Leonardo Meeting, San Francisco, May 12, 2008 Naughty Knotty Sculptures Carlo H. Séquin EECS,...
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Transcript of Leonardo Meeting, San Francisco, May 12, 2008 Naughty Knotty Sculptures Carlo H. Séquin EECS,...
![Page 1: Leonardo Meeting, San Francisco, May 12, 2008 Naughty Knotty Sculptures Carlo H. Séquin EECS, Computer Science University of CA, Berkeley.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d805503460f94a63a6b/html5/thumbnails/1.jpg)
Leonardo Meeting, San Francisco, May 12, 2008Leonardo Meeting, San Francisco, May 12, 2008
NaughtyKnotty Sculptures
Carlo H. Séquin
EECS, Computer Science
University of CA, Berkeley
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Technical Designs …Technical Designs …
CCD Camera, Bell Labs, 1973 Soda Hall, Berkeley, 1994
RISC chip, Berkeley, 1981 “Octa-Gear”, Berkeley, 2000
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Since 1994: Aesthetic Designs …Since 1994: Aesthetic Designs …
What is the role of the computer in:
aesthetic optimization,
the creative process ?
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Collaboration with Brent CollinsCollaboration with Brent Collins
“Hyperbolic Hexagon II”
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““Sculpture Generator I Sculpture Generator I ” GUI ” GUI
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Math-Art ConnectionMath-Art Connection
When does a mathematical model
become a piece of art ?
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Rapid Prototyping Model of the 24-CellRapid Prototyping Model of the 24-Cell
Noticethe 3-fold
permutationof colors
Made on the Z-corp machine.
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Hamiltonian Cycles on 4D Cross Polytope
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Sculptures Made from KnotsSculptures Made from Knots
2004 - 2007:Knots as constructive building blocks.
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Tetrahedral Trefoil Tangle Tetrahedral Trefoil Tangle (FDM)(FDM)
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Tetra Trefoil TanglesTetra Trefoil Tangles
Simple linking (1) -- Complex linking (2)
{over-over-under-under} {over-under-over-under}
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Tetra Trefoil TangleTetra Trefoil Tangle
Complex linking (two views)
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Platonic Trefoil TanglesPlatonic Trefoil Tangles
Take a Platonic polyhedron made from triangles,
Add a trefoil knot on every face,
Link with neighboring knots across shared edges.
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Icosahedral Trefoil TangleIcosahedral Trefoil Tangle
Simplest linking (type 1)
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Icosahedral Icosahedral Trefoil Trefoil TangleTangle(type 3)(type 3)
Doubly linked with each neighbor
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Arabic IcosahedronArabic Icosahedron
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Dodecahedral Pentafoil ClusterDodecahedral Pentafoil Cluster
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Realization: Extrude Hone - ProMetalRealization: Extrude Hone - ProMetal
Metal sintering and infiltration process
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Sculptures Made from KnotsSculptures Made from Knots
Generate knots & increase their complexity in a structured, procedural way
Make aesthetically pleasing artifacts
More recently, I have been looking for sculptureswhere the whole piece is just a single knot.
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Many Different Ways . . .Many Different Ways . . .
I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting
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The 2D Hilbert Curve (1891)The 2D Hilbert Curve (1891)
A plane-filling Peano curve
Do This In 3 D !
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““Hilbert” Curve in 3DHilbert” Curve in 3D
Start with Hamiltonian path on cube edges and recurse ...
Replaces an “elbow”
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Jane Yen: “Jane Yen: “Hilbert Radiator PipeHilbert Radiator Pipe” ” (2000)(2000)
Flaws( from a sculptor’s . point of view ):
4 coplanar segments
Not a closed loop
Broken symmetry
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Metal Sculpture at SIGGRAPH 2006Metal Sculpture at SIGGRAPH 2006
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A Knot Theorist’s ViewA Knot Theorist’s View
It is still just the un-knot !
Thus our construction element should use a “more knotted thing”:
e.g. an overhand knot:
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Recursion StepRecursion Step
Replace every 90° turn with a knotted elbow.
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Also: Start from a True KnotAlso: Start from a True Knot
e.g., a “cubist” trefoil knot.
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Recursive Cubist Trefoil KnotRecursive Cubist Trefoil Knot
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A Knot Theorist’s ViewA Knot Theorist’s View
This is just a compound-knot !
