Bridges 2012Bridges 2012

From Möbius Bands to Klein Knottles

EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley

Carlo H. Séquin

What is a What is a Möbius Band 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.

Deformation of a Möbius Band (ML)Deformation of a Möbius Band (ML)-- changing its apparent twist-- 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 ArtTwisted Möbius Bands in Art

Web Max Bill M.C. Escher M.C. Escher

The Two Different Möbius BandsThe Two Different Möbius Bands

ML and MR

are in two different regular homotopy classes!

What is a What is a Klein Bottle 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 )

How to Make a How to Make a Klein Bottle (1)Klein Bottle (1)

First make a “tube” by merging the horizontal edges of the rectangular domain

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

How to Make a How to Make a Klein Bottle (3)Klein Bottle (3)

Close ends smoothly by “inverting one sock”

LimerickLimerick

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 Bottle2 Möbius Bands Make a Klein Bottle

KOJ = MR + ML

Classical Classical ““Inverted-SockInverted-Sock”” Klein Bottle Klein Bottle

Figure-8 Klein BottleFigure-8 Klein Bottle

Making a Making a Figure-8Figure-8 Klein Bottle (1)Klein Bottle (1)

First make a “figure-8 tube” by merging the horizontal edges of the rectangular domain

Making a Making a Figure-8Figure-8 Klein Bottle (2)Klein Bottle (2)

Add a 180° flip to the tubebefore the ends are merged.

Two Different Figure-8 Klein BottlesTwo Different Figure-8 Klein Bottles

MR + MR = K8R

ML + ML = K8L

Yet Another Way to Match-up NumbersYet Another Way to Match-up Numbers

The New The New ““Double-SockDouble-Sock”” Klein Bottle Klein Bottle

The New The New ““Double-SockDouble-Sock”” Klein Bottle Klein Bottle

Rendered with Vivid 3D (Claude Mouradian)Rendered with Vivid 3D (Claude Mouradian)

http://netcyborg.free.fr/

The 4The 4thth Klein Bottle ?? 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.

LawsonLawson’’s Minimum Energy Klein Bottles Minimum Energy Klein Bottle

Klein Bottle AnalysisKlein 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.

““Double SockDouble Sock”” is NOT #4! 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.

Regular Homotopy Classes for ToriRegular Homotopy Classes for Tori

Decorated Klein BottlesDecorated Klein Bottles The 4th type can only be distinguished through

its surface decoration (parameterization)!

Arrows comeout of hole

Arrows gointo hole

Added collaron KB mouth

Klein Bottle: Regular Homotopy ClassesKlein Bottle: Regular Homotopy Classes

Which Type of Klein Bottle Do We Get?Which Type of Klein Bottle Do We Get?

It depends which of the two ends gets narrowed down.

Fancy Klein Bottles of Type KOJFancy Klein Bottles of Type KOJ

Cliff Stoll Klein bottles by Alan Bennett in the Science Museum in South Kensington, UK

Beyond Ordinary Klein BottlesBeyond Ordinary Klein Bottles

Glass sculptures by Alan Bennett Science Museum in South Kensington, UK

Klein Klein KnottlesKnottles Based on KOJ Based on KOJ

Always an odd number of “turn-back mouths”!

A Gridded Model of A Gridded Model of Trefoil KnottleTrefoil Knottle

Not a Klein Bottle – But a Torus !Not a Klein Bottle – But a Torus !

An even number of surface reversals renders the surface double-sided and orientable.

Klein Knottles with Fig.8 CrosssectionsKlein Knottles with Fig.8 Crosssections

A Gridded Model of A Gridded Model of Figure-8 TrefoilFigure-8 Trefoil

Rendered with Vivid 3D (Claude Mouradian)Rendered with Vivid 3D (Claude Mouradian)

http://netcyborg.free.fr/

Rendered with Vivid 3D (Claude Mouradian)Rendered with Vivid 3D (Claude Mouradian)

FDM Model

http://netcyborg.free.fr/

Summary of FindingsSummary 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 ).

ConclusionsConclusions

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 . . .

=== Questions ? ====== Questions ? ===