Bridges 2008 Leeuwarden Intricate Isohedral Tilings of 3

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

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

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

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

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

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.

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

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

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

Extruded “ 2. 5 D” Fish-Tiles Isohedral Fish-Tiles Go beyond 2. 5 D !

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

Tile from a Different Symmetry Group

Flat Extrusion of Quadfish

Modulating the Surface Height

Manufactured Tiles (FDM) Three tiles overlaid

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

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

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

Fabricated Tiles …

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

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

Movie on You. Tube ?

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

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

u Bird into fish … and back

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

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

A 3 D 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.

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

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 3 x 3 array of face vertices, making them quadratic Bézier patches.

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

Lessons Learned: u To make such a 3 D editing tool is hard. u To use it to make good 3 D 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 ?

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.

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

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

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

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

More “Sim-Fish” u At different resolutions

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

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

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

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

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

Diamond Lattice (8 cells shown)

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

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

Conway’s 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

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

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

Cluster of 2 Interlinked Key-Rings u 12 rings total

Honeycomb

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

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

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

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

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

3 -Rings in Triamond Lattice 0° 19. 5°

Skewed Tria-Tiles

Closed Chain of 10 Tria-Tiles

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

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

Parameterized Tile Description u Allows aesthetic optimization of the tile shape

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

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

Isohedral Toroidal Tiles u 4 -segments cubic lattice u 6 -segments diamond lattice u 10 -segments u 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.

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

Split Diamond 6 -Rings

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

Split Triamond 3 -Ring

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 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.

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

Array of Interlocking Cube Frames

Metropolis

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

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

Split Cube Cage for Assembly

Tetra-Cluster Built from 5 Cube Cages

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

Analogous Mis-Assembly in 2 D

Linking Topology of Cube-Cage Lattice

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

Genus-3 Cage made from Four 6 -Rings

Assembly of Diamond Lattice Cages

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

Diamond (Slice) Frame Lattice

With Complement Lattice Interspersed

With Actual FDM Parts … u “Some assembly required … “

Three 10 -rings Make a Triamond Cage

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

Genus-3 Cage Interlinked

Split 10 -Ring Frame

Some assembly with these parts

PART IV: Knot Tiles

Topological Arrangement of Knot-Tiles

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

Collection of Nearest-Neighbor Knots

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

Conclusions Many new and intriguing tiles …

Acknowledgments u Matthias u Mark Howison (2. 5 D & 3 D tile editors) u Adam Megacz (annealed fish) u Roman u John Goerner (interlocking 2. 5 D tiles) Fuchs (Voronoi cells) Sullivan (manuscript)

EXTRAS

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

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

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

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 …