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Valencia, November 2006Valencia, November 2006
Artistic Geometry
Carlo H. Séquin
U.C. Berkeley
Homage a Keizo UshioHomage a Keizo Ushio
Performance Art at ISAMA’99Performance Art at ISAMA’99San Sebastian 1999 San Sebastian 1999 (also in 2007)(also in 2007)
Keizo Ushio and his “OUSHI ZOKEI”
The Making of “Oushi Zokei”The Making of “Oushi Zokei”
The Making of “Oushi Zokei” (1)The Making of “Oushi Zokei” (1)
Fukusima, March’04 Transport, April’04
The Making of “Oushi Zokei” (2)The Making of “Oushi Zokei” (2)
Keizo’s studio, 04-16-04 Work starts, 04-30-04
The Making of “Oushi Zokei” (3)The Making of “Oushi Zokei” (3)
Drilling starts, 05-06-04 A cylinder, 05-07-04
The Making of “Oushi Zokei” (4)The Making of “Oushi Zokei” (4)
Shaping the torus with a water jet, May 2004
The Making of “Oushi Zokei” (5)The Making of “Oushi Zokei” (5)
A smooth torus, June 2004
The Making of “Oushi Zokei” (6)The Making of “Oushi Zokei” (6)
Drilling holes on spiral path, August 2004
The Making of “Oushi Zokei” (7)The Making of “Oushi Zokei” (7)
Drilling completed, August 30, 2004
The Making of “Oushi Zokei” (8)The Making of “Oushi Zokei” (8)
Rearranging the two parts, September 17, 2004
The Making of “Oushi Zokei” (9)The Making of “Oushi Zokei” (9)
Installation on foundation rock, October 2004
The Making of “Oushi Zokei” (10)The Making of “Oushi Zokei” (10)
Transportation, November 8, 2004
The Making of “Oushi Zokei” (11)The Making of “Oushi Zokei” (11)
Installation in Ono City, November 8, 2004
The Making of “Oushi Zokei” (12)The Making of “Oushi Zokei” (12)
Intriguing geometry – fine details !
Schematic of 2-Link TorusSchematic of 2-Link Torus
Small FDM (fused deposition model)
360°
Generalize to 3-Link TorusGeneralize to 3-Link Torus
Use a 3-blade “knife”
Generalize to 4-Link TorusGeneralize to 4-Link Torus
Use a 4-blade knife, square cross section
Generalize to 6-Link TorusGeneralize to 6-Link Torus
6 triangles forming a hexagonal cross section
Keizo Ushio’s Multi-LoopsKeizo Ushio’s Multi-Loops
If we change twist angle of the cutting knife, torus may not get split into separate rings.
180° 360° 540°
Cutting with a Multi-Blade KnifeCutting with a Multi-Blade Knife
Use a knife with b blades,
Rotate through t * 360°/b.
b = 2, t = 1; b = 3, t = 1; b = 3, t = 2.
Cutting with a Multi-Blade Knife ...Cutting with a Multi-Blade Knife ...
results in a(t, b)-torus link;
each component is a (t/g, b/g)-torus knot,
where g = GCD (t, b).
b = 4, t = 2 two double loops.
II. Borromean Torus ?II. Borromean Torus ?
Another Challenge:
Can a torus be split in such a way that a Borromean link results ?
Can the geometry be chosen so that the three links can be moved to mutually orthogonal positions ?
““Reverse Engineering”Reverse Engineering”
Make a Borromean Link from Play-Dough
Smash the Link into a toroidal shape.
Result: A Toroidal BraidResult: A Toroidal Braid
Three strands forming a circular braid
Cut-Profiles around the ToroidCut-Profiles around the Toroid
Splitting a Torus into Borromean RingsSplitting a Torus into Borromean Rings
Make sure the loops can be moved apart.
A First (Approximate) ModelA First (Approximate) Model
Individual parts made on the FDM machine.
Remove support; try to assemble 2 parts.
Assembled Borromean TorusAssembled Borromean Torus
With some fine-tuning, the parts can be made to fit.
A Better ModelA Better Model
Made on a Zcorporation 3D-Printer.
Define the cuts rather than the solid parts.
Separating the Three LoopsSeparating the Three Loops
A little widening of the gaps was needed ...
The Open Borromean TorusThe Open Borromean Torus
III. Focus on SPACE !III. Focus on SPACE !
Splitting a Torus for the sake of the resulting SPACE !
““Trefoil-Torso” by Nat FriedmanTrefoil-Torso” by Nat Friedman
Nat Friedman:
“The voids in sculptures may be as important as the material.”
Detail of Detail of “Trefoil-Torso”“Trefoil-Torso”
Nat Friedman:
“The voids in sculptures may be as important as the material.”
““Moebius Space” (SMoebius Space” (Sééquin, 2000)quin, 2000)
Keizo Ushio, 2004Keizo Ushio, 2004
Keizo’s “Fake” Split (2005)Keizo’s “Fake” Split (2005)
One solid piece ! -- Color can fool the eye !
Triply Twisted Moebius SpaceTriply Twisted Moebius Space
540°
Triply Twisted Moebius Space (2005)Triply Twisted Moebius Space (2005)
IV. Splitting Other StuffIV. Splitting Other Stuff
What if we started with something What if we started with something more intricate than a torus ?more intricate than a torus ?
... and then split it.... and then split it.
