1. Intricate Isohedral Tilings of 3D Euclidean Space
Granada 2003
Bridges 2008, Leeuwarden
Intricate Isohedral Tilings of 3D Euclidean Space
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley
Thanks for this opportunity to give a plenary lecture in this illustrious town.

2. My Fascination with Escher Tilings
Granada 2003
My Fascination with Escher Tilings
in the plane on the sphere on the torus
M.C. Escher Jane Yen, Young Shon, 2002
I have been fascinated with M.C. Escher’s work ever since I was a High-school student.
But I only started to work actively in tessellations in the last 10 years.

3. My Fascination with Escher Tilings
Granada 2003
My Fascination with Escher Tilings
on higher-genus surfaces:
London Bridges 2006
What next ?
Then in 2006 I carried such Escher tilings to higher-genus surfaces, in particular the genus 4 Tetrus.

4. Celebrating the Spirit of M.C. Escher
Granada 2003
Celebrating the Spirit of M.C. Escher
Try to do Escher-tilings in 3D …
A fascinating intellectual excursion !
So when this opportunity arose …
How better to celebrate the siprit of M.C.Escher than to try to carry his tiling work into the 3rd dimension.

5. A Very Large Domain ! A very large domain keep it somewhat limited
Granada 2003
A Very Large Domain !
… thus tried to keep the scope rather limited
-- and still this is what my office table looked like two weeks before I traveled to Europe.
A first major restriction is that I will only concern myself with isohedral tilings 
A very large domain
keep it somewhat limited

6. Monohedral vs. Isohedral
Granada 2003
Monohedral vs. Isohedral
Left an intrigiuing tiling I first saw in Gruenbaum and Shephards book: “Tilings and Patterns”.
It uses only a single tile shape, thus it is monohedral.
The same tile shape can be laid out differently …
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
Granada 2003
How to Make an Escher Tiling
… all edges around one gird unit == fundamental domain,
Observing all implied symmetries of the basic tiling.
Start from a regular tiling
Distort all equivalent edges in the same way

9. Genus 0: Simple Extrusions
Granada 2003
Genus 0: Simple Extrusions
A simple way to obtain 3D tiles …
Not very original, many people have done that, and there are many toys and puzzles using such shapes 
Start from one of Escher’s 2D tilings …
Add 3rd dimension by extruding shape.

10. Extruded “2.5D” Fish-Tiles
Granada 2003
Extruded “2.5D” Fish-Tiles
Isohedral Fish-Tiles
While these shapes are fun to play with, they are not truly 3D Escher tiles, at best they are 2.5 dimensional.
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
Granada 2003
Tile from a Different Symmetry Group
Another example …

13. Flat Extrusion of Quadfish

14. Modulating the Surface Height

15. Manufactured Tiles (FDM)
Three tiles overlaid

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

19. Fabricated Tiles …

20. Building Fish in Discrete Layers
Granada 2003
Building Fish in Discrete Layers
Another conceptual approach: building a 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.
Granada 2003
M. Goerner’s Tile
Well – it is not a good way of designing fish – but this approach has resulted in some rather intriguing tiles:
Mathias Goerner, a student in my Fall graduate course on solid modeling and RP has developed this neat tile:
Conceptually you start with a tiling pattern that has two different shapes. Then you line up two of these layers to maintain symmetry
-- in this case it means that the little red squares would go into the center of the yellow octagon on the layer below.
One such two-layer compound makes an isohedral tile. Now Mathias embellised the simple polygon shapes into twisted interlocking serrated wheel structures. They have to be assembled one at a time by inserting them with a twisting motion.
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. Granada 2003
A 3D Escher Tile Editor
… and at each vertex exactly 3 cells come together.
For more flexibility add an extra vertex in the center of each face.
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 designs is 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
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
Two different isohedral tilings.

