Bay Area Science Festival, 2013

1. Bay Area Science Festival, 2013 Magic of Klein Bottles Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

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

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

4. 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 !

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

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

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

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

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

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

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

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

13. 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.

14. (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.

15. When are 2 Klein Bottles the Same?

16. When are 2 Klein Bottles the Same?

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

18. 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."

19. A Twisted Klein Bottle Split it along a twisted longitudinal grid line . . .

20. Split Klein Bottle  Two Moebius Bands

21. Yet Another Way to Match-up Numbers

22. “Inverted Double-Sock” Klein Bottle

23. “Inverted Double-Sock” Klein Bottle

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

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

26. A Gridded Model of Trefoil Knottle