Simons Center, July 30, 2012 Carlo H. Séquin University of California, Berkeley Artistic Geometry...

Simons Center, July 30, 2012Simons Center, July 30, 2012

Carlo H. Séquin

University of California, Berkeley

Artistic Geometry -- The Math Behind the Art

ART ART MATH MATH

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).

My Conjecture ...My Conjecture ...

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

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

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

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

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

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

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

Shaping the torus with a water jet, May 2004

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

A smooth torus, June 2004

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

Drilling holes on spiral path, August 2004

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

Drilling completed, August 30, 2004

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

Rearranging the two parts, September 17, 2004

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

Installation on foundation rock, October 2004

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

Transportation, November 8, 2004

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

Installation in Ono City, November 8, 2004

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

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

Generalize to 6-Link TorusGeneralize to 6-Link Torus

6 triangles forming a hexagonal cross section

Keizo UshioKeizo Ushio’’s Multi-Loopss Multi-Loops 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:

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

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

(1994) Bondi, 2001

Splitting Knots …Splitting Knots …

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

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 Möbius band.

Split results in double-length strand.

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

““Infinite DualityInfinite Duality”” (S (Sééquin 2003)quin 2003)

Final ModelFinal Model

•Thicker beams•Wider gaps•Less slope

““Knot DividedKnot Divided”” by Team Minnesota by Team Minnesota

Splitting a Knotted Möbius BandSplitting a Knotted Möbius Band

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).

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.

Another 3-Way SplitAnother 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 3-fold symmetry

A Split TrefoilA Split Trefoil

To open: Rotate one half around central axis

Split Trefoil (side view, closed)Split Trefoil (side view, closed)

Split Trefoil (side view, open)Split Trefoil (side view, open)

Triple-Strand Trefoil (closed)Triple-Strand Trefoil (closed)

Triple-Strand Trefoil (opening up)Triple-Strand Trefoil (opening up)

Triple-Strand Trefoil (fully open)Triple-Strand Trefoil (fully open)

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

Brent Collins: Brent Collins: Hyperbolic HexagonHyperbolic Hexagon

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).

On More Very Special Twisted ToroidOn 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

How to Make a How to Make a Klein Bottle (1)Klein Bottle (1)

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

How to Make a How to Make a Klein Bottle (2)Klein Bottle (2) Join tube ends with reversed order:

How to Make a How to Make a Klein Bottle (3)Klein Bottle (3)

Close ends smoothly by “inverting one sock”

LimerickLimerick

A mathematician named Klein

thought Möbius bands are divine.

Said he: "If you glue

the edges of two,

you'll get a weird bottle like mine."

2 Möbius Bands Make a Klein Bottle2 Möbius Bands Make a Klein Bottle

KOJ = MR + ML

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

Klein Klein KnottlesKnottles Based on KOJ Based on KOJ

Always an odd number of “turn-back mouths”!

A Gridded Model of A Gridded Model of Trefoil KnottleTrefoil Knottle

Some More Klein Bottles . . .Some More Klein Bottles . . .

TopologyTopology

Shape does not matter -- only connectivity.

Surfaces can be deformed continuously.

SmoothlySmoothly Deforming Surfaces Deforming Surfaces

Surface may pass through itself.

It cannot be cut or torn; it cannot change connectivity.

It must never form any sharp creases or points of infinitely sharp curvature.

OK

Regular HomotopyRegular Homotopy

Two shapes are called regular homotopic, if they can be transformed into one anotherwith a continuous, smooth deformation(with no kinks or singularities).

Such shapes are then said to be:in the same regular homotopy class.

Regular Homotopic Torus EversionRegular Homotopic Torus Eversion

ThreeThree Structurally Different Structurally Different Klein BottlesKlein Bottles

All three are in different regular homotopy classes!

ConclusionsConclusions

Knotted and twisted structures play an important role in many areas of physics and the life sciences.

They also make fascinating art-objects . . .

2003: 2003: ““Whirled White WebWhirled White Web””

Inauguration Sutardja Dai Hall 2/27/09Inauguration Sutardja Dai Hall 2/27/09

Brent Collins and David LynnBrent Collins and David Lynn

Sculpture Generator #2Sculpture Generator #2

Is It Math ?Is It Math ?Is It Art ?Is It Art ?

it is:

“KNOT-ART”

QUESTIONS ?QUESTIONS ?

?