It does not really lead to a complex knot !
Thus our assembly step should cause a more serious entanglement:
adjacent knots should entangle one another, or
crossing strands should be knotted together . . .
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2.5D Celtic Knots – Basic Step2.5D Celtic Knots – Basic Step
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Celtic Knot – Denser ConfigurationCeltic Knot – Denser Configuration
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Celtic Knot – Second IterationCeltic Knot – Second Iteration
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Recursive 9-Crossing KnotRecursive 9-Crossing Knot
Is this really a 81-crossing knot ?
9 crossings
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OutlineOutline
I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting
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Knot-FusionKnot-Fusion
Combine 3 trefoils into a 9-crossing knot
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33rdrd Generation of Trefoil-Sierpinsky Generation of Trefoil-Sierpinsky
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From Paintings to SculpturesFrom Paintings to Sculptures
Do something like this in 3D !
Perhaps using two knotted strands(like your shoe laces).
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INTERMEZZO:INTERMEZZO:
Homage toHomage toFrank Smullin (1943 – 1983)Frank Smullin (1943 – 1983)
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Frank Smullin (1943 – 1983) Frank Smullin (1943 – 1983)
Tubular sculptures;
Apple II program for
calculating intersections.
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Frank Smullin (Nashville, 1981):Frank Smullin (Nashville, 1981):
“ The Granny-knot has more artistic merits than the square knot because it is more 3D;its ends stick out in tetrahedral fashion... ”
Square Knot Granny Knot
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Granny Knot as a Building BlockGranny Knot as a Building Block
Four tetrahedral links, like a carbon atom ...
can be assembled into diamond-lattice ...
... leads to the “Granny-Knot-Lattice”
Smullin: “TetraGranny”
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Strands in the Granny-Knot-LatticeStrands in the Granny-Knot-Lattice
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Granny-Knot-Lattice (SGranny-Knot-Lattice (Séquin, 1981)quin, 1981)
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A “Knotty” “3D” Recursion StepA “Knotty” “3D” Recursion Step
Use the Granny knot as a replacement element where two strands cross ...
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Next Recursion StepNext Recursion Step
Substitute the 8 crossings with 8 Granny-knots
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One More Recursion StepOne More Recursion Step
Now use eight of these composite elements;
connect;
beautify. Too much
com
plexity
!
Too much
com
plexity
!
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A Nice Symmetrical Starting KnotA Nice Symmetrical Starting Knot
Granny Knot with cross-connected ends
4-fold symmetric Knot 819
(3,4) Torus Knot
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Recursion StepRecursion Step
Placement of the 8 substitution knots
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Establishing ConnectivityEstablishing Connectivity
Grow knots until they almost touch
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Work in Progress ...Work in Progress ...
Connectors added to close the knot
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OutlineOutline
I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting
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Recursive Figure-8 Knot Recursive Figure-8 Knot (4 crossings)(4 crossings)
Recursion stepMark crossings over/under to form alternating knot
Result after 2 more recursion steps
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Recursive Figure-8 KnotRecursive Figure-8 Knot
Scale the stroke-width proportional to recursive reduction
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2.5D Recursive (Fractal) Knot2.5D Recursive (Fractal) Knot
Robert Fathauer: “Recursive Trefoil Knot”
Trefoil Recursion3 views step
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Recursion on a 7-crossing KnotRecursion on a 7-crossing Knot
Robert Fathauer, Bridges Conference, 2007
...
Map “the whole thing” into all meshes of similar shape
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From 2D Drawings to 3D SculptureFrom 2D Drawings to 3D Sculpture
Too flat ! Switch plane orientations
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Recursive Figure-8 Knot 3DRecursive Figure-8 Knot 3D
Maquette emerging from FDM machine
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Recursive Recursive Figure-8 KnotFigure-8 Knot
9 loop iterations
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OutlineOutline
I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting
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Split Trefoil (side view, closed)Split Trefoil (side view, closed)
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Split Trefoil (side view, open)Split Trefoil (side view, open)
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““Knot DividedKnot Divided” by Team Minnesota” by Team Minnesota
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Knotty ProblemKnotty Problem
How many crossings
does this Not-Divided Knot have ?
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Is It Math ?Is It Math ?Is It Art ?Is It Art ?
it is:
“KNOT-ART”