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

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
b = 2, t = 1;
through t * 360°/b.
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:
[email protected]
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).
a twist of ± 120° (break symmetry)
to yield a single connected strand.
 Add
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
1 Loop
4 Loops
3 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.
“Alter-Alterknot”
QUESTIONS
?
Inspired by Bathsheba Grossman
More Questions ?