séquin, 1995)

41. Toroidal Tiles,Variations 12 F Based on cubic lattice 24 facets 16 F

42. Square Wire Frames in BCC Lattice • Tiles are approx. Voronoi regions around wires

43. Diamond Lattice & “Triamond” Lattice • We can do the same with 2 other lattices !

44. Diamond Lattice (8 cells shown)

45. Triamond Lattice (8 cells shown) aka “(10,3)-Lattice”, A. F. Wells “3D Nets & Polyhedra” 1977

46. “Triamond” Lattice • Thanks to John Conway and Chaim Goodman Strauss ‘Knotting Art and Math’ Tampa, FL, Nov. 2007 Visit to Charles Perry’s “Solstice”

47. Conway’s Segmented Ring Construction • Find shortest edge-ring in primary lattice (n rim-edges) • One edge of complement lattice acts as a “hub”/“axle” • Form n tetrahedra between axle and each rim edge • Split tetrahedra with mid-plane between these 2 edges

48. Diamond Lattice: Ring Construction • Two complementary diamond lattices, • And two representative 6-segment rings

49. Diamond Lattice:  6-Segment Rings • 6 rings interlink with each “key ring” (grey)

50. Cluster of 2 Interlinked Key-Rings • 12 rings total