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

Bridges 2008, Leeuwarden

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Bridges 2008, Leeuwarden. Carlo H. S é quin EECS Computer Science Division University of California, Berkeley. Intricate Isohedral Tilings of 3D Euclidean Space. My Fascination with Escher Tilings. in the plane on the sphere on the torus - PowerPoint PPT Presentation

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Page 1: Bridges 2008, Leeuwarden

Bridges 2008, LeeuwardenBridges 2008, Leeuwarden

Intricate Isohedral TilingsIntricate Isohedral Tilings

of 3D Euclidean Spaceof 3D Euclidean Space

Carlo H. SCarlo H. Sééquinquin

EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley

Page 2: Bridges 2008, Leeuwarden

My Fascination with Escher TilingsMy Fascination with Escher Tilings

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

Page 3: Bridges 2008, Leeuwarden

My Fascination with Escher TilingsMy Fascination with Escher Tilings

on higher-genus surfaces:

London Bridges 2006

What next ?

Page 4: Bridges 2008, Leeuwarden

Celebrating the Spirit of M.C. EscherCelebrating the Spirit of M.C. EscherTry to do Escher-tilings in 3D …

A fascinating intellectual excursion !A fascinating intellectual excursion !

Page 5: Bridges 2008, Leeuwarden

A very large domainA very large domain keep it somewhat limitedkeep it somewhat limited

Page 6: Bridges 2008, Leeuwarden

Monohedral vs. Monohedral vs. IsohedralIsohedral

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.

Page 7: Bridges 2008, Leeuwarden

Still a Large Domain! Still a Large Domain! Outline Outline

Genus 0 Modulated extrusions Multi-layer tiles Metamorphoses 3D Shape Editing

Genus 1: “Toroids” Tiles of Higher Genus Interlinked Knot-Tiles

Page 8: Bridges 2008, Leeuwarden

How to Make an “Escher Tiling”How to Make an “Escher Tiling”

Start from a regular tiling Distort all equivalent edges in the same way

Page 9: Bridges 2008, Leeuwarden

Genus 0:Genus 0: Simple Extrusions Simple Extrusions

Start from one of Escher’s 2D tilings … Add 3rd dimension by extruding shape.

Page 10: Bridges 2008, Leeuwarden

Extruded “2.5D” Fish-TilesExtruded “2.5D” Fish-Tiles

Isohedral Fish-Tiles

Go beyond 2.5D !

Page 11: Bridges 2008, Leeuwarden

Modulated ExtrusionsModulated Extrusions Do something with top and bottom surfaces !

Tailor the surface height before extrusion.

Page 12: Bridges 2008, Leeuwarden

Tile from a Different Symmetry GroupTile from a Different Symmetry Group

Page 13: Bridges 2008, Leeuwarden

Flat Extrusion of QuadfishFlat Extrusion of Quadfish

Page 14: Bridges 2008, Leeuwarden

Modulating the Surface HeightModulating the Surface Height

Page 15: Bridges 2008, Leeuwarden

Red part is viewed from the bottom

Manufactured Tiles (FDM)Manufactured Tiles (FDM)

Three tiles overlaid

Page 16: Bridges 2008, Leeuwarden

Offset (Shifted) OverlayOffset (Shifted) Overlay

Let thick and thin areas complement each other: RED = Thick areas; BLUE = THIN areas;

Page 17: Bridges 2008, Leeuwarden

Shift Fish Outline to Desired PositionShift Fish Outline to Desired Position

CAD tool calculates intersections with underlying height map of repeated fish tiles.

Page 18: Bridges 2008, Leeuwarden

3D Shape is Saved in .STL Format3D Shape is Saved in .STL Format

As QuickSlice sees the shape …

Page 19: Bridges 2008, Leeuwarden

Fabricated Tiles …Fabricated Tiles …

Top and bottom view Snug fit in the plane …

Page 20: Bridges 2008, Leeuwarden

Adding Two More TilesAdding Two More Tiles

Page 21: Bridges 2008, Leeuwarden

Adding Tiles in a 2Adding Tiles in a 2ndnd Layer Layer

Snug fit also in the third dimension !

