Simons Center, July 30, 2012 Carlo H. Séquin University of California, Berkeley Artistic Geometry...
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Transcript of Simons Center, July 30, 2012 Carlo H. Séquin University of California, Berkeley Artistic Geometry...
![Page 1: Simons Center, July 30, 2012 Carlo H. Séquin University of California, Berkeley Artistic Geometry -- The Math Behind the Art.](https://reader033.fdocuments.in/reader033/viewer/2022051316/56649f505503460f94c7382e/html5/thumbnails/1.jpg)
Simons Center, July 30, 2012Simons Center, July 30, 2012
Carlo H. Séquin
University of California, Berkeley
Artistic Geometry -- The Math Behind the Art
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ART ART MATH MATH
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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).
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My Conjecture ...My Conjecture ...
Early art: Patterns on bones, pots, weavings...
Mathematics (geometry) to help make things fit:
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Geometry ! Geometry !
Descriptive Geometry – love since high school
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Descriptive GeometryDescriptive Geometry
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40 Years of Geometry and Design40 Years of Geometry and Design
CCD TV Camera Soda Hall
RISC 1 Computer Chip Octa-Gear (Cyberbuild)
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More Recent CreationsMore Recent Creations
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Homage a Keizo UshioHomage a Keizo Ushio
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ISAMA, San Sebastian 1999ISAMA, San Sebastian 1999
Keizo Ushio and his “OUSHI ZOKEI”
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The Making of The Making of ““Oushi ZokeiOushi Zokei””
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (1) (1)
Fukusima, March’04 Transport, April’04
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (2) (2)
Keizo’s studio, 04-16-04 Work starts, 04-30-04
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (3) (3)
Drilling starts, 05-06-04 A cylinder, 05-07-04
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (4) (4)
Shaping the torus with a water jet, May 2004
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (5) (5)
A smooth torus, June 2004
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (6) (6)
Drilling holes on spiral path, August 2004
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (7) (7)
Drilling completed, August 30, 2004
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (8) (8)
Rearranging the two parts, September 17, 2004
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (9) (9)
Installation on foundation rock, October 2004
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (10) (10)
Transportation, November 8, 2004
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (11) (11)
Installation in Ono City, November 8, 2004
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The Making of The Making of ““Oushi ZokeiOushi Zokei”” (12) (12)
Intriguing geometry – fine details !
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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°
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Slicing a Bagel . . .Slicing a Bagel . . .
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. . . and Adding Cream Cheese. . . and Adding Cream Cheese
From George Hart’s web page:http://www.georgehart.com/bagel/bagel.html
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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°
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Generalize this to 3-Link TorusGeneralize this to 3-Link Torus
Use a 3-blade “knife”
360°
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Generalization to 4-Link TorusGeneralization to 4-Link Torus
Use a 4-blade knife, square cross section
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Generalize to 6-Link TorusGeneralize to 6-Link Torus
6 triangles forming a hexagonal cross section
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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°
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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.
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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.
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““Moebius SpaceMoebius Space”” (S (Sééquin, 2000)quin, 2000)
ART:Focus on the
cutting space !Use “thick knife”.
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Anish KapoorAnish Kapoor’’s s ““BeanBean”” in Chicago in Chicago
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Keizo Ushio, 2004Keizo Ushio, 2004
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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:
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Twisted Möbius Bands in ArtTwisted Möbius Bands in Art
Web Max Bill M.C. Escher M.C. Escher
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Triply Twisted Möbius SpaceTriply Twisted Möbius Space
540°
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Triply Twisted Moebius Space (2005)Triply Twisted Moebius Space (2005)
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Triply Twisted Moebius Space (2005)Triply Twisted Moebius Space (2005)
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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 . . .
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Splitting Möbius Bands (not just tori)Splitting Möbius Bands (not just tori)
Keizo
Ushio
1990
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Splitting Möbius BandsSplitting Möbius Bands
M.C.Escher FDM-model, thin FDM-model, thick
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Splits of 1.5-Twist BandsSplits of 1.5-Twist Bandsby Keizo Ushioby Keizo Ushio
(1994) Bondi, 2001
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Splitting Knots …Splitting Knots …
Splitting a Möbius band comprising 3 half-twists results in a trefoil knot.
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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.
