Transcript PPT - Computer Science Division - University of California, Berkeley

Bay Area Science Festival, 2013
Magic of Klein Bottles
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley
Classical “Inverted-Sock” Klein Bottle
 Type
“KOJ”:
K: Klein bottle
 O:
 J:
tube profile
overall tube shape
Several Fancy Klein Bottles
Cliff Stoll
Klein bottles by Alan Bennett
in the Science Museum in South Kensington, UK
What is a Klein Bottle ?
A
single-sided surface
 with
 It
can be made made from a rectangle:
 with

no edges or punctures.
Euler characteristic: V – E + F = 0
It is always self-intersecting in 3D !
How to Make a Klein Bottle (1)
make a “tube”
by merging the horizontal edges
of the rectangular domain
 First
How to Make a Klein Bottle (2)
 Join
tube ends with reversed order:
How to Make a Klein Bottle (3)
 Close
ends smoothly
by “inverting sock end”
Figure-8 Klein Bottle
 Type
“K8L”:
K: Klein bottle
 8:
tube profile
 L:
left-twisting
Making a Figure-8 Klein Bottle (1)
make a “figure-8 tube”
by merging the horizontal edges
of the rectangular domain
 First
Making a Figure-8 Klein Bottle (2)
 Add
a 180° flip to the tube
before the ends are merged.
Two Different Figure-8 Klein Bottles
Right-twisting
Left-twisting
The Rules of the Game: Topology

Shape does not matter -- only connectivity.

Surfaces can be deformed continuously.
Smoothly 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) Homotopy
With these rules:

Two shapes are called homotopic,
if they can be transformed into one another
with a continuous smooth deformation
(with no kinks or singularities).

Such shapes are then said to be:
in the same homotopy class.
When are 2 Klein Bottles the Same?
When are 2 Klein Bottles the Same?
2 Möbius Bands Make a Klein Bottle
KOJ
=
MR
+
ML
Limerick
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."
A Twisted Klein Bottle
Split it along a twisted longitudinal grid line . . .
Split Klein Bottle  Two Moebius Bands
Yet Another Way to Match-up Numbers
“Inverted Double-Sock” Klein Bottle
“Inverted Double-Sock” Klein Bottle
Rendered with Vivid 3D (Claude Mouradian)
http://netcyborg.free.fr/
Klein Bottles Based on KOJ
(in the same class as the “Inverted Sock”)
Always an odd number of “turn-back mouths”!
A Gridded Model of Trefoil Knottle