Page 22: Bridges 2008, Leeuwarden

Building Fish in Discrete LayersBuilding 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.

Page 23: Bridges 2008, Leeuwarden

M. Goerner’s TileM. Goerner’s Tile

Glue together elements from two subsequent layers.

Page 24: Bridges 2008, Leeuwarden

Escher Night and DayEscher Night and Day

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

Page 25: Bridges 2008, Leeuwarden

M.C. Escher: MetamorphosisM.C. Escher: Metamorphosis

Do similar “morph”-transformation in the 3rd dim.

Page 26: Bridges 2008, Leeuwarden

Bird Bird Fish Fish

A sweep-morph from bird into fish … and back

Page 27: Bridges 2008, Leeuwarden

““FishFishBird”-Tile Fills 3D SpaceBird”-Tile Fills 3D Space

1 red + 1 yellow

isohedral tile

Page 28: Bridges 2008, Leeuwarden

True 3DTrue 3D Tiles 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

Page 29: Bridges 2008, Leeuwarden

A 3D Escher Tile EditorA 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.

Page 30: Bridges 2008, Leeuwarden

BCC Cell: Editing ResultBCC Cell: Editing Result

A fish-like tile shape that tessellates 3D space

Page 31: Bridges 2008, Leeuwarden

Another Fundamental CellAnother 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.

Page 32: Bridges 2008, Leeuwarden

Cell 2: Editing ResultCell 2: Editing Result

Can yield fish-like shapes Need more editing capabilities to add details …

Page 33: Bridges 2008, Leeuwarden

Adam Megacz’ Compound Cell EditorAdam Megacz’ Compound Cell Editor

“Hammerhead” starting configurationCan select and drag individual vertices Corresponding vertices will follow !

Page 34: Bridges 2008, Leeuwarden

Final Edited ShapeFinal Edited Shape

“Butterfly-Stingray” by Adam Megacz

Page 35: Bridges 2008, Leeuwarden

Snug fit in the plane …

The Fabricated Tiles …The Fabricated Tiles …

and between the planes!

Page 36: Bridges 2008, Leeuwarden

Lessons Learned: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 ?

Page 37: Bridges 2008, Leeuwarden

Iterative Shape ApproximationIterative 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 matchthe desired goal shape as close as possible.

Page 38: Bridges 2008, Leeuwarden

““Escherization” ResultsEscherization” Resultsby Kaplan and Salesin, 2000by Kaplan and Salesin, 2000

Two different isohedral tilings.

Page 39: Bridges 2008, Leeuwarden

Towards 3D EscherizationTowards 3D Escherization

The basic cell, based on a rhombic dodecahedron Each cell has 12 direct neighbors

Page 40: Bridges 2008, Leeuwarden

The Goal ShapeThe Goal Shape

Designed in a separate CAD program

Page 41: Bridges 2008, Leeuwarden

Simulated Annealing in ActionSimulated Annealing in Action

Basic cell and goal shape (wire frame) Subdivided and partially annealed 3D fish tile

Page 42: Bridges 2008, Leeuwarden

The Final ResultThe Final Result

made on a Fused Deposition Modeling Machine, then hand painted.

Page 43: Bridges 2008, Leeuwarden

More “Sim-Fish”More “Sim-Fish”

At different resolutions

Page 44: Bridges 2008, Leeuwarden

Part II:Part II: Tiles of Genus > 0 Tiles of Genus > 0

In 3D you can interlink tiles topologically !

Page 45: Bridges 2008, Leeuwarden

Genus 1: ToroidsGenus 1: Toroids

An assembly of 4-segment rings,based on the BCC lattice (Séquin, 1995)

Page 46: Bridges 2008, Leeuwarden

Toroidal Tiles,Toroidal Tiles,VariationsVariations

Based on cubic lattice

24 facets

12 F

16 F

Page 47: Bridges 2008, Leeuwarden

Square Wire Frames in BCC LatticeSquare Wire Frames in BCC Lattice

Tiles are approx. Voronoi regions around wire loops

Page 48: Bridges 2008, Leeuwarden

Diamond Lattice & “Triamond” LatticeDiamond Lattice & “Triamond” Lattice

We can do the same with two other lattices !

