Bridges 2008, Leeuwarden of 3D Euclidean Space Intricate Isohedral Tilings of 3D Euclidean Space Carlo H. Séquin EECS Computer Science Division University.
Post on 29-Mar-2015
216 Views
Preview:
Transcript
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
top related