Bridges 2008, Leeuwarden Intricate Isohedral Tilings of 3D Euclidean Space Intricate Isohedral Tilings of 3D Euclidean Space Carlo H. Séquin EECS Computer.

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

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

My Fascination with Escher TilingsMy Fascination with Escher Tilings

on higher-genus surfaces:

London Bridges 2006

What next ?

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 !

A very large domainA very large domain

keep it somewhat limitedkeep it somewhat limited

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.

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

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

Genus 0:Genus 0: Simple Extrusions Simple Extrusions

Start from one of Escher’s 2D tilings …

Add 3rd dimension by extruding shape.

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

Isohedral Fish-Tiles

Go beyond 2.5D !

Modulated ExtrusionsModulated Extrusions Do something with top and bottom surfaces !

Tailor the surface height before extrusion.

Tile from a Different Symmetry GroupTile from a Different Symmetry Group

Flat Extrusion of QuadfishFlat Extrusion of Quadfish

Modulating the Surface HeightModulating the Surface Height

Red part is viewed from the bottom

Manufactured Tiles (FDM)Manufactured Tiles (FDM)

Three tiles overlaid

Offset (Shifted) OverlayOffset (Shifted) Overlay

Let thick and thin areas complement each other:

RED = Thick areas; BLUE = THIN areas;

Shift Fish Outline to Desired PositionShift Fish Outline to Desired Position

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

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

As QuickSlice sees the shape …

Fabricated Tiles …Fabricated Tiles …

Top and bottom view Snug fit in the plane …

Adding Two More TilesAdding Two More Tiles

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

Snug fit also in the third dimension !

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.

M. Goerner’s TileM. Goerner’s Tile

Glue together elements from two subsequent layers.

Escher Night and DayEscher Night and Day

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

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

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

Bird Bird Fish Fish

A sweep-morph from bird into fish … and back

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

1 red + 1 yellow

isohedral tile

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

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.

BCC Cell: Editing ResultBCC Cell: Editing Result

A fish-like tile shape that tessellates 3D space

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.

Cell 2: Editing ResultCell 2: Editing Result

Can yield fish-like shapes

Need more editing capabilities to add details …

Adam Megacz’ Compound Cell EditorAdam Megacz’ Compound Cell Editor

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

Final Edited ShapeFinal Edited Shape

“Butterfly-Stingray” by Adam Megacz

Snug fit in the plane …

The Fabricated Tiles …The Fabricated Tiles …

and between the planes!

Lessons Learned:Lessons Learned:

To make such a 3D editing tool is hard.

To use it to make good 3D tile designsis 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 ?

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.

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

Two different isohedral tilings.

Towards 3D EscherizationTowards 3D Escherization

The basic cell, based on a rhombic dodecahedron

Each cell has 12 direct neighbors

The Goal ShapeThe Goal Shape

Designed in a separate CAD program

Simulated Annealing in ActionSimulated Annealing in Action

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

The Final ResultThe Final Result

made on a Fused Deposition Modeling Machine,

then hand painted.

More “Sim-Fish”More “Sim-Fish”

At different resolutions

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

In 3D you can interlink tiles topologically !

Genus 1: ToroidsGenus 1: Toroids

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

Toroidal Tiles,Toroidal Tiles,VariationsVariations

Based on cubic lattice

24 facets

12 F

16 F

Square Wire Frames in BCC LatticeSquare Wire Frames in BCC Lattice

Tiles are approx. Voronoi regions around wire loops

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

We can do the same with two other lattices !

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

Diamond LatticeDiamond Lattice

SLS model

by George Hart

Double (Interlinked) Diamond LatticeDouble (Interlinked) Diamond Lattice

computer model

by George Hart

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

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

““Triamond” LatticeTriamond” Lattice

computer model

by George Hart

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

computer model

by George Hart

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

SLS model

by George Hart

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

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

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.

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

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

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

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

12 rings total

HoneycombHoneycomb

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

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

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

“Front” and “Back”

Two more symmetrical views !

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 ?

3-Segment Ring ?3-Segment Ring ?

NO – that does not work !

3-Rings in Triamond Lattice3-Rings in Triamond Lattice

0°19.5°

Skewed Tria-TilesSkewed Tria-Tiles

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

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 ?

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.

Parameterized Tile DescriptionParameterized Tile Description

Allows aesthetic optimization of the tile shape

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

A single tile Two interlinked tiles

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

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

-------- Rotatate 45° -------

A

A

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.

Sliced Cubic 4-RingsSliced Cubic 4-Rings

Each ring interlinks with 8 others

Sliced Diamond 6-RingsSliced Diamond 6-Rings

Slicing the 10-Segment RingSlicing the 10-Segment Ring

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

“Front” view “Back” view

All possible color pairs are present !

Slicing the Tria-TileSlicing the Tria-Tile

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

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.

Simplest Genus-5 Cube FrameSimplest Genus-5 Cube Frame

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

Array of Interlocking Cube FramesArray of Interlocking Cube Frames

MetropolisMetropolis

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

Note: Every cube face has two wire squares along it

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!

Split Cube Cage for AssemblySplit Cube Cage for Assembly

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

Linear Array of Cube CagesLinear Array of Cube Cages

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

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

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

CagesCages and Frames in and Frames in Diamond LatticeDiamond Lattice

four 6-segment rings form a genus-3 cage

6-ring keychain …

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

Assembly of Diamond Lattice CagesAssembly of Diamond Lattice Cages

4-Ring Diamond 4-Ring Diamond FrameFrame

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

Diamond (Slice) Frame LatticeDiamond (Slice) Frame Lattice

With Complement Lattice InterspersedWith Complement Lattice Interspersed

With Actual FDM Parts …With Actual FDM Parts …

“Some assembly required … “

Assembly of Diamond Lattice FramesAssembly of Diamond Lattice Frames

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

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 !

Assembling Triamond CagesAssembling Triamond Cages

7 cages hook into the green central cage

Adding More Triamond CagesAdding More Triamond Cages

More green cages at the bottom.

Three blue cages on top.

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”

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

FDM parts designed for the assembly of complex clusters.

Assembling Triamond 3-Ring FramesAssembling Triamond 3-Ring Frames

7 frames hooked into white half-frame

Adding Upper Half of White FrameAdding Upper Half of White Frame

A total of 14 frames hook into each frame

Completed Cluster AssemblyCompleted Cluster Assembly

PART IV:PART IV: Knot Tiles Knot Tiles

Topological Arrangement of Knot-TilesTopological Arrangement of Knot-Tiles

Important Geometrical ConsiderationsImportant Geometrical Considerations Critical point:

prevent fusion into higher-genus object!

Collection of Nearest-Neighbor KnotsCollection of Nearest-Neighbor Knots

Finding Voronoi Zone for Wire KnotsFinding Voronoi Zone for Wire Knots

2 Solutions for different knot parameters

ConclusionsConclusions

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

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)

E X T R A SE X T R A S