Bay Area Science Festival, 2013
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Transcript of Bay Area Science Festival, 2013
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Bay Area Science Festival, 2013Bay Area Science Festival, 2013
Magic of Klein Bottles
EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley
Carlo H. Séquin
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Type “KOJ”:
K: Klein bottle
O: tube profile
J: overall tube shape
Classical Classical ““Inverted-SockInverted-Sock”” Klein Bottle Klein Bottle
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Several Fancy Klein BottlesSeveral Fancy Klein Bottles
Cliff Stoll Klein bottles by Alan Bennett in the Science Museum in South Kensington, UK
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What is a What is a Klein Bottle Klein Bottle ?? A single-sided surface
with no edges or punctures.
It can be made made from a rectangle:
with Euler characteristic: V – E + F = 0
It is always self-intersecting in 3D !
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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 sock end”
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Type “K8L”:
K: Klein bottle
8: tube profile
L: left-twisting
Figure-8 Klein BottleFigure-8 Klein Bottle
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Making a Making a Figure-8Figure-8 Klein Bottle (1)Klein Bottle (1)
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 (2)Klein Bottle (2)
Add a 180° flip to the tubebefore the ends are merged.
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Two Different Figure-8 Klein BottlesTwo Different Figure-8 Klein Bottles
Right-twisting Left-twisting
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The Rules of the Game: The Rules of the Game: TopologyTopology
Shape does not matter -- only connectivity.
Surfaces can be deformed continuously.
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Smoothly Deforming SurfacesSmoothly 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) Homotopy(Regular) Homotopy
Two shapes are called 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 homotopy class.
With these rules:
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When are 2 Klein Bottles the Same?When are 2 Klein Bottles the Same?
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When are 2 Klein Bottles the Same?When are 2 Klein Bottles the Same?
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2 Möbius Bands Make a Klein Bottle2 Möbius Bands Make a Klein Bottle
KOJ = MR + ML
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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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A Twisted Klein BottleA Twisted Klein Bottle
Split it along a twisted longitudinal grid line . . .
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Split Klein Bottle Split Klein Bottle Two Moebius Bands Two Moebius Bands
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Yet Another Way to Match-up NumbersYet Another Way to Match-up Numbers
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““Inverted Double-SockInverted Double-Sock”” Klein Bottle Klein Bottle
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““Inverted Double-SockInverted Double-Sock”” Klein Bottle Klein Bottle
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Rendered with Vivid 3D (Claude Mouradian)Rendered with Vivid 3D (Claude Mouradian)
http://netcyborg.free.fr/
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Klein Bottles Based on KOJKlein Bottles Based on KOJ(in the same class as the “Inverted Sock”)(in the same class as the “Inverted Sock”)
Always an odd number of “turn-back mouths”!
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A Gridded Model of A Gridded Model of Trefoil KnottleTrefoil Knottle