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

36. Simulated Annealing in Action
Subdivided and partially annealed fish tile
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. Granada 2003
Genus 1: Toroids
On the right the geometry of each ring
An assembly of 4-segment rings, based on the BCC lattice (Séquin, 1995)

41. Toroidal Tiles, Variations
Granada 2003
Toroidal Tiles, Variations
12 F
Variations are possible – all just different polyhedral approximations of the Voronoi zones of certain arrays of interlinked wire frames.
24 facets
16 F
Based on cubic lattice

42. Square Wire Frames in BCC Lattice
Granada 2003
Square Wire Frames in BCC Lattice
Square tubular frames act as “post-office” sites for their respective Voronoi zones.
Each point in space that is closer to a red frame than to any other frame becomes part of that red toroidal tile
And the same for the two other colors.
Tiles are approx. Voronoi regions around wires

43. Diamond Lattice & “Triamond” Lattice
Granada 2003
Diamond Lattice & “Triamond” Lattice
On left: diamond lattice – focus on small blue atoms – each atom is linked to 4 nearest neighbors.
This lattice is an enhancement of the face-centered cubic lattice FCC. (Blue in center of 4 green atoms on a face of the cube)
On right: triamond lattice. Each atom is linked to only 3 nearest neighbors.
This lattice is an enhancement of the body-centered cubic lattice: BCC. (yellow in center of 8 green atoms at the corners of the cube)
We can do the same with 2 other lattices !

44. Diamond Lattice (8 cells shown)
Granada 2003
Diamond Lattice (8 cells shown)
Again, Look at small blue atoms – linked to 8 nearest neighbors

45. Triamond Lattice (8 cells shown)
Granada 2003
Triamond Lattice (8 cells shown)
Compare look at big yellow atoms: only 3 NN.
And in January of 2008 rediscovered once again by Mr. Sunada, Meiji Universit, Japan
== 8 months ago, I did not know about this lattice either !
aka “(10,3)-Lattice”, A. F. Wells “3D Nets & Polyhedra” 1977

46. “Triamond” Lattice Thanks to John Conway and Chaim Goodman Strauss
Granada 2003
“Triamond” Lattice
Thanks to John Conway and Chaim Goodman Strauss
In the car to downtown Tampa, John explained to me the mysteries of the Triamond lattice
And the generic construction that would lead to the segmented ring tiles for all three lattices mentioned so far.
OK – now we want to see, how we can use these lattices to form isohedral segmented ring tiles.
‘Knotting Art and Math’ Tampa, FL, Nov. 2007
Visit to Charles Perry’s
“Solstice”

47. Conway’s Segmented Ring Construction
Granada 2003
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
Here is Conway’s recipe:
Let’s do it for the cubic lattice first, since it is easiest to understand …

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

49. Diamond Lattice:  6-Segment Rings
Granada 2003
Diamond Lattice:  6-Segment Rings
Left, the way rings interlock
Right: rings have been slimmed down by about 50% for better visibility.
6 rings interlink with each “key ring” (grey)

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

51. Honeycomb
Granada 2003
A larger assembly of 6-segment rings – driven by mostly aesthetic considerations.
-- is in art exhibit.

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

53. Triamond Lattice:  10-Segment Rings
Two chiral ring versions from complement lattices
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 ??
We have now seen the three rings that follow from the Conway construction.
Are there other rings ?
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 ?
NO – that does not work !

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

58. Skewed Tria-Tiles

59. Closed Chain of 10 Tria-Tiles

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

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

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

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

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

65. Isohedral Toroidal Tiles
4-segments  cubic lattice
6-segments  diamond lattice
10-segments  triamond lattice
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
Each ring interlinks with 8 others

67. Split Diamond 6-Rings

68. Key-Ring with Twenty 10-segment Rings
Granada 2003
Key-Ring with Twenty 10-segment Rings
Look for the silver key ring ! Amazing that this works. Took me a while to put it together!
But the insane thing is, that each of the 20 rings hooked into the key-ring also acts as key-ring for 20 other rings !
“Front” view “Back” view
All possible color pairs are present !

69. Split Triamond 3-Ring

70. PART III: Tiles Of Higher Genus
No need to limit ourselves to simple genus_1 toroids !
We can use handle-bodies of higher genus that interlink with neighboring tiles with separate handle-loops.
Again the possibilities seem endless, so let’s 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
“Frame” built from six split 4-rings

72. Array of Interlocking Cube Frames
Granada 2003
Array of Interlocking Cube Frames
It is kind of obvious how these elements interlink: you can clearly see the structure of the BCC in this arrangement.
Note there a re two separate interpenetrating cubic lattices here:
one is composed from alternating yellow and blue cubes,
The other is built from red and green cubes.

