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

Splitting Tori, Knots, and Moebius Bands

Carlo H. Séquin

U.C. Berkeley

Homage a Keizo UshioHomage a Keizo Ushio

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

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:conrad@valett.de

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

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 ?

More Questions ?More Questions ?