Celebration of the Mind MSRI 2013 Weird Klein
Celebration of the Mind, MSRI, 2013 Weird Klein Bottles Carlo H. Séquin EECS Computer Science Division University of California, Berkeley
Several Fancy Klein Bottles Cliff Stoll Klein bottles by Alan Bennett in the Science Museum in South Kensington, UK
Not a Klein Bottle – But a Torus ! u An even number of surface reversals renders the surface double-sided and orientable.
Which Ones are Klein Bottles ? ? Glass sculptures by Alan Bennett Science Museum in South Kensington, UK
What is a Klein Bottle ? u. A single-sided surface u with u It can be made from a rectangle: u with è no edges or punctures. Euler characteristic: V – E + F = 0 It is always self-intersecting in 3 D !
How to Make a Klein Bottle (1) u First make a “tube” by merging the horizontal edges of the rectangular domain
How to Make a Klein Bottle (2) u Join tube ends with reversed order:
How to Make a Klein Bottle (3) u Close ends smoothly by “inverting sock end”
Classical “Inverted-Sock” Klein Bottle u Type “KOJ”: K: Klein bottle u O: u J: tube profile overall tube shape
Figure-8 Klein Bottle u Type “K 8 L”: K: Klein bottle u 8: tube profile u L: left-twisting
Making a Figure-8 Klein Bottle (1) u First make a “figure-8 tube” by merging the horizontal edges of the rectangular domain
Making a Figure-8 Klein Bottle (2) u 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 u Shape does not matter -- only connectivity. u Surfaces can be deformed continuously.
Smoothly Deforming Surfaces u Surface may pass through itself. u It cannot be cut or torn; it cannot change connectivity. u It must never form any sharp creases or points of infinitely sharp curvature. OK
(Regular) Homotopy With these rules: u Two shapes are called homotopic, if they can be transformed into one another with a continuous smooth deformation (with no kinks or singularities). u 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. "
Deformation of a Möbius Band (ML) -- changing its apparent twist +180°(ccw), 0°, – 180°(cw) – 540°(cw) Apparent twist, compared to a rotation-minimizing frame (RMF) Measure the built-in twist when sweep path is a circle!
The Two Different Möbius Bands ML and MR are in two different regular homotopy classes!
Two Different Figure-8 Klein Bottles MR + MR = K 8 R ML + ML = K 8 L
Klein Bottle Analysis u Cut the Klein bottle into two Möbius bands and look at the twists of the two. u Now we can see that there must be at least 3 different types of Klein bottles in 3 different regular homotopy classes: ML + MR; ML + ML; MR + MR.
“Inverted Double-Sock” Klein Bottle
Rendered with Vivid 3 D (Claude Mouradian) http: //netcyborg. free. fr/
Yet Another Way to Match-up Numbers
“Inverted Double-Sock” Klein Bottle
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
Klein Knottles with Fig. 8 Crosssections Triply twisted trefoil knot 6 -pointed fig. 8 star Twisted fig. 8 zig-zag
A Gridded Model of Figure-8 Trefoil
Decorated Klein Bottles: 4 TYPES ! u The 4 th type can only be distinguished through its surface decoration (parameterization)! Arrows come out of hole Added collar on KB mouth Arrows go into hole
Which Type of Klein Bottle Do We Get? u It depends which of the two ends gets narrowed down.
Klein Bottle: Regular Homotopy Classes
QUESTIONS ?
- Slides: 36
