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Splitting Tori, Knots, and Moebius Bands
Florida 1999 BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley
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Homage a Keizo Ushio
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Performance Art at ISAMA’99
Keizo Ushio and his “OUSHI ZOKEI”
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The Making of “Oushi Zokei”
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The Making of “Oushi Zokei” (1)
Fukusima, March’ Transport, April’04
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The Making of “Oushi Zokei” (2)
Keizo’s studio, Work starts,
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The Making of “Oushi Zokei” (3)
Drilling starts, A cylinder,
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The Making of “Oushi Zokei” (4)
Shaping the torus with a water jet, May 2004
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The Making of “Oushi Zokei” (5)
A smooth torus, June 2004
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The Making of “Oushi Zokei” (6)
Drilling holes on spiral path, August 2004
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The Making of “Oushi Zokei” (7)
Drilling completed, August 30, 2004
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The Making of “Oushi Zokei” (8)
Rearranging the two parts, September 17, 2004
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The Making of “Oushi Zokei” (9)
Installation on foundation rock, October 2004
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The Making of “Oushi Zokei” (10)
Transportation, November 8, 2004
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The Making of “Oushi Zokei” (11)
Installation in Ono City, November 8, 2004
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The Making of “Oushi Zokei” (12)
Intriguing geometry – fine details !
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Schematic of 2-Link Torus
360° Small FDM (fused deposition model)
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Generalize to 3-Link Torus
Use a 3-blade “knife”
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Generalize to 4-Link Torus
Use a 4-blade knife, square cross section
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Generalize to 6-Link Torus
6 triangles forming a hexagonal cross section
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Keizo Ushio’s Multi-Loops
If we change twist angle of the cutting knife, torus may not get split into separate rings. 180° ° °
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Cutting 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.
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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 = two double loops.
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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 ?
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“Reverse Engineering”
Make a Borromean Link from Play-Dough Smash the Link into a toroidal shape.
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Result: A Toroidal Braid
Three strands forming a circular braid
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Cut-Profiles around the Toroid
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Splitting a Torus into Borromean Rings
Make sure the loops can be moved apart.
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A First (Approximate) Model
Individual parts made on the FDM machine. Remove support; try to assemble 2 parts.
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Assembled Borromean Torus
With some fine-tuning, the parts can be made to fit.
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A Better Model Made on a Zcorporation 3D-Printer.
Define the cuts rather than the solid parts.
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Separating the Three Loops
A little widening of the gaps was needed ...
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The Open Borromean Torus
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III. Focus on SPACE ! Splitting a Torus for the sake of the resulting SPACE !
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“Trefoil-Torso” by Nat Friedman
Nat Friedman: “The voids in sculptures may be as important as the material.”
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Detail of “Trefoil-Torso”
Nat Friedman: “The voids in sculptures may be as important as the material.”
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“Moebius Space” (Séquin, 2000)
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Keizo Ushio, 2004
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Keizo’s “Fake” Split (2005)
One solid piece ! -- Color can fool the eye !
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Triply Twisted Moebius Space
540°
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Triply Twisted Moebius Space (2005)
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IV. Splitting Other Stuff
What if we started with something more intricate than a torus ? and then split it.
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Splitting Moebius Bands
Keizo Ushio 1990
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Splitting Moebius Bands
M.C.Escher FDM-model, thin FDM-model, thick
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Splits of 1.5-Twist Bands by Keizo Ushio
(1994) Bondi, 2001
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Another Way to Split the Moebius Band
Metal band available from Valett Design:
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Splitting Knots Splitting a Moebius band comprising 3 half-twists results in a trefoil knot.
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Splitting 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).
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Splitting 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.
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Splitting 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.
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Split Moebius Trefoil (Séquin, 2003)
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“Infinite Duality” (Séquin 2003)
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Final Model Thicker beams Wider gaps Less slope
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“Knot Divided” by Team Minnesota
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V. 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!
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Splitting a Junction For every one of N arms of a junction, there will be a passage thru the junction.
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Flipping Double Links To avoid breaking up into individual loops.
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Splitting the Tetrahedron Edge-Graph
3 Loops 1 Loop 4 Loops
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“Alter-Knot” by Bathsheba Grossman
Has some T-junctions
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Turn this into a pure ribbon configuration!
Some of the links had to be twisted.
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Inspired by Bathsheba Grossman
“Alter-Alterknot” QUESTIONS ? Inspired by Bathsheba Grossman
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More Questions ?
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