73. Metropolis
Granada 2003
From the proper angle it looks like a monstruous cubistic sky scraper right out of the movie Metropolis,
So that gave it its name. This assembly is also in the art show.
This view also gives you a good chance to see the offset between the two interspersed cubic lattices.

74. Linking Topology of “Metropolis”
Granada 2003
Linking Topology of “Metropolis”
Now we can form a different tiype of higher-genus tile …
Note: Every cube face has two wire squares along it

75. Cube Cage Built from Six 4-Rings
Granada 2003
Cube Cage Built from Six 4-Rings
But alternatively we can take six of our original, fat, non-split rings and assemble them into something I call a “cage”.
These cages will stack together and interlink in a somewhat different manner!
“Cages” built from the original non-split rings.

76. Split Cube Cage for Assembly
Granada 2003
Split Cube Cage for Assembly
This shows some of the actual elements I constructed to demonstrate these assemblies.

77. Tetra-Cluster Built from 5 Cube Cages
Granada 2003
Tetra-Cluster Built from 5 Cube Cages
That shows the basic interlink structure: each cube-cage links with four neighboring ones in a tetrahedral manner.
(take for example the silver one in the middle).

78. Linear Array of Cube Cages
Granada 2003
Linear Array of Cube Cages
THIS DOES NOT TILE 3D SPACE !
But you have to be careful how you start assembling these genus-5 tiles. A linear array of tiles does not work!
You get stuck and will not be able to fill space completely with an isohedral tile arrangement!
An interlinking chain along the space diagonal

79. Analogous Mis-Assembly in 2D

80. Linking Topology of Cube-Cage Lattice
Granada 2003
Linking Topology of Cube-Cage Lattice
Every tile interlocks with 4 from the complement lattice in tetrahedral fashion,
And touches 12 tiles of its own sparsely populated type of lattice along its 12 edges.

81. Cages and Frames in Diamond Lattice
Granada 2003
Cages and Frames in Diamond Lattice
6-ring keychain
Remember the 6-ring keychain ? – Add a couple of rings on top to form a 4-ring cage of genus 3 with tetra symmetry.
Four 6-segment rings form a genus-3 cage

82. Genus-3 Cage made from Four 6-Rings
Granada 2003
Genus-3 Cage made from Four 6-Rings
Here is the actual cage built on the FDM machine.
Here again I use the original, fat, non-split rings to form a “cage” with tetrahedral symmetry.
Its convex hull is the rhombic dodecahedron, and thus each tile touches 12 neighbors on the outside.
Its 6 bent tetrahedral arms interlink with 6 nearest neighbor tiles.

83. Assembly of Diamond Lattice Cages
Granada 2003
Assembly of Diamond Lattice Cages
6 cages link with the blue cage in the center in a cluster with octahedral symmetry.

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

85. Diamond (Slice) Frame Lattice

86. With Complement Lattice Interspersed
Granada 2003
With Complement Lattice Interspersed
The two complementary lattices interspersed.

87. With Actual FDM Parts … “Some assembly required … “ 
Granada 2003
With Actual FDM Parts …
PHOTO OF PARTS __ UNASSEMBLED ??
“Some assembly required … “ 

88. Three 10-rings Make a Triamond Cage

89. Cages in the Triamond Lattice
Two genus-3 cages == compound of three 10-rings
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
Granada 2003
Topological Arrangement of Knot-Tiles
Interlink them so that they hold space together !
All loops need to accommodate one or more strands going through it, so that that cannot collaps when we try to fill space densely.

95. Important Geometrical Considerations
Granada 2003
Important Geometrical Considerations
Critical point: prevent fusion into higher-genus object!
But there is also a second problem ; crossing loops must not touch so as not to fuse and change the genus of the handle-body.

96. Collection of Nearest-Neighbor Knots

97. Finding Voronoi Zone for Wire Knots
Granada 2003
Finding Voronoi Zone for Wire Knots
Work to be done is to turn this into a nice polyhedral approximation in the same way as was done for the skewed triatile.
2 Solutions for different knot parameters

98. Conclusions
Many new and intriguing tiles …

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

100. E X T R A S

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

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

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

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

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

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

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