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BRIDGES, Banff, August 2005 BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley
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BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

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Page 1: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

BRIDGES, Banff, August 2005BRIDGES, Banff, August 2005

Splitting Tori, Knots, and Moebius Bands

Carlo H. Séquin

U.C. Berkeley

Page 2: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Homage a Keizo UshioHomage a Keizo Ushio

Page 3: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Performance Art at ISAMA’99Performance Art at ISAMA’99

Keizo Ushio and his “OUSHI ZOKEI”

Page 4: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei”The Making of “Oushi Zokei”

Page 5: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (1)The Making of “Oushi Zokei” (1)

Fukusima, March’04 Transport, April’04

Page 6: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (2)The Making of “Oushi Zokei” (2)

Keizo’s studio, 04-16-04 Work starts, 04-30-04

Page 7: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (3)The Making of “Oushi Zokei” (3)

Drilling starts, 05-06-04 A cylinder, 05-07-04

Page 8: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (4)The Making of “Oushi Zokei” (4)

Shaping the torus with a water jet, May 2004

Page 9: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (5)The Making of “Oushi Zokei” (5)

A smooth torus, June 2004

Page 10: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (6)The Making of “Oushi Zokei” (6)

Drilling holes on spiral path, August 2004

Page 11: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (7)The Making of “Oushi Zokei” (7)

Drilling completed, August 30, 2004

Page 12: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (8)The Making of “Oushi Zokei” (8)

Rearranging the two parts, September 17, 2004

Page 13: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (9)The Making of “Oushi Zokei” (9)

Installation on foundation rock, October 2004

Page 14: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (10)The Making of “Oushi Zokei” (10)

Transportation, November 8, 2004

Page 15: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (11)The Making of “Oushi Zokei” (11)

Installation in Ono City, November 8, 2004

Page 16: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Making of “Oushi Zokei” (12)The Making of “Oushi Zokei” (12)

Intriguing geometry – fine details !

Page 17: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Schematic of 2-Link TorusSchematic of 2-Link Torus

Small FDM (fused deposition model)

360°

Page 18: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Generalize to 3-Link TorusGeneralize to 3-Link Torus

Use a 3-blade “knife”

Page 19: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Generalize to 4-Link TorusGeneralize to 4-Link Torus

Use a 4-blade knife, square cross section

Page 20: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Generalize to 6-Link TorusGeneralize to 6-Link Torus

6 triangles forming a hexagonal cross section

Page 21: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

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°

Page 22: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

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.

Page 23: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

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.

Page 24: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

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 ?

Page 25: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

““Reverse Engineering”Reverse Engineering”

Make a Borromean Link from Play-Dough

Smash the Link into a toroidal shape.

Page 26: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Result: A Toroidal BraidResult: A Toroidal Braid

Three strands forming a circular braid

Page 27: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Cut-Profiles around the ToroidCut-Profiles around the Toroid

Page 28: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Splitting a Torus into Borromean RingsSplitting a Torus into Borromean Rings

Make sure the loops can be moved apart.

Page 29: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

A First (Approximate) ModelA First (Approximate) Model

Individual parts made on the FDM machine.

Remove support; try to assemble 2 parts.

Page 30: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Assembled Borromean TorusAssembled Borromean Torus

With some fine-tuning, the parts can be made to fit.

Page 31: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

A Better ModelA Better Model

Made on a Zcorporation 3D-Printer.

Define the cuts rather than the solid parts.

Page 32: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Separating the Three LoopsSeparating the Three Loops

A little widening of the gaps was needed ...

Page 33: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

The Open Borromean TorusThe Open Borromean Torus

Page 34: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

III. Focus on SPACE !III. Focus on SPACE !

Splitting a Torus for the sake of the resulting SPACE !

Page 35: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

““Trefoil-Torso” by Nat FriedmanTrefoil-Torso” by Nat Friedman

Nat Friedman:

“The voids in sculptures may be as important as the material.”

Page 36: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Detail of Detail of “Trefoil-Torso”“Trefoil-Torso”

Nat Friedman:

“The voids in sculptures may be as important as the material.”

Page 37: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

““Moebius Space” (SMoebius Space” (Sééquin, 2000)quin, 2000)

Page 38: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.
Page 39: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Keizo Ushio, 2004Keizo Ushio, 2004

Page 40: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Keizo’s “Fake” Split (2005)Keizo’s “Fake” Split (2005)

One solid piece ! -- Color can fool the eye !

Page 41: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Triply Twisted Moebius SpaceTriply Twisted Moebius Space

540°

Page 42: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Triply Twisted Moebius Space (2005)Triply Twisted Moebius Space (2005)

Page 43: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.
Page 44: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

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.

Page 45: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Splitting Moebius BandsSplitting Moebius Bands

Keizo

Ushio

1990

Page 46: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Splitting Moebius BandsSplitting Moebius Bands

M.C.Escher FDM-model, thin FDM-model, thick

Page 47: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Splits of 1.5-Twist BandsSplits of 1.5-Twist Bandsby Keizo Ushioby Keizo Ushio

(1994) Bondi, 2001

Page 48: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Another Way to Split the Moebius BandAnother Way to Split the Moebius Band

Metal band available from Valett Design:[email protected]

Page 49: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Splitting KnotsSplitting Knots

Splitting a Moebius band comprising 3 half-twists results in a trefoil knot.

Page 50: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

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).

Page 51: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

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.

Page 52: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

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.

Page 53: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Split Moebius Trefoil (SSplit Moebius Trefoil (Sééquin, 2003)quin, 2003)

Page 54: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

““Infinite Duality” (SInfinite Duality” (Sééquin 2003)quin 2003)

Page 55: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Final ModelFinal Model

•Thicker beams•Wider gaps•Less slope

Page 56: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

““Knot Divided” by Team MinnesotaKnot Divided” by Team Minnesota

Page 57: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

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!

Page 58: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Splitting a JunctionSplitting a Junction

For every one of N arms of a junction,there will be a passage thru the junction.

Page 59: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Flipping Double LinksFlipping Double Links

To avoid breaking up into individual loops.

Page 60: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Splitting the Tetrahedron Edge-GraphSplitting the Tetrahedron Edge-Graph

4 Loops

3 Loops

1 Loop

Page 61: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

““Alter-Knot” by Bathsheba GrossmanAlter-Knot” by Bathsheba Grossman

Has some T-junctions

Page 62: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

Turn this into a pure ribbon configuration!Turn this into a pure ribbon configuration!

Some of the links had to be twisted.

Page 63: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

“ “Alter-Alterknot”Alter-Alterknot”

Inspired by Bathsheba Grossman

QUESTIONS ?

Page 64: BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley.

More Questions ?More Questions ?