1. Celebration of the Mind, MSRI, 2013
ISAMA 2004
Celebration of the Mind, MSRI, 2013
Weird Klein Bottles
From previous talk by Cliff Stoll you should already have a good understanding of what a Klein bottle may look like.
These are all honest Klein bottles!
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley

2. Several Fancy Klein Bottles
ISAMA 2004
Several Fancy Klein Bottles
These are all honest Klein bottles!
Cliff Stoll Klein bottles by Alan Bennett in the Science Museum in South Kensington, UK

3. Not a Klein Bottle – But a Torus !
ISAMA 2004
Not a Klein Bottle – But a Torus !
You may also have learned that not every inverted sock structure is a Klein bottle.
Each “turn-back mouth” produces an inside-out surface reversals.
With an even number of such reversals, the surface stays orientable, so it is then a torus !
>>> This is also a torus !
An even number of surface reversals renders the surface double-sided and orientable.

4. Which Ones are Klein Bottles ??
ISAMA 2004
Which Ones are Klein Bottles ??
But Alan Bennett has made many more, rather fancy glass sculptures. Are these real Klein bottles? – Not really!
The first image shows three Klein bottles of the classical inverted sock type, nested within one another.
What about the other two images ? How can we tell ?
(They show 2-handle and 3-handle KBs, respectively; These are really objects with a genus higher than 2 – that of a regular Klein bottle! )
Glass sculptures by Alan Bennett Science Museum in South Kensington, UK

5. What is a Klein Bottle ? A single-sided surface
ISAMA 2004
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 !
So lets review: What exactly is a Klein bottle?
Most importantly: KB is a single sided surface. --But not everything that is single sided is a Klein bottle: Moebius bands, Projective plane, Boy’s surface, Steiner Surface . . .
The surface must have no edges or punctures. It can always be constructed from a rectangular domain by joining its edges as indicated by the colored arrows.
If we draw some kind of mesh on this surface and then count the vertices V, the edges E and the facets F and form the expression V – E + F then the sum must be zero.
No embeddings of this object in 3D are possible! There will always be some self-intersections.
--- But this may be a little too abstract …

6. How to Make a Klein Bottle (1)
ISAMA 2004
How to Make a Klein Bottle (1)
First make a “tube” by merging the horizontal edges of the rectangular domain
Let’s visualize that construction process:
Start with the rectangular domain shown at left. First we form a tube by joining the horizontal (green) edges:
Then we need to join the vertical (brown) edges -- but they have reversed orientation! So we cannot just join the two ends into a donut!

7. How to Make a Klein Bottle (2)
ISAMA 2004
How to Make a Klein Bottle (2)
Join tube ends with reversed order:
[Then we need to join the vertical (brown) edges -- but they have reversed orientation! Note that the number sequences run in opposite directions!]
-- But we can properly join matching numbers to one another by narrowing down the left end of the tube and sticking it in sideways into the larger right tube end. This allows us to properly line up all the number labels (as shown on the right).

8. How to Make a Klein Bottle (3)
ISAMA 2004
How to Make a Klein Bottle (3)
Now we merge these two concentric ends smoothly by “inverting the end of the sock”; i.e., by turning the inner tube inside-out and fusing it to the outer tube.
Close ends smoothly by “inverting sock end”

9. Classical “Inverted-Sock” Klein Bottle
ISAMA 2004
Classical “Inverted-Sock” Klein Bottle
Type “KOJ”: K: Klein bottle
O: tube profile
J: overall tube shape
This construction results in the classical, “inverted sock” Klein Bottle.
Here one small ribbon on its surface has been singled out, painted in green and orange, and enhanced with surface-normal arrows to show that it is indeed a single-sided, “non-orientable” Möbius band. I give it the formal type specification KOJ.
Most of you are probably familiar with this geometry.

10. Figure-8 Klein Bottle Type “K8L”: K: Klein bottle 8: tube profile
ISAMA 2004
Figure-8 Klein Bottle
Type “K8L”: K: Klein bottle
8: tube profile
L: left-twisting
But there are other ways to construct a Klein bottle – and this is also perfectly good Klein bottle.
Let’s see how this was constructed:

11. Making a Figure-8 Klein Bottle (1)
ISAMA 2004
Making a Figure-8 Klein Bottle (1)
First make a “figure-8 tube” by merging the horizontal edges of the rectangular domain
When merging the two horizontal green edges, we are not forced to form a round, circular tube. Instead we may give it a figure-8 cross section. Note what has happened to the labeling: The label “2” is on top on the left, but at the bottom on the right. So to bring these numbers together, we have to twist this tube . . .

12. Making a Figure-8 Klein Bottle (2)
ISAMA 2004
Making a Figure-8 Klein Bottle (2)
After we have given the tube a 180° twist, the two ends can readily be merged so that the numbers match up.
The result is a twisted ring with a figure-8 profile.
Add a 180° flip to the tube before the ends are merged.

13. Two Different Figure-8 Klein Bottles
ISAMA 2004
Two Different Figure-8 Klein Bottles
Depending on whether we twist the tube to the left or to the right before closure, we obtain two different Klein bottles.
Should we consider those two to be the same ?? But they certainly look quite different from the inverted sock Klein bottle !
Right-twisting Left-twisting

14. The Rules of the Game: Topology
ISAMA 2004
The Rules of the Game: Topology
We have to establish the rules of the game! We look at these bottles from a topology point of view!
>>> In topology detailed shape is not important -- only connectivity.
Surfaces can be deformed continuously. So, topologically, the donut and the coffee mug are equivalent.
Shape does not matter -- only connectivity.
Surfaces can be deformed continuously. 14

15. Smoothly Deforming Surfaces
ISAMA 2004
Smoothly 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
For our analysis we go even further: We allow a piece of surface to be stretched arbitrarily, and it can even pass through itself !
However, - it cannot be torn or cut to change its connectivity, and it must never form a sharp crease or any point of infinitely sharp curvature.
Surfaces that can be transformed into each other in this way are said to be in the same regular homotopy class.

