deyoung museum, june12, 2013 carlo h. séquin university of california, berkeley tracking twisted...
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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).
Regular, Geometric ArtRegular, Geometric Art
Early art: Patterns on bones, pots, weavings...
Mathematics (geometry) to help make things fit:
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)
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”” (6) (6)
Drilling holes on spiral path, August 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”” (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°
. . . 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°
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”.
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.
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 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
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.
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).
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.
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.
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”
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.
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
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: