Splitting Tori, Knots, and Moebius Bands Florida 1999 BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley

Homage a Keizo Ushio

Performance Art at ISAMA’99 Keizo Ushio and his “OUSHI ZOKEI”

The Making of “Oushi Zokei”

The Making of “Oushi Zokei” (1) Fukusima, March’04 Transport, April’04

The Making of “Oushi Zokei” (2) Keizo’s studio, 04-16-04 Work starts, 04-30-04

The Making of “Oushi Zokei” (3) Drilling starts, 05-06-04 A cylinder, 05-07-04

The Making of “Oushi Zokei” (4) Shaping the torus with a water jet, May 2004

The Making of “Oushi Zokei” (5) A smooth torus, June 2004

The Making of “Oushi Zokei” (6) Drilling holes on spiral path, August 2004

The Making of “Oushi Zokei” (7) Drilling completed, August 30, 2004

The Making of “Oushi Zokei” (8) Rearranging the two parts, September 17, 2004

The Making of “Oushi Zokei” (9) Installation on foundation rock, October 2004

The Making of “Oushi Zokei” (10) Transportation, November 8, 2004

The Making of “Oushi Zokei” (11) Installation in Ono City, November 8, 2004

The Making of “Oushi Zokei” (12) Intriguing geometry – fine details !

Schematic of 2-Link Torus 360° Small FDM (fused deposition model)

Generalize to 3-Link Torus Use a 3-blade “knife”

Generalize to 4-Link Torus Use a 4-blade knife, square cross section

Generalize to 6-Link Torus 6 triangles forming a hexagonal cross section

Keizo 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 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 ... 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 ? 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” Make a Borromean Link from Play-Dough Smash the Link into a toroidal shape.

Result: A Toroidal Braid Three strands forming a circular braid

Cut-Profiles around the Toroid

Splitting a Torus into Borromean Rings Make sure the loops can be moved apart.

A First (Approximate) Model Individual parts made on the FDM machine. Remove support; try to assemble 2 parts.

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

A Better Model Made on a Zcorporation 3D-Printer. Define the cuts rather than the solid parts.

Separating the Three Loops A little widening of the gaps was needed ...

The Open Borromean Torus

III. Focus on SPACE ! Splitting a Torus for the sake of the resulting SPACE !

“Trefoil-Torso” by Nat Friedman Nat Friedman: “The voids in sculptures may be as important as the material.”

Detail of “Trefoil-Torso” Nat Friedman: “The voids in sculptures may be as important as the material.”

“Moebius Space” (Séquin, 2000)

Keizo Ushio, 2004

Keizo’s “Fake” Split (2005) One solid piece ! -- Color can fool the eye !

Triply Twisted Moebius Space 540°

Triply Twisted Moebius Space (2005)

IV. Splitting Other Stuff What if we started with something more intricate than a torus ? ... and then split it.

Splitting Moebius Bands Keizo Ushio 1990

Splitting Moebius Bands M.C.Escher FDM-model, thin FDM-model, thick

Splits of 1.5-Twist Bands by Keizo Ushio (1994) Bondi, 2001

Another Way to Split the Moebius Band Metal band available from Valett Design: conrad@valett.de

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

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).

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.

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.

Split Moebius Trefoil (Séquin, 2003)

“Infinite Duality” (Séquin 2003)

Final Model Thicker beams Wider gaps Less slope

“Knot Divided” by Team Minnesota

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!

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

Flipping Double Links To avoid breaking up into individual loops.

Splitting the Tetrahedron Edge-Graph 3 Loops 1 Loop 4 Loops

“Alter-Knot” by Bathsheba Grossman Has some T-junctions

Turn this into a pure ribbon configuration! Some of the links had to be twisted.

Inspired by Bathsheba Grossman “Alter-Alterknot” QUESTIONS ? Inspired by Bathsheba Grossman

More Questions ?