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Knotting Mathematics and Art University of Southern Florida, Nov.3, 2007. Carlo H. Séquin U.C. Berkeley. Knotty problems in knot theory. Naughty Knotty Sculptures. Sculptures Made from Knots (1). 2004 - 2007: Knots as constructive building blocks. Tetrahedral Trefoil Tangle (FDM). - PowerPoint PPT Presentation
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Knotting Mathematics and ArtKnotting Mathematics and Art University of Southern Florida, Nov.3, 2007University of Southern Florida, Nov.3, 2007
NaughtyKnotty Sculptures
Carlo H. Séquin
U.C. Berkeley
Knotty problems in knot theory
Sculptures Made from Knots (1)Sculptures Made from Knots (1)
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 Tangle (2)Tetra Trefoil Tangle (2)
Complex linking -- two different views
Tetra Trefoil TangleTetra Trefoil Tangle
Complex linking (two views)
Octahedral Trefoil TangleOctahedral Trefoil Tangle
Octahedral Trefoil Tangle (1)Octahedral Trefoil Tangle (1)
Simplest linking
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.
Tetrahedron, Octahedron, ... done !
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 Knots (2)Sculptures Made from Knots (2)
Generate knots & increase their complexity in a structured, procedural way:
I. Bottom-up assembly of knots
II. Top-down mesh infilling
III. Longitudinal knot splitting
Make aesthetically pleasing artifacts
For this conference I have been looking for sculptureswhere the whole piece is just a single knot and
which also involve some “interesting” knots.
OutlineOutline
I. Bottom-up assembly of knots
II. Top-down mesh infilling
III. 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:
Perhaps knotting together crossing strands . . .
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
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
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 assembly of knots
II. Top-down mesh infilling
III. Longitudinal knot splitting
Recursive Figure-8 KnotRecursive Figure-8 Knot
Recursion stepMark crossings over/under to form alternating knot
Result after 2 more recursion steps
Recursive Figure-8 KnotRecursive Figure-8 Knot
Scale stroke-width proportional to recursive reduction
2.5D Recursive (Fractal) Knot2.5D Recursive (Fractal) Knot
Robert Fathauer: “Recursive Trefoil Knot”
Trefoil Recursion
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 assembly of knots
II. Top-down mesh infilling
III. Longitudinal knot splitting
A Split TrefoilA Split Trefoil
To open: Rotate around z-axis
Split Trefoil (side view, closed)Split Trefoil (side view, closed)
Split Trefoil (side view, open)Split Trefoil (side view, open)
Another Split TrefoilAnother Split Trefoil
How much “wiggle room” is there ?
Trefoil “Harmonica”Trefoil “Harmonica”
An Iterated Trefoil-Path of TrefoilsAn Iterated Trefoil-Path of Trefoils
Splitting Moebius BandsSplitting Moebius Bands
Litho by FDM-model FDM-modelM.C.Escher thin, colored thick
Split Moebius Trefoil (SSplit Moebius Trefoil (Sééquin, 2003)quin, 2003)
““Knot DividedKnot Divided” by Team Minnesota” by Team Minnesota
Knotty ProblemKnotty Problem
How many crossings
does this Not-Divided Knot have ?
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 ?
ConclusionsConclusions
Knots are mathematically intriguing and they are inspiring artistic elements.
They can be used as building blocks for sophisticated constellations.
They can be extended recursively to form much more complicated knots.
They can be split lengthwise to make interesting knots and tangles.
Is It Math ?Is It Math ?Is It Art ?Is It Art ?
it is:
“KNOT-ART”