Page 49: Bridges 2008, Leeuwarden

Diamond Lattice Diamond Lattice (8 cells shown)(8 cells shown)

Page 50: Bridges 2008, Leeuwarden

Diamond LatticeDiamond Lattice

SLS modelby George Hart

Page 51: Bridges 2008, Leeuwarden

Double (Interlinked) Diamond LatticeDouble (Interlinked) Diamond Lattice

computer modelby George Hart

Page 52: Bridges 2008, Leeuwarden

Triamond Lattice Triamond Lattice (8 cells shown)(8 cells shown)

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

Page 53: Bridges 2008, Leeuwarden

““Triamond” LatticeTriamond” Lattice

computer modelby George Hart

Page 54: Bridges 2008, Leeuwarden

Double (interlinked) “Triamond” LatticeDouble (interlinked) “Triamond” Lattice

computer modelby George Hart

Page 55: Bridges 2008, Leeuwarden

Double (interlinked) “Triamond” LatticeDouble (interlinked) “Triamond” Lattice

SLS modelby George Hart

Page 56: Bridges 2008, Leeuwarden

““Triamond” LatticeTriamond” Lattice Thanks to John Conway and Chaim Goodman Strauss

‘Knotting Art and Math’ Tampa, FL, Nov. 2007Visit to Charles Perry’s “Solstice”

Page 57: Bridges 2008, Leeuwarden

Conway’s Segmented Ring ConstructionConway’s Segmented Ring Construction Find shortest edge-ring in primary lattice (4 cyan tubes) One edge of complement lattice acts as “axle” (yellow tube) Form n tetrahedra between axle and one rim edge each (black)

Split tetrahedra with mid-plane between these two edges.

Do this for the next ring edge.

Do this for all four ring edges: yields a 4-segment ring.

Page 58: Bridges 2008, Leeuwarden

Diamond Lattice: Ring ConstructionDiamond Lattice: Ring Construction

One diamond lattice cellComplement diamond lattice cell6-ring of edges + corr. “axle”6-segment ring in “red” latticeComplementary 6-segment ring

Page 59: Bridges 2008, Leeuwarden

Diamond Lattice: Diamond Lattice: 6-Segment Rings 6-Segment Rings

6 rings interlink with each “key ring” (grey)

Page 60: Bridges 2008, Leeuwarden

Cluster of 2 Interlinked Key-RingsCluster of 2 Interlinked Key-Rings

12 rings total

Page 61: Bridges 2008, Leeuwarden

HoneycombHoneycomb

Page 62: Bridges 2008, Leeuwarden

Triamond Lattice RingsTriamond Lattice Rings Thanks to John Conway and

Chaim Goodman-Strauss

A single triamond lattice cellAdd a second lattice cellTwo 10-rings in the primary lattice5 interlinked complementary ringsAdding the same set of 5 in the 2nd cell

Page 63: Bridges 2008, Leeuwarden

Triamond Lattice: Triamond Lattice: 10-Segment Rings 10-Segment Rings

Two chiral ring versions from complement lattices Key-ring of one kind links 10 rings of the other kind

Page 64: Bridges 2008, Leeuwarden

Key-Ring with Ten 10-segment RingsKey-Ring with Ten 10-segment Rings

“Front” and “Back”

Two more symmetrical views !

Page 65: Bridges 2008, Leeuwarden

Are There Other Rings ??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 ?

Page 66: Bridges 2008, Leeuwarden

3-Segment Ring ?3-Segment Ring ?

NO – that does not work !

Page 67: Bridges 2008, Leeuwarden

3-Rings in Triamond Lattice3-Rings in Triamond Lattice

0°19.5°

Page 68: Bridges 2008, Leeuwarden

Skewed Tria-TilesSkewed Tria-Tiles

Page 69: Bridges 2008, Leeuwarden

Closed Chain of 10 Tria-TilesClosed Chain of 10 Tria-Tiles

Page 70: Bridges 2008, Leeuwarden

Closed Chain of 10 Tria-Tiles (FDM)Closed Chain of 10 Tria-Tiles (FDM)

• This pointy corner bothers me …

• Can we re-design the tile and get rid of it ?

