DeYoung Museum, June12, 2013 Carlo H. Séquin University of California, Berkeley Tracking Twisted Toroids MATHEMATICAL TREASURE HUNTS

DeYoung Museum, June12, 2013DeYoung Museum, June12, 2013

Carlo H. Séquin

University of California, Berkeley

Tracking Twisted Toroids

MATHEMATICAL TREASURE HUNTS

What came first: Art or Mathematics ?What came first: Art or Mathematics ?

Question posed Nov. 16, 2006 by Dr. Ivan Sutherland“father” of computer graphics (SKETCHPAD, 1963).

Early “Free-Form” ArtEarly “Free-Form” Art

Cave paintings, Lascaux Venus von Willendorf

Regular, Geometric ArtRegular, Geometric Art

Early art: Patterns on bones, pots, weavings...

Mathematics (geometry) to help make things fit:

Geometry ! Geometry !

Descriptive Geometry – love since high school

Descriptive GeometryDescriptive Geometry

40 Years of Geometry and Design40 Years of Geometry and Design

CCD TV Camera Soda Hall (for CS)

RISC 1 Computer Chip Octa-Gear (Cyberbuild)

More Recent CreationsMore Recent Creations

Homage a Keizo UshioHomage a Keizo Ushio

ISAMA, San Sebastian 1999ISAMA, San Sebastian 1999

Keizo Ushio and his “OUSHI ZOKEI”

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

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

Fukusima, March’04 Transport, April’04


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


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


Shaping the torus with a water jet, May 2004


A smooth torus, June 2004


Drilling holes on spiral path, August 2004


Drilling completed, August 30, 2004


Rearranging the two parts, September 17, 2004


Installation on foundation rock, October 2004


Transportation, November 8, 2004


Installation in Ono City, November 8, 2004


Intriguing geometry – fine details !

Schematic Model of 2-Link TorusSchematic Model of 2-Link Torus

Knife blades rotate through 360 degreesas it sweep once around the torus ring.

360°

Slicing a Bagel . . .Slicing a Bagel . . .

. . . and Adding Cream Cheese. . . and Adding Cream Cheese

From George Hart’s web page:http://www.georgehart.com/bagel/bagel.html

Schematic Model of 2-Link TorusSchematic Model of 2-Link Torus

2 knife blades rotate through 360 degreesas they sweep once around the torus ring.

360°

Generalize this to 3-Link TorusGeneralize this to 3-Link Torus

Use a 3-blade “knife”

360°

Generalization to 4-Link TorusGeneralization to 4-Link Torus

Use a 4-blade knife, square cross section

Generalization to 6-Link TorusGeneralization to 6-Link Torus

6 triangles forming a hexagonal cross section

Keizo UshioKeizo Ushio’’s Multi-Loop Cutss Multi-Loop Cuts There is a second parameter:

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,

Twist knife 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.

““Moebius SpaceMoebius Space”” (S (Sééquin, 2000)quin, 2000)

ART:Focus on the

cutting space !Use “thick knife”.

Anish KapoorAnish Kapoor’’s s ““BeanBean”” in Chicago in Chicago

Keizo Ushio, 2004Keizo Ushio, 2004

It is a It is a Möbius Band Möbius Band !!

A closed ribbon with a 180° flip;

A single-sided surface with a single edge:

+180°(ccw), 0°, –180°, –540°(cw)

Apparent twist (compared to a rotation-minimizing frame)

Changing Shapes of a Möbius BandChanging Shapes of a Möbius Band

Regular Homotopies

Using a “magic” surface material that can pass through itself.

Twisted Möbius Bands in ArtTwisted Möbius Bands in Art

Web Max Bill M.C. Escher M.C. Escher

Triply Twisted Möbius SpaceTriply Twisted Möbius Space

540°

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

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

Splitting Other StuffSplitting 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 that shape . . .. . . and then split that shape . . .

