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BID Lunch, February 25, 2014 “LEGO ® ” Knots EECS Computer Science Division University of...
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Transcript of BID Lunch, February 25, 2014 “LEGO ® ” Knots EECS Computer Science Division University of...
![Page 1: BID Lunch, February 25, 2014 “LEGO ® ” Knots EECS Computer Science Division University of California, Berkeley Carlo H. Séquin Michelle Galemmo.](https://reader035.fdocuments.net/reader035/viewer/2022062500/56649efe5503460f94c123e7/html5/thumbnails/1.jpg)
BID Lunch, February 25, 2014BID Lunch, February 25, 2014
“LEGO®” Knots
EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley
Carlo H. SéquinMichelle Galemmo
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The Bridges ConferenceThe Bridges Conference
Mathematical Connections in Art, Music, and Science.
Annually held in July/August since 1998 (4-5 days).
Attracts: mathematicians, scientists, artists, educators, musicians, writers, dancers, weavers, model builders, and computer scientists.
Conference Venues: Formal paper presentations: regular, short, invited/plenary.
Also: workshops, art gallery, and informal show-and-tell/sell.
Evenings: concerts, theater, movies (with a math connection).
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Bridges in Many Wonderful PlacesBridges in Many Wonderful Places
Cities: Winfield, Towson, London, Coimbra, Granada, Donostia, Leeuwarden, Enschede, Pecs, …
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My Favorite Annual Conference: 2014My Favorite Annual Conference: 2014
BRIDGES Art …
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My First 20 Bridges PapersMy First 20 Bridges Papers
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Inspiration: Henk van PuttenInspiration: Henk van Putten
“Borsalino” “Interaction”
Sculptural forms put together from a few modular shapes
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Geometry of the Geometry of the BorsalinoBorsalino
Just 2 geometrical components: 3 semi-circular end-caps (orange) 6 curved connectors, bending through 45º== a square cross section swept along 9 circular arcs.
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The Wonders of Rapid-PrototypingThe Wonders of Rapid-Prototyping
Two modular components can form the Borsalino
C R=2.4142
E R=1.0
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Playing with those Two ComponentsPlaying with those Two Components
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The The ““LooseLoose”” Borsalino Borsalino
Two larger types of parts required: 3 end-caps; 6 connectors.
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Rapid Prototyping with FDMRapid Prototyping with FDM
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A Look Into the FDM MachineA Look Into the FDM Machine
Galapagos 6 sculpture in progress
2 NOZZLES
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Inexpensive FDM MachineInexpensive FDM Machine
Afinia-H-Series 3D printer
Has only ONE type of filament.
Support structures: not desirable: are hard to remove,
leave part surface scarred.
Limited cantilevering is possibleto form overhangs and bridges:( 45º max).
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SUPPORT
Fabrication Issues:Fabrication Issues:Optimize Build OrientationOptimize Build Orientation
Horizontal tube (flat):filled with support
Tube on edge:may tip over,gets squashed
Vertical tube:support at flanges
Can we orient part so no supports are needed? (All surfaces must be steeper than 45º)
SUPPORT
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Square flanges Tapered flanges Upside-down
support no support no support
Fabrication Issues: Fabrication Issues: FlangesFlanges
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Fabrication Issues: Fabrication Issues: Curved ConnectorsCurved Connectors
Ignore Problem Built-in Wall Extended Taper
Chosen Solution!
No face steeper than 45º !
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Fabrication Issues: Fabrication Issues: End-CapsEnd-Caps
Form a 45º slanted “cathedral ceiling” internally:
needs support needs no support
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Needs some support for central arch !
We wanted to keep inner tube open: Use scaffolding and take the trouble of doing some clean-up!
Fabrication Issues: Fabrication Issues:
Enlarged End-CapsEnlarged End-Caps
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Parts Catalog So FarParts Catalog So Far
2 types of end-caps; 3 curved connectors
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What can we do with those 5 Parts?What can we do with those 5 Parts?
Wild and crazy “snakes”, e.g., a Hilbert curve;
Emulation of other Henk van Putten sculptures …
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EmulationsEmulations
Interaction
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Many More Possibilities …Many More Possibilities …
Mix and match …
D
CB
A
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2 End-Caps plus 3 Connector Types ...2 End-Caps plus 3 Connector Types ...
allow us to build these allow us to build these Twisted 2-Lobe BorsalinosTwisted 2-Lobe Borsalinos
The mathematics does not really work out quite right;they are off by 6%, 15%, and 6% in the radius ratios.
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Flipped-end BorsalinoFlipped-end Borsalino
Put extension between connectors
movebackward
stretched
move up,forward
needsdiagonalend-cap
moveforward
movebackward
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Flipped-end BorsalinoFlipped-end Borsalino
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The New Rhombic End-cap …The New Rhombic End-cap …… yields new possibilities:
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Rhombic BorsalinosRhombic Borsalinos We just need to make a
new connector part: bending again through 45º, but in diagonal direction!
This Borsalino now has a loose enough geometry,so that there is roomto introduce twisted legs:
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Twisted Connector PiecesTwisted Connector Pieces
Enlarged connector pieces with 45º twist:
results in two different pieces (azimuth!)
4 pairs make a nice twisted ring (360º )
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Twisted BorsalinosTwisted Borsalinos
1 twisted branch; 3 twisted branches.