Page 71: Bridges 2008, Leeuwarden

Optimizing the Tile GeometryOptimizing the Tile Geometry

Finding the true geometry of the Voronoi zoneby sampling 3D space and calculating distancesfrom a set of given wire frames;

Then making suitable planar approximations.

Page 72: Bridges 2008, Leeuwarden

Parameterized Tile DescriptionParameterized Tile Description

Allows aesthetic optimization of the tile shape

Page 73: Bridges 2008, Leeuwarden

““Optimized” Skewed Tria-TilesOptimized” Skewed Tria-Tiles Got rid of the pointy protrusions !

A single tile Two interlinked tiles

Page 74: Bridges 2008, Leeuwarden

Key-Ring of Optimized Tria-TilesKey-Ring of Optimized Tria-Tiles

And they still fit together snugly ! (red tiles consist of only two shanks)

C

B

BA

A

Page 75: Bridges 2008, Leeuwarden

Larger Assembly of Optimized Tria-TilesLarger Assembly of Optimized Tria-Tiles

-------- Rotatate 45° -------

A

A

Page 76: Bridges 2008, Leeuwarden

Isohedral Toroidal TilesIsohedral Toroidal Tiles Cubic lattice 4-segment rings Diamond lattice 6-segment rings Triamond lattice 10-segment rings Triamond lattice 3-segment rings

These rings are linking 4, 6, 10, 3 other rings.

The linking numbers can be doubled, if the rings are sliced longitudinally.

Page 77: Bridges 2008, Leeuwarden

Sliced Cubic 4-RingsSliced Cubic 4-Rings

Each ring interlinks with 8 others

Page 78: Bridges 2008, Leeuwarden

Sliced Diamond 6-RingsSliced Diamond 6-Rings

Page 79: Bridges 2008, Leeuwarden

Slicing the 10-Segment RingSlicing the 10-Segment Ring

Page 80: Bridges 2008, Leeuwarden

Key-Ring with Twenty Sliced 10-RingsKey-Ring with Twenty Sliced 10-Rings

“Front” view “Back” view

All possible color pairs are present !

Page 81: Bridges 2008, Leeuwarden

Slicing the Tria-TileSlicing the Tria-Tile

6 sliced Tria-Tiles hook into the white key-ring

Page 82: Bridges 2008, Leeuwarden

PART III: Tiles Of Higher GenusPART 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.

Page 83: Bridges 2008, Leeuwarden

Simplest Genus-5 Cube FrameSimplest Genus-5 Cube Frame

“Frame” built from six sliced 4-segment-rings

Page 84: Bridges 2008, Leeuwarden

Array of Interlocking Cube FramesArray of Interlocking Cube Frames

Page 85: Bridges 2008, Leeuwarden

MetropolisMetropolis

Page 86: Bridges 2008, Leeuwarden

Linking Topology of “Metropolis”Linking Topology of “Metropolis”

Note: Every cube face has two wire squares along it

Page 87: Bridges 2008, Leeuwarden

Cube “Cage” Built from Six 4-RingsCube “Cage” Built from Six 4-Rings

“Cages” built from the original non-sliced rings.

Only one “Voronoi-generator-square” per face!

Page 88: Bridges 2008, Leeuwarden

Split Cube Cage for AssemblySplit Cube Cage for Assembly

Page 89: Bridges 2008, Leeuwarden

Tetra-Cluster Built from 5 Cube Cages Tetra-Cluster Built from 5 Cube Cages

Page 90: Bridges 2008, Leeuwarden

Linear Array of Cube CagesLinear Array of Cube Cages

An interlinking chain along the space diagonalTHIS DOES NOT TILE 3D SPACE !