Splitting Möbius Bands (not just tori)Splitting Möbius Bands (not just tori)

Keizo

Ushio

1990

Splitting Möbius BandsSplitting Möbius Bands

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

Splitting a Band Splitting a Band with a Twist of 540°with a Twist of 540°by Keizo Ushioby Keizo Ushio

(1994) Bondi, 2001

Another Way to Split the Möbius BandAnother Way to Split the Möbius Band

Metal band available from Valett Design:[email protected]

SOME HANDS-ON ACTIVITIESSOME HANDS-ON ACTIVITIES

1. Splitting Möbius Strips

2. Double-layer Möbius Strips

3. Escher’s Split Möbius Band

Activity #1: Möbius StripsActivity #1: Möbius StripsFor people who have not previously played

with physical Möbius strips.

Take an 11” long white paper strip; bend it into a loop;

But before joining the end, flip one end an odd number of times: +/– 180°or 540°;

Compare results among students:How many different bands do you find?

Take a marker pen and draw a line ¼” offfrom one of the edges . . .Continue the line until it closes (What happens?)

Cut the strip lengthwise down the middle . . .(What happens? -- Discuss with neighbors!)

Activity #2: Double Möbius StripsActivity #2: Double Möbius Strips Take TWO 11” long, 2-color paper strips;

put them on top of each other so touching colors match;bend sandwich into a loop; join after 1 or 3 flips( tape the two layers individually! ).

Convince yourself that strips are separate by passing a pencil or small paper piece around the whole loop.

Separate (open-up) the two loops.

Put the configuration back together.

Activity #3: Escher’s Split Möbius BandActivity #3: Escher’s Split Möbius Band Take TWO 11”-long, 2-color paper strips;

tape them together into a 22”-long paper strip (match color).

Try to form this shape inspired by MC Escher’s drawing:

After you have succeeded, can you reconfigure your modelinto something that looks like picture #3 ?

MUSEUM or WEB ACTIVITIESMUSEUM or WEB ACTIVITIES

1. Find pictures or sculptures of twisted toroids.

2. Find earliest depiction of aMöbius Band.

Twisted PrismsTwisted Prisms

An n-sided prismatic ribbon can be end-to-end connected in at least n different ways

Helaman Ferguson: Umbilic TorusHelaman Ferguson: Umbilic Torus

Splitting a Trefoil into 3 StrandsSplitting a Trefoil into 3 Strands Trefoil with a triangular cross section

(twist adjusted to close smoothly, maintain 3-fold symmetry).

3-way split results in 3 separate intertwined trefoils.

Add a twist of ± 120° (break symmetry) to yield a single connected strand.

Symmetrical 3-Way SplitSymmetrical 3-Way Split

Parts are different, but maintain 3-fold symmetry

Split into 3 Congruent PartsSplit into 3 Congruent Parts

Change the twist of the configuration!

Parts no longer have C3 symmetry, but are congruent.

More Ways to Split a TrefoilMore Ways to Split 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).

TopologyTopology

Shape does not matter -- only connectivity.

Surfaces can be deformed continuously.

The Genus of an ObjectThe Genus of an Object Number of tunnels through a solid blob.

Number of handles glued onto a sphere.

Number of cuts needed to break all loops,but still keep object hanging together.

Objects with Different GenusObjects with Different Genus

g=1 g=2 g=3 g=4 g=5

ACTIVITIES related to GENUSACTIVITIES related to GENUS

1. Find museum artifacts of genus 1, 2, 3 ….

If you cannot find physical artifacts,pictures of appropriate objects are OK too.

2. Determine the genus of a select sculpture.

3. Find a highly complex object of genus 0.

Twisted “Chains”Twisted “Chains”

““Millennium Arch” Millennium Arch” (Hole-Saddle Toroid)(Hole-Saddle Toroid)

Brent Collins: Brent Collins: Hyperbolic HexagonHyperbolic Hexagon

A Special Kind of Toroidal StructuresA Special Kind of Toroidal Structures

Collaboration with sculptor Brent Collins: “Hyperbolic Hexagon” (1994) “Hyperbolic Hexagon II” (1996) “Heptoroid” (1998)

= = > “Scherk-towers” wound up into a loop.

