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SLS/CS Seminar, USF, Nov.4, 2008SLS/CS Seminar, USF, Nov.4, 2008
NaughtyKnotty Sculptures
Carlo H. Séquin
U.C. Berkeley
Knotty problems in knot theory
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NOT This:NOT This:
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But This: But This: Sculptures Made from Knots Sculptures Made from Knots
Knots as constructive sculptural building blocks.
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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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3 Hamiltonian Cycles on 4D Cross Polytope
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Hamiltonian Cycles on 4D Cross Polytope
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PART APART A
Knots as Constructive Building BlocksKnots 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;explore several different methods… -->
Make aesthetically pleasing artifacts!
More recently, I have been looking for sculptureswhere the whole piece is just a single knot.
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PART BPART BWays to Make Complicated KnotsWays to Make Complicated Knots
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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Sierpinski Trefoil KnotSierpinski Trefoil Knot
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Close-up of Sierpinski Trefoil KnotClose-up of Sierpinski Trefoil Knot
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33rdrd Generation of Sierpinski Knot Generation of Sierpinski Knot
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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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A Split TrefoilA Split Trefoil
To open: Rotate one half around z-axis
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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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Splitting Moebius BandsSplitting Moebius Bands
Litho by FDM-model FDM-modelM.C.Escher thin, colored thick
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Split Moebius Trefoil (SSplit Moebius Trefoil (Sééquin, 2003)quin, 2003)
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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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A More General QuestionA More General Question
Take any knot made from an n-sided prismatic cord.
Split that cord lengthwise into n strands.
Cut the bundle of strands at one point and reconnect,after giving the bundle of n strands a twistequivalent of t strand-spacings (where n, t are mutually prime).
How complex is the resulting knot ?
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PART CPART C
Space-filling KnotsSpace-filling Knots
Can we pack knots so tightly
that they fill all of 3D space ?
Ian Stewart, Mathematical Recreations,
Scientific American, Nov. 1995
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4 (convoluted) Trefoils Make a Cube4 (convoluted) Trefoils Make a Cube
Cubes stack up to fill space
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A Simpler, More Elegant SolutionA Simpler, More Elegant Solution
Three congruent
interlocking trefoils
make a hexagonal
prismatic block.
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Isohedral Trefoil-Knot Tile of 3D SpaceIsohedral Trefoil-Knot Tile of 3D Space
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What We Would Really Like ...What We Would Really Like ...
Stacking cubes or prisms in 3D space …
is a “cheap” way to fill space with knots!
Neighboring knots should mutually link,
so that the “fabric of space holds together.”
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Toroidal Tile (Linking Unknots)Toroidal Tile (Linking Unknots)
Assembly of 5 Tiles
The Basic Tile
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Extensible Linkage of Toroidal TilesExtensible Linkage of Toroidal Tiles
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The Use of Figure-8 KnotsThe Use of Figure-8 Knots
Figure-8 knot can also have 4 lobes
sticking out in tetrahedral directions.
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Figure-8 Knots in Diamond Lattice CellFigure-8 Knots in Diamond Lattice Cell
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A Denser Lattice of Figure-8 KnotsA Denser Lattice of Figure-8 Knots
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Dense Figure-8 Knot LatticeDense Figure-8 Knot Lattice
Model made with selective laser sintering.
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My Conceptual 3D-CAD ToolsMy Conceptual 3D-CAD Tools
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ConclusionsConclusions
Knots are mathematically intriguing and they are also inspiring artistic elements.
They can be used as building blocks for sophisticated aesthetic assemblies.
They can be extended recursively to form much more complicated knots.
They can be split lengthwise to make interesting knots and tangles.
They can be used to tile 3D space.
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Is It Math ?Is It Math ?Is It Art ?Is It Art ?
it is:
“KNOT-ART”
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Figure-8 KnotFigure-8 KnotBronze, Dec. 2007Bronze, Dec. 2007
Carlo SCarlo Sééquinquin
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ChineseChineseButton KnotButton Knot
(Knot 9(Knot 94040))
Bronze, Dec. 2007Bronze, Dec. 2007
Carlo SCarlo Sééquinquin
cast & patina bycast & patina bySteve ReinmuthSteve Reinmuth