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Split Moebius Trefoil (SSplit Moebius Trefoil (Sééquin, 2003)quin, 2003)
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““Infinite DualityInfinite Duality”” (S (Sééquin 2003)quin 2003)
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Final ModelFinal Model
•Thicker beams•Wider gaps•Less slope
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““Knot DividedKnot Divided”” by Team Minnesota by Team Minnesota
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Splitting a Knotted Möbius BandSplitting a Knotted Möbius Band
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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).
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Twisted PrismsTwisted Prisms
An n-sided prismatic ribbon can be end-to-end connected in at least n different ways
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Helaman Ferguson: Umbilic TorusHelaman Ferguson: Umbilic Torus
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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.
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Another 3-Way SplitAnother 3-Way Split
Parts are different, but maintain 3-fold symmetry
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Split into 3 Congruent PartsSplit into 3 Congruent Parts
Change the twist of the configuration!
Parts no longer have 3-fold symmetry
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A Split TrefoilA Split Trefoil
To open: Rotate one half around central axis
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Split Trefoil (side view, closed)Split Trefoil (side view, closed)
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Split Trefoil (side view, open)Split Trefoil (side view, open)
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Triple-Strand Trefoil (closed)Triple-Strand Trefoil (closed)
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Triple-Strand Trefoil (opening up)Triple-Strand Trefoil (opening up)
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Triple-Strand Trefoil (fully open)Triple-Strand Trefoil (fully open)
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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
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Brent Collins: Brent Collins: Hyperbolic HexagonHyperbolic Hexagon
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ScherkScherk’’s 2nd Minimal Surfaces 2nd Minimal Surface
2 planes the central core 4 planesbi-ped saddles 4-way saddles
= “Scherk tower”
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ScherkScherk’’s 2nd Minimal Surfaces 2nd Minimal Surface
Normal“biped”saddles
Generalization to higher-order saddles(monkey saddle)“Scherk Tower”
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V-artV-art(1999)(1999)
VirtualGlassScherkTowerwithMonkeySaddles
(Radiance 40 hours)
Jane Yen
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Closing the LoopClosing the Loop
straight
or
twisted
“Scherk Tower” “Scherk-Collins Toroids”
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Sculpture Generator 1Sculpture Generator 1, GUI , GUI
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Shapes from Shapes from Sculpture Generator 1Sculpture Generator 1
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The Finished The Finished HeptoroidHeptoroid
at Fermi Lab Art Gallery (1998).
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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
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Making a Making a Figure-8Figure-8 Klein Bottle Klein Bottle
Add a 180° flip to the tubebefore the ends are merged.
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Figure-8 Klein BottleFigure-8 Klein Bottle
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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
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Classical Classical ““Inverted-SockInverted-Sock”” Klein Bottle Klein Bottle
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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
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How to Make a How to Make a Klein Bottle (2)Klein Bottle (2) Join tube ends with reversed order:
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How to Make a How to Make a Klein Bottle (3)Klein Bottle (3)
Close ends smoothly by “inverting one sock”
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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."
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2 Möbius Bands Make a Klein Bottle2 Möbius Bands Make a Klein Bottle
KOJ = MR + ML
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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
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Klein Klein KnottlesKnottles Based on KOJ Based on KOJ
Always an odd number of “turn-back mouths”!
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A Gridded Model of A Gridded Model of Trefoil KnottleTrefoil Knottle
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Some More Klein Bottles . . .Some More Klein Bottles . . .
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TopologyTopology
Shape does not matter -- only connectivity.
Surfaces can be deformed continuously.
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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
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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.
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Regular Homotopic Torus EversionRegular Homotopic Torus Eversion
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ThreeThree Structurally Different Structurally Different Klein BottlesKlein Bottles
All three are in different regular homotopy classes!
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ConclusionsConclusions
Knotted and twisted structures play an important role in many areas of physics and the life sciences.
They also make fascinating art-objects . . .
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2003: 2003: ““Whirled White WebWhirled White Web””
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Inauguration Sutardja Dai Hall 2/27/09Inauguration Sutardja Dai Hall 2/27/09
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Brent Collins and David LynnBrent Collins and David Lynn
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Sculpture Generator #2Sculpture Generator #2
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Is It Math ?Is It Math ?Is It Art ?Is It Art ?
it is:
“KNOT-ART”
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QUESTIONS ?QUESTIONS ?
?