ScherkScherk’’s 2nd Minimal Surfaces 2nd Minimal Surface

2 planes the central core 4 planesbi-ped saddles 4-way saddles

= “Scherk tower”

ScherkScherk’’s 2nd Minimal Surfaces 2nd Minimal Surface

Normal“biped”saddles

Generalization to higher-order saddles(monkey saddle)“Scherk Tower”

V-artV-art(1999)(1999)

VirtualGlassScherkTowerwithMonkeySaddles

(Radiance 40 hours)

Jane Yen

Closing the LoopClosing the Loop

straight

or

twisted

“Scherk Tower” “Scherk-Collins Toroids”

Sculpture Generator 1Sculpture Generator 1, GUI , GUI

Shapes from Shapes from Sculpture Generator 1Sculpture Generator 1

The Finished The Finished HeptoroidHeptoroid

at Fermi Lab Art Gallery (1998).

Sculpture Generator #2Sculpture Generator #2

One More Very Special Twisted ToroidOne More Very Special Twisted Toroid

First make a “figure-8 tube” by merging the horizontal edges of the rectangular domain

Making a Making a Figure-8Figure-8 Klein Bottle Klein Bottle

Add a 180° flip to the tubebefore the ends are merged.

Figure-8 Klein BottleFigure-8 Klein Bottle

What is a What is a Klein Bottle Klein Bottle ??

A single-sided surface

with no edges or punctures

with Euler characteristic: V – E + F = 0

corresponding to: genus = 2

Always self-intersecting in 3D

Classical Classical ““Inverted-SockInverted-Sock”” Klein Bottle Klein Bottle

The Two Klein Bottles Side-by-SideThe Two Klein Bottles Side-by-Side

Both are composed from two Möbius bands !

Fancy Klein Bottles of Type KOJFancy Klein Bottles of Type KOJ

Cliff Stoll Klein bottles by Alan Bennet in the Science Museum in South Kensington, UK

LimerickLimerick

Mathematicians try hard to floor us

with a non-orientable torus.

The bottle of Klein,

they say, is divine.

But it is so exceedingly porus!

by Cliff Stoll

ACTIVITIES with Twisted ToroidsACTIVITIES with Twisted Toroids

1. Twisted prismatic toroids.

2. Making a figure-8 Klein bottle.

Activity #1: Twisted Prismatic ToroidsActivity #1: Twisted Prismatic Toroids

Mark one face of the square-profile foam-rubber prismwith little patches of masking tape;

Bend the foam-rubber prism into a loop;(or combine two sticks and make a trefoil knot);

Twist the prism, join the ends; fix with tape: Avoid matching the 2 ends of the marked face to obtain a true Möbius prism.

With your finger, continue to trace the marked faceuntil it closes back to itself (perhaps add tape patches)(How many passes through the loop does it make?)(Have all the prism faces been marked?)

Discuss with neighbors!

Activity #2: Figure-8 Klein BottleActivity #2: Figure-8 Klein Bottle

Take a 2.8” x 11” 2-colored paper strip;

Length-wise crease down the middle, and ¼ width;

Give the whole strip a zig-zag Z-shaped profile(assume that the ends that touch the middle crease are connected

through the middle to form a figure-8 profile);

Connect the ends of the figure-8 tube after a 180°flip.

Draw a longitudinal line with a marker pen.

Why is this a Klein bottle? -- Discuss with neighbors!

PROFILE:

QUESTIONS ?QUESTIONS ?

?

Documents

DeYoung Museum, June12, 2013 Carlo H. Séquin University of California, Berkeley Tracking Twisted Toroids MATHEMATICAL TREASURE HUNTS