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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
Technical Designs …Technical Designs …
CCD Camera, Bell Labs, 1973 Soda Hall, Berkeley, 1994
RISC chip, Berkeley, 1981 “Octa-Gear”, Berkeley, 2000
Since 1994: Aesthetic Designs …Since 1994: Aesthetic Designs …
What is the role of the computer in:
aesthetic optimization,
the creative process ?
Collaboration with Brent CollinsCollaboration with Brent Collins
“Hyperbolic Hexagon II”
““Sculpture Generator I Sculpture Generator I ” GUI ” GUI
Math-Art ConnectionMath-Art Connection
When does a mathematical model
become a piece of art ?
Rapid Prototyping Model of the 24-CellRapid Prototyping Model of the 24-Cell
Noticethe 3-fold
permutationof colors
Made on the Z-corp machine.
Hamiltonian Cycles on 4D Cross Polytope
Sculptures Made from KnotsSculptures Made from Knots
2004 - 2007:Knots as constructive building blocks.
Tetrahedral Trefoil Tangle Tetrahedral Trefoil Tangle (FDM)(FDM)
Tetra Trefoil TanglesTetra Trefoil Tangles
Simple linking (1) -- Complex linking (2)
{over-over-under-under} {over-under-over-under}
Tetra Trefoil TangleTetra Trefoil Tangle
Complex linking (two views)
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.
Icosahedral Trefoil TangleIcosahedral Trefoil Tangle
Simplest linking (type 1)
Icosahedral Icosahedral Trefoil Trefoil TangleTangle(type 3)(type 3)
Doubly linked with each neighbor
Arabic IcosahedronArabic Icosahedron
Dodecahedral Pentafoil ClusterDodecahedral Pentafoil Cluster
Realization: Extrude Hone - ProMetalRealization: Extrude Hone - ProMetal
Metal sintering and infiltration process
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.
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
The 2D Hilbert Curve (1891)The 2D Hilbert Curve (1891)
A plane-filling Peano curve
Do This In 3 D !
““Hilbert” Curve in 3DHilbert” Curve in 3D
Start with Hamiltonian path on cube edges and recurse ...
Replaces an “elbow”
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
Metal Sculpture at SIGGRAPH 2006Metal Sculpture at SIGGRAPH 2006
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:
Recursion StepRecursion Step
Replace every 90° turn with a knotted elbow.
Also: Start from a True KnotAlso: Start from a True Knot
e.g., a “cubist” trefoil knot.
Recursive Cubist Trefoil KnotRecursive Cubist Trefoil Knot
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 . . .
2.5D Celtic Knots – Basic Step2.5D Celtic Knots – Basic Step
Celtic Knot – Denser ConfigurationCeltic Knot – Denser Configuration
Celtic Knot – Second IterationCeltic Knot – Second Iteration
Recursive 9-Crossing KnotRecursive 9-Crossing Knot
Is this really a 81-crossing knot ?
9 crossings
OutlineOutline
I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting
Knot-FusionKnot-Fusion
Combine 3 trefoils into a 9-crossing knot
33rdrd Generation of Trefoil-Sierpinsky Generation of Trefoil-Sierpinsky
From Paintings to SculpturesFrom Paintings to Sculptures
Do something like this in 3D !
Perhaps using two knotted strands(like your shoe laces).
INTERMEZZO:INTERMEZZO:
Homage toHomage toFrank Smullin (1943 – 1983)Frank Smullin (1943 – 1983)
Frank Smullin (1943 – 1983) Frank Smullin (1943 – 1983)
Tubular sculptures;
Apple II program for
calculating intersections.
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
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”
Strands in the Granny-Knot-LatticeStrands in the Granny-Knot-Lattice
Granny-Knot-Lattice (SGranny-Knot-Lattice (Séquin, 1981)quin, 1981)
A “Knotty” “3D” Recursion StepA “Knotty” “3D” Recursion Step
Use the Granny knot as a replacement element where two strands cross ...
Next Recursion StepNext Recursion Step
Substitute the 8 crossings with 8 Granny-knots
One More Recursion StepOne More Recursion Step
Now use eight of these composite elements;
connect;
beautify. Too much
com
plexity
!
Too much
com
plexity
!
A Nice Symmetrical Starting KnotA Nice Symmetrical Starting Knot
Granny Knot with cross-connected ends
4-fold symmetric Knot 819
(3,4) Torus Knot
Recursion StepRecursion Step
Placement of the 8 substitution knots
Establishing ConnectivityEstablishing Connectivity
Grow knots until they almost touch
Work in Progress ...Work in Progress ...
Connectors added to close the knot
OutlineOutline
I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting
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
Recursive Figure-8 KnotRecursive Figure-8 Knot
Scale the stroke-width proportional to recursive reduction
2.5D Recursive (Fractal) Knot2.5D Recursive (Fractal) Knot
Robert Fathauer: “Recursive Trefoil Knot”
Trefoil Recursion3 views step
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
From 2D Drawings to 3D SculptureFrom 2D Drawings to 3D Sculpture
Too flat ! Switch plane orientations
Recursive Figure-8 Knot 3DRecursive Figure-8 Knot 3D
Maquette emerging from FDM machine
Recursive Recursive Figure-8 KnotFigure-8 Knot
9 loop iterations
OutlineOutline
I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting
Split Trefoil (side view, closed)Split Trefoil (side view, closed)
Split Trefoil (side view, open)Split Trefoil (side view, open)
““Knot DividedKnot Divided” by Team Minnesota” by Team Minnesota
Knotty ProblemKnotty Problem
How many crossings
does this Not-Divided Knot have ?
Is It Math ?Is It Math ?Is It Art ?Is It Art ?
it is:
“KNOT-ART”
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