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Triply Twisted Rhombic BorsalinoTriply Twisted Rhombic Borsalino
Sculpture!
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Inspiration: Paul BlochInspiration: Paul Bloch
“After Wright” (Guggenheim, NYC)
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Helical PiecesHelical Pieces
Another useful component!
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Spiral SculpturesSpiral Sculptures
Using the helical pieces
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A Look Behind the SceneA Look Behind the Scene
It does not really close smoothly!
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Inspiration: Paul BlochInspiration: Paul Bloch
“After Wright” (Guggenheim, NYC)
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““CoccoonCoccoon””
Return path through the center of the helix.
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Inspiration: Bruce BeasleyInspiration: Bruce Beasley
Autodesk exhibition, December 2013
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Inspiration: Jon KrawczykInspiration: Jon Krawczyk
303 2nd Street, San Francisco
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““Pas de DeuxPas de Deux””
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Real LEGOReal LEGO®® Knots ? Knots ?
Beginning of the table of knots …
This is all you have seen so far. It is not really a knot!
Unknot
Trefoil
Figure-8
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Real Knots: Trefoil (3_1)Real Knots: Trefoil (3_1)
One new piece (magenta) for smooth closure
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Trefoil KnotTrefoil Knot
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Real Knots: Figure-8 Knot (4_1)Real Knots: Figure-8 Knot (4_1)
Two new pieces (magenta, red) for smooth closure
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Figure-8 KnotFigure-8 Knot
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LEGOLEGO®® DUPLO DUPLO
Match interface
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LEGOLEGO®® DUPLO DUPLO
Just playing around . . .
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LEGOLEGO®® DUPLO DUPLO
Borromean Link Hopf Link
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What Is the Right Interface ?What Is the Right Interface ?
Open tubes
or
LEGO nibs ?
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Making Sculptures Glow …Making Sculptures Glow …
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Making Sculptures Glow …Making Sculptures Glow …
testing
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Making Sculptures Glow …Making Sculptures Glow …
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Hands-on SculptingHands-on Sculpting
Xmas break 2013
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Branching Out ?Branching Out ?
Junction pieces arbitrary graphs
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““OrganicOrganic”” Looking Objects Looking Objects
Trees or corrals ?
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Making Graph StructuresMaking Graph Structures
Tetrahedral graph:
4 valence-3 vertices
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Next Attempt: More compact, but . . .Next Attempt: More compact, but . . .
This is not the tetrahedral graph!
Not this:
But this:
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Tetra GraphTetra Graph
OK tetrahedral graph.
But needed one extra custom-made piece!
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Other Attempts ?Other Attempts ?
What about the cubical edge graph ?
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Cubical Edge Graph: 2Cubical Edge Graph: 2ndnd Attempt Attempt New stub placement in square frame:
Place 2 Y-component close together, pointing in opposite directions.
Put two such frames on top of one another and connect pairs of stubs with the rhombic end-caps.
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Cubical Edge Graph -- SolutionCubical Edge Graph -- Solution
Stack the two frames in an offset manner,so that the connecting arcs run at a 45º angle.
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Cubical Edge Graph as a SculptureCubical Edge Graph as a Sculpture
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Max Bill Sculpture, and beyond . . .Max Bill Sculpture, and beyond . . .
What is needed to emulate this?
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But there are other ways But there are other ways
of of ““branching outbranching out”” . . . . . .
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BID Lunch, February 25, 2014BID Lunch, February 25, 2014
Tria-Tubes
EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley
Michelle GalemmoCarlo H. Séquin
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A New ProfileA New Profile
All sculptures shown were based on a square cross section:
What will happen if we try a different profile,e.g. a triangular one ?
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Designing a Designing a ““Tri-BorsalinoTri-Borsalino”” This is the tightest turn without self-intersections of
a circular tube containing a triangular cross section in an arbitrary azimuth (angle*) orientation.
Two different alignments result in two different end-caps:
*
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Dimensioning the TRIA-TUBESDimensioning the TRIA-TUBES
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Two Triangular End-CapsTwo Triangular End-Caps
Type #1 Type #2
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Two Two ““Tri-BorsalinosTri-Borsalinos”” We keep the 3D space curve of the classical Borsalino
as the sweep path for our new triangular cross section.
The two end-caps produce two different Tri-Borsalinos.
The connector pieces bending through 45º need to have a twist of 15º to make smooth connections.
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Our Initial Parts ListOur Initial Parts List 2 types of end-caps:
Curved connectors: for type #1 for type #2
Connector sleeve
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Some Possible Assemblies:Some Possible Assemblies:Using just End-CapsUsing just End-Caps
two hexagonal rings and a triangular loop
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Some Possible Assemblies:Some Possible Assemblies:from only curved connector piecesfrom only curved connector pieces
A variety of twisted Möbius rings
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What Next ?What Next ?
More twisted connectors?
Twisted end-caps?
Straight extension pieces for flip-over Borsalino?
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Tri-Borsalino DerivativesTri-Borsalino Derivatives
r = 0.57735
Extend the link (by √8) between the connectors.
R(endcap) = sqrt(4/3) = 2r = 1.1547 = cos(30)*4/3
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ConclusionsConclusions
LEGO®-Knots plus Tria-Tubes:
An ever expanding modular system to do hands-on geometrical sculpture for people who do not want to do math or touch a computer.
The result is a mixture of:-- creative design decisions and-- practical fabrication considerations.
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