Bay Area Science Festival, 2013Bay Area Science Festival, 2013

Magic of Klein Bottles

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

Carlo H. Séquin

Type “KOJ”:

K: Klein bottle

O: tube profile

J: overall tube shape

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

Several Fancy Klein BottlesSeveral Fancy Klein Bottles

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

What is a What is a Klein Bottle 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 !

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 sock end”

Type “K8L”:

K: Klein bottle

8: tube profile

L: left-twisting

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

Right-twisting Left-twisting

The Rules of the Game: The Rules of the Game: TopologyTopology

Shape does not matter -- only connectivity.

Surfaces can be deformed continuously.

Smoothly Deforming SurfacesSmoothly Deforming Surfaces

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.

OK

(Regular) Homotopy(Regular) Homotopy

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.

With these rules:

When are 2 Klein Bottles the Same?When are 2 Klein Bottles the Same?

When are 2 Klein Bottles the Same?When are 2 Klein Bottles the Same?

2 Möbius Bands Make a Klein Bottle2 Möbius Bands Make a Klein Bottle

KOJ = MR + ML

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."

A Twisted Klein BottleA Twisted Klein Bottle

Split it along a twisted longitudinal grid line . . .

Split Klein Bottle Split Klein Bottle Two Moebius Bands Two Moebius Bands

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

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

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

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

http://netcyborg.free.fr/

Klein Bottles Based on KOJKlein Bottles Based on KOJ(in the same class as the “Inverted Sock”)(in the same class as the “Inverted Sock”)

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

A Gridded Model of A Gridded Model of Trefoil KnottleTrefoil Knottle