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

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 u on higher-genus surfaces: London Bridges 2006 u What next ?

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

5. A Very Large Domain ! u A very large domain u 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 u Genus 0 l Modulated extrusions l Multi-layer tiles l Metamorphoses l 3D Shape Editing u Genus 1: Toroids u Tiles of Higher Genus u Interlinked Knot-Tiles

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

9. Genus 0: Simple Extrusions u Start from one of Eschers 2D tilings … u Add 3 rd dimension by extruding shape.

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

11. Modulated Extrusions u 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. Manufactured Tiles (FDM) Three tiles overlaid

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

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

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

19. Fabricated Tiles …

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

21. M. Goerners Tile u Glue elements of the two layers together.

22. Movie on YouTube ?

23. Escher Night and Day u Inspiration: Eschers wonderful shape transformations (more by C. Kaplan…)

24. Escher Metamorphosis u Do the morph-transformation in the 3 rd dim.

25. u Bird into fish … and back

26. Fish Bird-Tile Fills 3D Space 1 red + 1 yellow isohedral tile

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

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

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

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

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

32. Lessons Learned: u To make such a 3D editing tool is hard. u To use it to make good 3D tile designs is tedious and difficult. u 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!). u Can we let a program do the editing ?

33. Iterative Shape Approximation u Try simulated annealing to find isohedral shape: Escherization, Kaplan and Salesin, SIGGRAPH 2000 ). 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.

34. Escherization Results by Kaplan and Salesin, 2000 u Two different isohedral tilings.

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

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

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

38. More Sim-Fish u At different resolutions

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

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

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

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

43. Diamond Lattice & Triamond Lattice u 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 u Thanks to John Conway and Chaim Goodman Strauss Knotting Art and Math Tampa, FL, Nov. 2007 Visit to Charles Perrys Solstice

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

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

49. Diamond Lattice: 6-Segment Rings u 6 rings interlink with each key ring (grey)

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

51. Honeycomb

52. Triamond Lattice Rings u Thanks to John Conway and Chaim Goodman-Strauss

53. Triamond Lattice: 10-Segment Rings u Two chiral ring versions from complement lattices u Key-ring of one kind links 10 rings of the other kind

54. Key-Ring with Ten 10-segment Rings Front and Back more symmetrical views

55. Are There Other Rings ?? u We have now seen the three rings that follow from the Conway construction. u Are there other rings ? u In particular, it is easily possible to make a key-ring of order 3 -- does this lead to a lattice with isohedral tiles ?

56. 3-Segment Ring ? u NO – that does not work !

57. 3-Rings in Triamond Lattice 0°0° 19.5°

58. Skewed Tria-Tiles

59. Closed Chain of 10 Tria-Tiles

60. Loop of 10 Tria-Tiles (FDM) u This pointy corner bothers me … u Can we re-design the tile and get rid of it ?

61. Optimizing the Tile Geometry u Finding the true geometry of the Voronoi zone by sampling 3D space and calculating distaces from a set of given wire frames; u Then making suitable planar approximations.

62. Parameterized Tile Description u Allows aesthetic optimization of the tile shape

63. Optimized Skewed Tria-Tiles u Got rid of the pointy protrusions ! A single tile Two interlinked tiles

64. Key-Ring of Optimized Tria-Tiles u And they still go together !

65. Isohedral Toroidal Tiles u 4-segments cubic lattice u 6-segments diamond lattice u 10-segments triamond lattice u 3-segments triamond lattice These rings are linking 4, 6, 10, 3 other rings. These numbers can be doubled, if the rings are split longitudinally.

66. Split Cubic 4-Rings u Each ring interlinks with 8 others

67. Split Diamond 6-Rings

68. Key-Ring with Twenty 10-segment Rings Front view Back view All possible color pairs are present !

69. Split Triamond 3-Ring

70. PART III: Tiles Of Higher Genus u No need to limit ourselves to simple genus_1 toroids ! u We can use handle-bodies of higher genus that interlink with neighboring tiles with separate handle-loops. u Again the possibilities seem endless, so lets take a structures approach and focus on regular tiles derived from the 3 lattices that we have discussed so far in this talk.

71. Simplest Genus-5 Cube Frame u Frame built from six split 4-rings

72. Array of Interlocking Cube Frames

73. Metropolis

74. Linking Topology of Metropolis u Note: Every cube face has two wire squares along it

75. Cube Cage Built from Six 4-Rings Cages built from the original non-split rings.

76. Split Cube Cage for Assembly

77. Tetra-Cluster Built from 5 Cube Cages

78. Linear Array of Cube Cages u An interlinking chain along the space diagonal THIS DOES NOT TILE 3D SPACE !

79. Analogous Mis-Assembly in 2D

80. Linking Topology of Cube-Cage Lattice

81. Cages and Frames in Diamond Lattice u Four 6-segment rings form a genus-3 cage 6-ring keychain

82. Genus-3 Cage made from Four 6-Rings

83. Assembly of Diamond Lattice Cages

84. Assembling Split 6-Rings u 4 RINGS Forming a diamond-frame

85. Diamond (Slice) Frame Lattice

86. With Complement Lattice Interspersed

87. With Actual FDM Parts … u Some assembly required …

88. Three 10-rings Make a Triamond Cage

89. Cages in the Triamond Lattice u Two genus-3 cages == compound of three 10-rings u They come in two different chiralities !

90. Genus-3 Cage Interlinked

91. Split 10-Ring Frame

92. Some assembly with these parts

93. PART IV: Knot Tiles

94. Topological Arrangement of Knot-Tiles

95. Important Geometrical Considerations u Critical point: prevent fusion into higher-genus object!

96. Collection of Nearest-Neighbor Knots

97. Finding Voronoi Zone for Wire Knots u 2 Solutions for different knot parameters

98. Conclusions Conclusions Many new and intriguing tiles …

99. Acknowledgments u Matthias Goerner (interlocking 2.5D tiles) u Mark Howison (2.5D & 3D tile editors) u Adam Megacz (annealed fish) u Roman Fuchs (Voronoi cells) u John Sullivan (manuscript)

100. E X T R A S

101. What Linking Numbers are Possible? u We have: 4, 6, 10, 3 u And by splitting: 8, 12, 20, 6 u Lets go for the low missing numbers: 1, 2, 5, 7, 9 …

102. Linking Number =1 u Cube with one handle that interlocks with one neighbor

103. Linking Number =2 u Long chains of interlinked rings, packed densely side by side.

104. Linking Number =5 u Idea: take every second one in the triamond lattice with L=10 u But try this first on Honecomb where it is easier to see what is going on …

105. Linking Number =3 u But derived from Diamond lattice by taking only every other ring… u the unit cell:

106. An Array of such Cells u Has the connectivity of the Triamond Lattice !

107. Array of Five Rings Interlinked ?? u Does not seem to lead to an isohedral tiling