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

Bridges 2008, LeeuwardenBridges 2008, Leeuwarden

Intricate 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 Very Large Domain !A Very Large Domain !

A very large domain

keep it somewhat limited

Monohedral vs. IsohedralMonohedral 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! 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 TilingHow 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 !

Shape height of surface 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

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 …

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 elements of the two layers together.

Movie on YouTube ?Movie on YouTube ?

Escher Night and DayEscher Night and Day

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

Escher MetamorphosisEscher Metamorphosis

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

Bird into fish … and back

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

1 red + 1 yellow

isohedral tile

True 3D TilesTrue 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

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.

Cell 1: Editing ResultCell 1: 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

Fish-like shapes …

Need more diting capabilities to add details …

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 – and the goal shape

Simulated Annealing in ActionSimulated Annealing in Action

Basic cell and goal shape (wire frame) Subdivided and partially annealed 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 wires

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

We can do the same with 2 other lattices !

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

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

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

““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 (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

Diamond Lattice: Ring ConstructionDiamond Lattice: Ring Construction

Two complementary diamond lattices,

And two representative 6-segment rings

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

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”

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

Loop of 10 Tria-Tiles (FDM)Loop 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 distacesfrom 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 go together !

Isohedral Toroidal TilesIsohedral 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.

Split Cubic 4-RingsSplit Cubic 4-Rings

Each ring interlinks with 8 others

Split Diamond 6-RingsSplit Diamond 6-Rings

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

“Front” view “Back” view

All possible color pairs are present !

Split Triamond 3-RingSplit Triamond 3-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 in this talk.

Simplest Genus-5 Cube FrameSimplest Genus-5 Cube Frame

“Frame” built from six split 4-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-split rings.

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

Assembling Split 6-RingsAssembling Split 6-Rings

4 RINGS Forming a “diamond-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 … “

Three 10-rings Make a Triamond CageThree 10-rings Make a Triamond Cage

Cages in the Triamond LatticeCages in the Triamond Lattice

Two genus-3 cages == compound of three 10-rings

They come in two different chiralities !

Genus-3 Cage InterlinkedGenus-3 Cage Interlinked

Split 10-Ring FrameSplit 10-Ring Frame

Some assembly with these partsSome assembly with these parts

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)

Roman Fuchs (Voronoi cells)

John Sullivan (manuscript)

E X T R A SE X T R A S

What Linking Numbers are Possible?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 …

Linking Number =1Linking Number =1

Cube with one handle that interlocks with one neighbor

Linking Number =2Linking Number =2

Long chains of interlinked rings,packed densely side by side.

Linking Number =5Linking 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 …

Linking Number =3Linking Number =3 But derived from Diamond lattice by taking

only every other ring…

the unit cell:

An Array of such CellsAn Array of such Cells

Has the connectivity of the Triamond Lattice !

Array of Five Rings Interlinked ??Array of Five Rings Interlinked ??

Does not seem to lead to an isohedral tiling