bridges 2008, leeuwarden intricate isohedral tilings of 3d euclidean space intricate isohedral...
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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
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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
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My Fascination with Escher TilingsMy Fascination with Escher Tilings
on higher-genus surfaces:
London Bridges 2006
What next ?
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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 !
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A very large domainA very large domain
keep it somewhat limitedkeep it somewhat limited
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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.
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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
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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
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Genus 0:Genus 0: Simple Extrusions Simple Extrusions
Start from one of Escher’s 2D tilings …
Add 3rd dimension by extruding shape.
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Extruded “2.5D” Fish-TilesExtruded “2.5D” Fish-Tiles
Isohedral Fish-Tiles
Go beyond 2.5D !
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Modulated ExtrusionsModulated Extrusions Do something with top and bottom surfaces !
Tailor the surface height before extrusion.
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Tile from a Different Symmetry GroupTile from a Different Symmetry Group
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Flat Extrusion of QuadfishFlat Extrusion of Quadfish
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Modulating the Surface HeightModulating the Surface Height
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Red part is viewed from the bottom
Manufactured Tiles (FDM)Manufactured Tiles (FDM)
Three tiles overlaid
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Offset (Shifted) OverlayOffset (Shifted) Overlay
Let thick and thin areas complement each other:
RED = Thick areas; BLUE = THIN areas;
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Shift Fish Outline to Desired PositionShift Fish Outline to Desired Position
CAD tool calculates intersections with underlying height map of repeated fish tiles.
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3D Shape is Saved in .STL Format3D Shape is Saved in .STL Format
As QuickSlice sees the shape …
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Fabricated Tiles …Fabricated Tiles …
Top and bottom view Snug fit in the plane …
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Adding Two More TilesAdding Two More Tiles
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Adding Tiles in a 2Adding Tiles in a 2ndnd Layer Layer
Snug fit also in the third dimension !
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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.
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M. Goerner’s TileM. Goerner’s Tile
Glue together elements from two subsequent layers.
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Escher Night and DayEscher Night and Day
Inspiration: Escher’s wonderful shape transformations (more by C. Kaplan…)
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M.C. Escher: MetamorphosisM.C. Escher: Metamorphosis
Do similar “morph”-transformation in the 3rd dim.
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Bird Bird Fish Fish
A sweep-morph from bird into fish … and back
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““FishFishBird”-Tile Fills 3D SpaceBird”-Tile Fills 3D Space
1 red + 1 yellow
isohedral tile
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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
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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.
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BCC Cell: Editing ResultBCC Cell: Editing Result
A fish-like tile shape that tessellates 3D space
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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.
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Cell 2: Editing ResultCell 2: Editing Result
Can yield fish-like shapes
Need more editing capabilities to add details …
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Adam Megacz’ Compound Cell EditorAdam Megacz’ Compound Cell Editor
“Hammerhead” starting configurationCan select and drag individual vertices Corresponding vertices will follow !
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Final Edited ShapeFinal Edited Shape
“Butterfly-Stingray” by Adam Megacz
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Snug fit in the plane …
The Fabricated Tiles …The Fabricated Tiles …
and between the planes!
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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 ?
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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.
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““Escherization” ResultsEscherization” Resultsby Kaplan and Salesin, 2000by Kaplan and Salesin, 2000
Two different isohedral tilings.
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Towards 3D EscherizationTowards 3D Escherization
The basic cell, based on a rhombic dodecahedron
Each cell has 12 direct neighbors
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The Goal ShapeThe Goal Shape
Designed in a separate CAD program
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Simulated Annealing in ActionSimulated Annealing in Action
Basic cell and goal shape (wire frame) Subdivided and partially annealed 3D fish tile
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The Final ResultThe Final Result
made on a Fused Deposition Modeling Machine,
then hand painted.
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More “Sim-Fish”More “Sim-Fish”
At different resolutions
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Part II:Part II: Tiles of Genus > 0 Tiles of Genus > 0
In 3D you can interlink tiles topologically !
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Genus 1: ToroidsGenus 1: Toroids
An assembly of 4-segment rings,based on the BCC lattice (Séquin, 1995)
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Toroidal Tiles,Toroidal Tiles,VariationsVariations
Based on cubic lattice
24 facets
12 F
16 F
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Square Wire Frames in BCC LatticeSquare Wire Frames in BCC Lattice
Tiles are approx. Voronoi regions around wire loops
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Diamond Lattice & “Triamond” LatticeDiamond Lattice & “Triamond” Lattice
We can do the same with two other lattices !
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Diamond Lattice Diamond Lattice (8 cells shown)(8 cells shown)
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Diamond LatticeDiamond Lattice
SLS model
by George Hart
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Double (Interlinked) Diamond LatticeDouble (Interlinked) Diamond Lattice
computer model
by George Hart
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Triamond Lattice Triamond Lattice (8 cells shown)(8 cells shown)
aka “(10,3)-Lattice”, A. F. Wells “3D Nets & Polyhedra” 1977
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““Triamond” LatticeTriamond” Lattice
computer model
by George Hart
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Double (interlinked) “Triamond” LatticeDouble (interlinked) “Triamond” Lattice
computer model
by George Hart
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Double (interlinked) “Triamond” LatticeDouble (interlinked) “Triamond” Lattice
SLS model
by George Hart
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““Triamond” LatticeTriamond” Lattice Thanks to John Conway and Chaim Goodman Strauss
‘Knotting Art and Math’ Tampa, FL, Nov. 2007Visit to Charles Perry’s “Solstice”
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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.