Splitting Moebius BandsSplitting Moebius Bands
Keizo
Ushio
1990
Splitting Moebius BandsSplitting Moebius Bands
M.C.Escher FDM-model, thin FDM-model, thick
Splits of 1.5-Twist BandsSplits of 1.5-Twist Bandsby Keizo Ushioby Keizo Ushio
(1994) Bondi, 2001
Another Way to Split the Moebius BandAnother Way to Split the Moebius Band
Metal band available from Valett Design:[email protected]
Splitting KnotsSplitting Knots
Splitting a Moebius band comprising 3 half-twists results in a trefoil knot.
Splitting a TrefoilSplitting a Trefoil
This trefoil seems to have no “twist.”
However, the Frenet frame undergoes about 270° of torsional rotation.
When the tube is split 4 ways it stays connected, (forming a single strand that is 4 times longer).
Splitting a Trefoil into 3 StrandsSplitting a Trefoil into 3 Strands Trefoil with a triangular cross section
(Twist adjusted to close smoothly and maintain 3-fold symmetry).
Add a twist of ± 120° (break symmetry) to yield a single connected strand.
Splitting a Trefoil into 2 StrandsSplitting a Trefoil into 2 Strands Trefoil with a rectangular cross section
Maintaining 3-fold symmetry makes this a single-sided Moebius band.
Split results in double-length strand.
Split Moebius Trefoil (SSplit Moebius Trefoil (Sééquin, 2003)quin, 2003)
““Infinite Duality” (SInfinite Duality” (Sééquin 2003)quin 2003)
Final ModelFinal Model
•Thicker beams•Wider gaps•Less slope
““Knot Divided” by Team MinnesotaKnot Divided” by Team Minnesota
What would happen What would happen if the original band were double-sided?if the original band were double-sided?
==> True split into two knots ! Probably tangled result
How tangled is it ?
How much can the 2 parts move ?
Explore these issues, and others ...
Splitting the Knot into 3 StrandsSplitting the Knot into 3 Strands
3-deep stack
Another 3-Way SplitAnother 3-Way Split
Parts are different, but maintain 3-fold symmetry
Split into 3 Congruent PartsSplit into 3 Congruent Parts
Change the twist of the configuration!
Parts no longer have C3 symmetry
Split Trefoil (closed)Split Trefoil (closed)
Split Trefoil (open)Split Trefoil (open)
Triple-Strand Trefoil (closed)Triple-Strand Trefoil (closed)
Triple-Strand Trefoil (opening up)Triple-Strand Trefoil (opening up)
Triple-Strand Trefoil (fully open)Triple-Strand Trefoil (fully open)
How Much Wiggle Room ?How Much Wiggle Room ?
Take a simple trefoil knot
Split it lengthwise
See what happens ...
Trefoil StackTrefoil Stack
An Iterated Trefoil-Path of TrefoilsAn Iterated Trefoil-Path of Trefoils
Linking Knots ...Linking Knots ...
Use knots as constructive building blocks !
Tetrahedral Trefoil Tangle (FDM)Tetrahedral Trefoil Tangle (FDM)
Tetra Trefoil TanglesTetra Trefoil Tangles
Simple linking (1) -- Complex linking (2)
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 !
Arabic IcosahedronArabic Icosahedron
Icosahedral Trefoil TangleIcosahedral Trefoil Tangle
Simplest linking (type 1)
Icosahedral Icosahedral Trefoil Trefoil TangleTangle(Type 3)(Type 3)
Doubly linked with each neighbor
Arabic Icosahedron, UniGrafix, 1983Arabic Icosahedron, UniGrafix, 1983
Arabic IcosahedronArabic Icosahedron
Is It Math ?Is It Math ?Is It Art ?Is It Art ?
It is:
“KNOT-ART”
Space-filling SculpturesSpace-filling Sculptures
Can we pack knots so tightly
that they fill all of 3D space ?
First: Review of Space-Filling Curves
The 2D Hilbert Curve (1891)The 2D Hilbert Curve (1891)A plane-filling Peano curve
Fall 1983: CS Graduate Course: “Creative Geometric Modeling”
Do This In 3 D !
Construction of the 2D Hilbert CurveConstruction of the 2D Hilbert Curve
112233
Construction of 3D Hilbert CurveConstruction of 3D Hilbert Curve
““Hilbert” Curve in 3DHilbert” Curve in 3D
Start with Hamiltonian Path on Cube Edges
““Hilbert_512_3D”Hilbert_512_3D”
ProMetal Division of Ex One Company ProMetal Division of Ex One Company Headquarters in Irwin, Pennsylvania, USA.Headquarters in Irwin, Pennsylvania, USA.
Questions ?Questions ?
SparesSpares
V. Splitting GraphsV. Splitting Graphs
Take a graph with no loose ends
Split all edges of that graph
Reconnect them, so there are no junctions
Ideally, make this a single loop!
Splitting a JunctionSplitting a Junction
For every one of N arms of a junction,there will be a passage thru the junction.
Flipping Double LinksFlipping Double Links
To avoid breaking up into individual loops.
Splitting the Tetrahedron Edge-GraphSplitting the Tetrahedron Edge-Graph
4 Loops
3 Loops
1 Loop
““Alter-Knot” by Bathsheba GrossmanAlter-Knot” by Bathsheba Grossman
Has some T-junctions
Turn this into a pure ribbon configuration!Turn this into a pure ribbon configuration!
Some of the links had to be twisted.
“ “Alter-Alterknot”Alter-Alterknot”
Inspired by Bathsheba Grossman
QUESTIONS ?