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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

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

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

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

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A Very Large Domain ! u A very large domain u keep it somewhat limited

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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.

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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

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

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

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

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

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Tile from a Different Symmetry Group

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Flat Extrusion of Quadfish

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Modulating the Surface Height

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Manufactured Tiles (FDM) Three tiles overlaid

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

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

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

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Fabricated Tiles …

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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.

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

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Movie on YouTube ?

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

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

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u Bird into fish … and back

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

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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

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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.

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

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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.

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

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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 ?

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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.

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

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

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

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

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More Sim-Fish u At different resolutions

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

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

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

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

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Diamond Lattice & Triamond Lattice u We can do the same with 2 other lattices !

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Diamond Lattice (8 cells shown)

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

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

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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

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

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

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Cluster of 2 Interlinked Key-Rings u 12 rings total

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Honeycomb

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

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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

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

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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 ?

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

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

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Skewed Tria-Tiles

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Closed Chain of 10 Tria-Tiles

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

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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.

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Parameterized Tile Description u Allows aesthetic optimization of the tile shape

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

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

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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.

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

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Split Diamond 6-Rings

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

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Split Triamond 3-Ring

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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.

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

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Array of Interlocking Cube Frames

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Metropolis

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

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

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Split Cube Cage for Assembly

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Tetra-Cluster Built from 5 Cube Cages

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

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Analogous Mis-Assembly in 2D

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Linking Topology of Cube-Cage Lattice

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

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Genus-3 Cage made from Four 6-Rings

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Assembly of Diamond Lattice Cages

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

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Diamond (Slice) Frame Lattice

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With Complement Lattice Interspersed

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With Actual FDM Parts … u Some assembly required …

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Three 10-rings Make a Triamond Cage

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

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Genus-3 Cage Interlinked

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Split 10-Ring Frame

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Some assembly with these parts

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PART IV: Knot Tiles

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Topological Arrangement of Knot-Tiles

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

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Collection of Nearest-Neighbor Knots

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

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Conclusions Conclusions Many new and intriguing tiles …

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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)

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E X T R A S

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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 …

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

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

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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 …

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

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

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

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