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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
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Diamond Lattice: Diamond Lattice: 6-Segment Rings 6-Segment Rings
6 rings interlink with each “key ring” (grey)
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Cluster of 2 Interlinked Key-RingsCluster of 2 Interlinked Key-Rings
12 rings total
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HoneycombHoneycomb
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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
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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
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Key-Ring with Ten 10-segment RingsKey-Ring with Ten 10-segment Rings
“Front” and “Back”
Two more symmetrical views !
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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 ?
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3-Segment Ring ?3-Segment Ring ?
NO – that does not work !
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3-Rings in Triamond Lattice3-Rings in Triamond Lattice
0°19.5°
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Skewed Tria-TilesSkewed Tria-Tiles
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Closed Chain of 10 Tria-TilesClosed Chain of 10 Tria-Tiles
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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 ?
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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.
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Parameterized Tile DescriptionParameterized Tile Description
Allows aesthetic optimization of the tile shape
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““Optimized” Skewed Tria-TilesOptimized” Skewed Tria-Tiles Got rid of the pointy protrusions !
A single tile Two interlinked tiles
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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
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Larger Assembly of Optimized Tria-TilesLarger Assembly of Optimized Tria-Tiles
-------- Rotatate 45° -------
A
A
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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.
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Sliced Cubic 4-RingsSliced Cubic 4-Rings
Each ring interlinks with 8 others
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Sliced Diamond 6-RingsSliced Diamond 6-Rings
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Slicing the 10-Segment RingSlicing the 10-Segment Ring
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Key-Ring with Twenty Sliced 10-RingsKey-Ring with Twenty Sliced 10-Rings
“Front” view “Back” view
All possible color pairs are present !
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Slicing the Tria-TileSlicing the Tria-Tile
6 sliced Tria-Tiles hook into the white key-ring
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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.
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Simplest Genus-5 Cube FrameSimplest Genus-5 Cube Frame
“Frame” built from six sliced 4-segment-rings
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Array of Interlocking Cube FramesArray of Interlocking Cube Frames
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MetropolisMetropolis
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Linking Topology of “Metropolis”Linking Topology of “Metropolis”
Note: Every cube face has two wire squares along it
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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!
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Split Cube Cage for AssemblySplit Cube Cage for Assembly
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Tetra-Cluster Built from 5 Cube Cages Tetra-Cluster Built from 5 Cube Cages
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Linear Array of Cube CagesLinear Array of Cube Cages
An interlinking chain along the space diagonalTHIS DOES NOT TILE 3D SPACE !
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Analogous Mis-Assembly in 2DAnalogous Mis-Assembly in 2D
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Linking Topology of Cube-Cage LatticeLinking Topology of Cube-Cage Lattice
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CagesCages and Frames in and Frames in Diamond LatticeDiamond Lattice
four 6-segment rings form a genus-3 cage
6-ring keychain …
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Genus-3 Cage made from Four 6-RingsGenus-3 Cage made from Four 6-Rings
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Assembly of Diamond Lattice CagesAssembly of Diamond Lattice Cages
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4-Ring Diamond 4-Ring Diamond FrameFrame
Four sliced 6-segment ringsTogether they form a genus-3 frame
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Diamond (Slice) Frame LatticeDiamond (Slice) Frame Lattice
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With Complement Lattice InterspersedWith Complement Lattice Interspersed
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With Actual FDM Parts …With Actual FDM Parts …
“Some assembly required … “
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Assembly of Diamond Lattice FramesAssembly of Diamond Lattice Frames
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Three 10-rings Yield a Three 10-rings Yield a Triamond CageTriamond Cage
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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 !
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Assembling Triamond CagesAssembling Triamond Cages
7 cages hook into the green central cage
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Adding More Triamond CagesAdding More Triamond Cages
More green cages at the bottom.
Three blue cages on top.
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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”
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Split 3-Ring Triamond Frame (FDM)Split 3-Ring Triamond Frame (FDM)
FDM parts designed for the assembly of complex clusters.
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Assembling Triamond 3-Ring FramesAssembling Triamond 3-Ring Frames
7 frames hooked into white half-frame
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Adding Upper Half of White FrameAdding Upper Half of White Frame
A total of 14 frames hook into each frame
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Completed Cluster AssemblyCompleted Cluster Assembly
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PART IV:PART IV: Knot Tiles Knot Tiles
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Topological Arrangement of Knot-TilesTopological Arrangement of Knot-Tiles
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Important Geometrical ConsiderationsImportant Geometrical Considerations Critical point:
prevent fusion into higher-genus object!
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Collection of Nearest-Neighbor KnotsCollection of Nearest-Neighbor Knots
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Finding Voronoi Zone for Wire KnotsFinding Voronoi Zone for Wire Knots
2 Solutions for different knot parameters
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ConclusionsConclusions
Many new and intriguing tiles …Many new and intriguing tiles …
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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)
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E X T R A SE X T R A S