Bridges 2012 From Mbius Bands to Klein Knottles

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

What is a Möbius Band ? u 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.

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!

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

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

What is a Klein Bottle ? u. A single-sided surface u with no edges or punctures u with Euler characteristic: V – E + F = 0 u corresponding u always to: genus = 2 self-intersecting in 3 D ( only immersions, no embeddings )

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 one sock”

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. "

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

Classical “Inverted-Sock” Klein Bottle

Figure-8 Klein Bottle

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 MR + MR = K 8 R ML + ML = K 8 L

Yet Another Way to Match-up Numbers

The New “Double-Sock” Klein Bottle

The New “Double-Sock” Klein Bottle

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

The 4 th Klein Bottle ? ? u There are 22 -χ distinct regular homotopy classes of immersions of a surface of Euler characteristic χ into R 3. u 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. u The first 3 Klein bottles presented clearly belong to three different regular homotopy classes.

Lawson’s Minimum Energy Klein Bottle

Klein Bottle Analysis u A regular homotopy cannot change the twist of a MB. Thus, left-twisting bands stay left-twisting, and right-twisting ones stay right-twisting! u K 8 L and K 8 R have chirality. They are mirror images of one another! u But so does the Lawson KB! Thus, there are two different Lawson KBs. u So – if the Lawson Klein bottle were something new, then there would be TWO new bottle types. u But this cannot be; there are only four types total; thus the Lawson bottles transform into K 8 R and K 8 L.

“Double Sock” is NOT #4! u It turns out the “Double-Sock K. B. ” also has chirality! u And thus it also comes in two forms that transform into the respective K 8 R or K 8 L. u Thus is cannot play the role of #4. u Therefore, we need to look for a K. B. made of ML + MR to serve as #4. u Thus #4 structurally belongs into the class KOJ. u It can only be distinguished from the classical KOJ, if we place some markings on its surface.

Regular Homotopy Classes for Tori

Decorated Klein Bottles 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

Klein Bottle: Regular Homotopy Classes

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

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

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

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

A Gridded Model of Trefoil Knottle

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

Klein Knottles with Fig. 8 Crosssections

A Gridded Model of Figure-8 Trefoil

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

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

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

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

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