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Exploring Möbius Bands and Klein Bottles: Unique Surfaces Unveiled

Dive into the intriguing world of Möbius bands and Klein bottles, unique mathematical surfaces with complex properties and topological distinctions. Learn about their deformations, constructions, homotopy classes, and artistic representations. Delve into the structural intricacies and classifications of these captivating surfaces. Discover the fascinating connections between Möbius bands and Klein bottles, unravelling the mysteries of their twisted geometries in the realm of mathematics and art.

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Exploring Möbius Bands and Klein Bottles: Unique Surfaces Unveiled

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  1. Bridges 2012 From Möbius Bands to Klein Knottles Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

  2. What is a Möbius Band ? • A single-sided surface with a single edge: A closed ribbon with a 180° flip. The “Sue-Dan-ese” M.B., a “bottle” with circular rim.

  3. Deformation of a Möbius Band (ML)-- changing its apparent twist +180°(ccw), 0°, –180°, –540°(cw) Apparent twist, compared to a rotation-minimizing frame (RMF) Measure the built-in twist when sweep path is a circle!

  4. Twisted Möbius Bands in Art Web Max Bill M.C. Escher M.C. Escher

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

  6. What is a 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( only immersions, no embeddings )

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

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

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

  10. 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."

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

  12. Classical “Inverted-Sock” Klein Bottle

  13. Figure-8 Klein Bottle

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

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

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

  17. Yet Another Way to Match-up Numbers

  18. The New “Double-Sock” Klein Bottle

  19. The New “Double-Sock” Klein Bottle

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

  21. The 4th Klein Bottle ?? • There are 22-χ distinct regular homotopy classes of immersions of a surface of Euler characteristic χ into R3. • Thus there must be 4 distinct Klein bottle types that cannot be transformed smoothly into one another. J. Hass and J. Hughes, Immersions of Surfaces in 3-Manifolds. Topology, Vol.24, No.1, pp 97-112, 1985. • The first 3 Klein bottles presented clearly belong to three different regular homotopy classes.

  22. Lawson’s Minimum Energy Klein Bottle

  23. Klein Bottle Analysis • A regular homotopy cannot change the twist of a MB. Thus, left-twisting bands stay left-twisting, and right-twisting ones stay right-twisting! • K8L and K8R have chirality. They are mirror images of one another! • But so does the Lawson KB! Thus, there are two different Lawson KBs. • So – if the Lawson Klein bottle were something new,then there would be TWO new bottle types. • But this cannot be; there are only four types total; thus the Lawson bottles transform into K8R and K8L.

  24. “Double Sock” is NOT #4! • It turns out the “Double-Sock K.B.” also has chirality! • And thus it also comes in two forms that transform into the respective K8R or K8L. • Thus is cannot play the role of #4. • Therefore, we need to look for a K.B. made of ML + MR to serve as #4. • Thus #4 structurally belongs into the class KOJ. • It can only be distinguished from the classical KOJ, if we place some markings on its surface.

  25. Regular Homotopy Classes for Tori

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

  27. Klein Bottle: Regular Homotopy Classes

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

  29. Fancy Klein Bottles of Type KOJ Cliff Stoll Klein bottles by Alan Bennett in the Science Museum in South Kensington, UK

  30. Beyond Ordinary Klein Bottles Glass sculptures by Alan Bennett Science Museum in South Kensington, UK

  31. Klein Knottles Based on KOJ Always an odd number of “turn-back mouths”!

  32. A Gridded Model of Trefoil Knottle

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

  34. Klein Knottles with Fig.8 Crosssections

  35. A Gridded Model of Figure-8 Trefoil

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

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

  38. Summary of Findings • Klein bottles are closely related to Möbius bands:every bottle is composed of two bands. • Structurally, there are three different types of K-Bsthat can’t be smoothly transformed into one another. • When considering marked (textured) surfaces, “inverted sock” Klein bottle splits into 2 different types:( arrows going into, or coming out of its mouth ).

  39. Conclusions • Klein bottles are fascinating surfaces. • They come in a wide variety of shapes,which are not always easy to analyze. • Many of these shapes make attractive constructivist sculptures . . .

  40. === Questions ? ===

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