simons center, july 30, 2012 carlo h. séquin university of california, berkeley artistic geometry...
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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 ?
?