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Weird and Wonderful World of Klein Bottles: A Celebration of Mathematical Artistry

Explore the mind-bending realm of Klein Bottles at MSRI 2013. Discover fanciful designs by Carlo H. Séquin and Alan Bennett, from the University of California, Berkeley to the Science Museum in UK. Unravel the mysteries of these single-sided, self-intersecting surfaces. Learn how to create Classical and Figure-8 Klein Bottles. Delve into the rules of topology and homotopy, with limericks and analyses to help you understand these unique mathematical objects. Experience the artistry of Klein Bottles through intricate glass sculptures and gridded models. Find out how subtle twists and turns can lead to intriguing variations. Join us as we celebrate the mathematical beauty of Klein Bottles in all their odd, intriguing glory.

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Weird and Wonderful World of Klein Bottles: A Celebration of Mathematical Artistry

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  1. Celebration of the Mind, MSRI, 2013 Weird Klein Bottles Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

  2. Several Fancy Klein Bottles Cliff Stoll Klein bottles by Alan Bennett in the Science Museum in South Kensington, UK

  3. An even number of surfacereversalsrenders the surface double-sided and orientable. Not a Klein Bottle – But a Torus !

  4. Which Ones are Klein Bottles ?? Glass sculptures by Alan Bennett Science Museum in South Kensington, UK

  5. What is a 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 !

  6. First make a “tube”by merging the horizontal edges of the rectangular domain How to Make a Klein Bottle (1)

  7. Join tube ends with reversed order: How to Make a Klein Bottle (2)

  8. How to Make a Klein Bottle (3) • Close ends smoothly by “inverting sock end”

  9. Classical “Inverted-Sock” Klein Bottle • Type “KOJ”:K: Klein bottle • O: tube profile • J: overall tube shape

  10. Figure-8 Klein Bottle • Type “K8L”:K: Klein bottle • 8: tube profile • L: left-twisting

  11. First make a “figure-8 tube”by merging the horizontal edges of the rectangular domain Making a Figure-8Klein Bottle (1)

  12. Making a Figure-8Klein Bottle (2) • Add a 180° flip to the tubebefore the ends are merged.

  13. Two Different Figure-8 Klein Bottles Right-twisting Left-twisting

  14. The Rules of the Game: Topology • Shape does not matter -- only connectivity. • Surfaces can be deformed continuously.

  15. Smoothly Deforming Surfaces OK • 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.

  16. (Regular) Homotopy With these rules: 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.

  17. When are 2 Klein Bottles the Same?

  18. When are 2 Klein Bottles the Same?

  19. KOJ = MR + ML 2 Möbius Bands Make a Klein Bottle

  20. 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."

  21. 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!

  22. The Two Different Möbius Bands ML and MR are in two different regular homotopy classes!

  23. Two Different Figure-8 Klein Bottles MR + MR = K8R ML + ML = K8L

  24. Klein Bottle Analysis • Cut the Klein bottle into two Möbius bandsand look at the twists of the two. • Now we can see that there must be at least 3 different types of Klein bottlesin 3 different regular homotopy classes:ML + MR; ML + ML; MR + MR.

  25. “Inverted Double-Sock” Klein Bottle

  26. Rendered with Vivid 3D (Claude Mouradian) http://netcyborg.free.fr/

  27. Yet Another Way to Match-up Numbers

  28. “Inverted Double-Sock” Klein Bottle

  29. Klein Bottles Based on KOJ(in the same class as the “Inverted Sock”) Always an odd number of “turn-back mouths”!

  30. A Gridded Model of Trefoil Knottle

  31. Klein Knottles with Fig.8 Crosssections Triply twistedtrefoil knot 6-pointedfig.8 star Twisted fig.8zig-zag

  32. A Gridded Model of Figure-8 Trefoil

  33. Decorated Klein Bottles: 4 TYPES ! • The 4th type can only be distinguished through its surface decoration (parameterization)! Arrows comeout of hole Added collaron KB mouth Arrows gointo hole

  34. Which Type of Klein Bottle Do We Get? • It depends which of the two ends gets narrowed down.

  35. Klein Bottle: Regular Homotopy Classes

  36. Q U E S T I O N S ?

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