Page 91: Bridges 2008, Leeuwarden

Analogous Mis-Assembly in 2DAnalogous Mis-Assembly in 2D

Page 92: Bridges 2008, Leeuwarden

Linking Topology of Cube-Cage LatticeLinking Topology of Cube-Cage Lattice

Page 93: Bridges 2008, Leeuwarden

CagesCages and Frames in and Frames in Diamond LatticeDiamond Lattice

four 6-segment rings form a genus-3 cage

6-ring keychain …

Page 94: Bridges 2008, Leeuwarden

Genus-3 Cage made from Four 6-RingsGenus-3 Cage made from Four 6-Rings

Page 95: Bridges 2008, Leeuwarden

Assembly of Diamond Lattice CagesAssembly of Diamond Lattice Cages

Page 96: Bridges 2008, Leeuwarden

4-Ring Diamond 4-Ring Diamond FrameFrame

Four sliced 6-segment ringsTogether they form a genus-3 frame

Page 97: Bridges 2008, Leeuwarden

Diamond (Slice) Frame LatticeDiamond (Slice) Frame Lattice

Page 98: Bridges 2008, Leeuwarden

With Complement Lattice InterspersedWith Complement Lattice Interspersed

Page 99: Bridges 2008, Leeuwarden

With Actual FDM Parts …With Actual FDM Parts …

“Some assembly required … “

Page 100: Bridges 2008, Leeuwarden

Assembly of Diamond Lattice FramesAssembly of Diamond Lattice Frames

Page 101: Bridges 2008, Leeuwarden

Three 10-rings Yield a Three 10-rings Yield a Triamond CageTriamond Cage

Page 102: Bridges 2008, Leeuwarden

Split 3-Ring Cages (Triamond Lattice)Split 3-Ring Cages (Triamond Lattice)

Genus-2 Triamond cages == compound of three 10-rings They come in two different chiralities !

Page 103: Bridges 2008, Leeuwarden

Assembling Triamond CagesAssembling Triamond Cages

7 cages hook into the green central cage

Page 104: Bridges 2008, Leeuwarden

Adding More Triamond CagesAdding More Triamond Cages

More green cages at the bottom.

Three blue cages on top.

Page 105: Bridges 2008, Leeuwarden

3 3 SlicedSliced Rings Yield Triamond Rings Yield Triamond FrameFrame

The two halves of a sliced 10-ring put together with their two “outer” faces yield 2/3 of a “frame”

Page 106: Bridges 2008, Leeuwarden

Split 3-Ring Triamond Frame (FDM)Split 3-Ring Triamond Frame (FDM)

FDM parts designed for the assembly of complex clusters.

Page 107: Bridges 2008, Leeuwarden

Assembling Triamond 3-Ring FramesAssembling Triamond 3-Ring Frames

7 frames hooked into white half-frame

Page 108: Bridges 2008, Leeuwarden

Adding Upper Half of White FrameAdding Upper Half of White Frame

A total of 14 frames hook into each frame

Page 109: Bridges 2008, Leeuwarden

Completed Cluster AssemblyCompleted Cluster Assembly

Page 110: Bridges 2008, Leeuwarden

PART IV:PART IV: Knot Tiles Knot Tiles

Page 111: Bridges 2008, Leeuwarden

Topological Arrangement of Knot-TilesTopological Arrangement of Knot-Tiles

Page 112: Bridges 2008, Leeuwarden

Important Geometrical ConsiderationsImportant Geometrical Considerations Critical point:

prevent fusion into higher-genus object!

Page 113: Bridges 2008, Leeuwarden

Collection of Nearest-Neighbor KnotsCollection of Nearest-Neighbor Knots

Page 114: Bridges 2008, Leeuwarden

Finding Voronoi Zone for Wire KnotsFinding Voronoi Zone for Wire Knots

2 Solutions for different knot parameters

Page 115: Bridges 2008, Leeuwarden

ConclusionsConclusions

Many new and intriguing tiles …Many new and intriguing tiles …

Page 116: Bridges 2008, Leeuwarden

AcknowledgmentsAcknowledgments

Matthias Goerner (interlocking 2.5D tiles) Mark Howison (2.5D & 3D tile editors) Adam Megacz (annealed fish & 3D tile editor) Roman Fuchs (Voronoi cell constructions) John Sullivan (review of my manuscript)

Page 117: Bridges 2008, Leeuwarden

E X T R A SE X T R A S