16. (Regular) Homotopy With these rules:
ISAMA 2004
(Regular) Homotopy
With these rules:
Two shapes are called homotopic, if they can be transformed into one another with a continuous smooth deformation (with no kinks or singularities).
Such shapes are then said to be: in the same homotopy class.
Homotopy is the one important math-word for this talk! You want to remember this so you can go and impress your friends!
Two shapes are called regular homotopic, if they can be transformed into one another with a continuous smooth deformation forming no kinks or singularities.
Such shapes are then said to be in the same homotopy class. 16

17. When are 2 Klein Bottles the Same?
ISAMA 2004
When are 2 Klein Bottles the Same?
Under those rules, all of the Klein bottles depicted here are actually in the same regular homotopy class!

18. When are 2 Klein Bottles the Same?
ISAMA 2004
When are 2 Klein Bottles the Same?
But these are not all the same. Exactly one of them is different – but which one ? -- and why?

19. 2 Möbius Bands Make a Klein Bottle
ISAMA 2004
2 Möbius Bands Make a Klein Bottle
Here is a very useful insight for our analysis: A Klein bottle can always be split into two Moebius bands!
-- as demonstrated here on the example of the classical Klein bottle shown on the left.
The lower yellow half is a right-twisted Moebius band.
The upper blue half – here shown flipped open to the right -- is a left-twisted Moebius band.
The two are mirror images of one another.
KOJ = MR ML

20. ISAMA 2004
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."
Every Klein bottle can be de-composed into two Moebius bands as expressed in this Limerick:

21. Deformation of a Möbius Band (ML) -- changing its apparent twist
ISAMA 2004
Deformation of a Möbius Band (ML) -- changing its apparent twist
+180°(ccw), °, –180°(cw) –540°(cw)
Apparent twist, compared to a rotation-minimizing frame (RMF)
Lets look closer at the regular homotopy transformations that we can apply to a Moebius band: Here is a left-twisting Moebius band made out of some magical, rubbery material, that can pass through itself, but cannot bend very sharply. In the first figure the 180° left twist can be clearly seen. Now as we change the sweep path of this band, the observed twist – as compared to a RMF – seems to change: By the time the path forms a self-intersecting figure-8 loop, the twist seems to have changed to clock-wise 180°! And if we complete the figure-8 crossover move to the right-most figure, we observe 540° of clockwise twist. >>> To obtain unambiguous results about the built-in twist, we should always measure twist when the sweep path is circular. We should measure twist modulo 720°; because we can always add or subtract that much twist with this figure-8 cross-over move. But we cannot change twist by 360 degrees! Thus a left-turning band cannot be transformed into a right turning band. They are in 2 different homotopy classes.
Measure the built-in twist when sweep path is a circle!

22. The Two Different Möbius Bands
ISAMA 2004
The Two Different Möbius Bands
The key point: if we take the mirror image of a Moebius band, then we obtain a new shape that is in a different regular homotopy class.
These two shapes cannot be smoothly transformed into one another! And that is the crucial point for our analysis!
ML and MR are in two different regular homotopy classes!

23. Two Different Figure-8 Klein Bottles
ISAMA 2004
Two Different Figure-8 Klein Bottles
MR MR = K8R
Now we apply this insight to the Fig8 Klein bottles. On top you see that the right-twisting KB splits into 2 right-twisting MBs, and the left-twisting KB, decomposes into 2 left-twisting MBs. Thus these two bottle types are in different regular homotopy classes. They cannot be smoothly transformed into one another.
ML ML = K8L

24. ISAMA 2004
Klein Bottle Analysis
Cut the Klein bottle into two Möbius bands and look at the twists of the two.
Now we can see that there must be at least 3 different types of Klein bottles in 3 different regular homotopy classes: ML + MR; ML + ML; MR + MR.
So when we encounter some unknown Klein bottle we can see which class it belongs into by analyzing its two Moebius bands.
We can state immediately that there must be at least 3 different types of bottles given by the 3 different combinations of 2 Moebius bands.
Let’s test this on some crazy new Klein bottle . . .

25. “Inverted Double-Sock” Klein Bottle
ISAMA 2004
“Inverted Double-Sock” Klein Bottle
Here is a Klein bottle that you probably have never seen before. I call it the “Inverted Double Sock” Klein bottle.

26. Rendered with Vivid 3D (Claude Mouradian)
ISAMA 2004
Rendered with Vivid 3D (Claude Mouradian)
And here is the beautiful rendering of it by Claude Mouradian.

27. Yet Another Way to Match-up Numbers
ISAMA 2004
Yet Another Way to Match-up Numbers
Let’s see how this was constructed: We again start with a tube with a figure-8 shaped profile. But now rather than twisting tis tube through 180 degrees to make the numbers match up, we can form an “inverted double-sock” to bring all labels into alignment. But to make a nice rounded closure cap we need to add one more modification: We make one lobe of the figure-8 somewhat smaller, and the other one somewhat larger. Gradually we let the larger end morph into the smaller one as we travel along the tube, and vice versa. Now the two asymmetrical figure-8 profiles can be nested nicely with some substantial separation between them . . .

28. “Inverted Double-Sock” Klein Bottle
ISAMA 2004
“Inverted Double-Sock” Klein Bottle
And now we can close off this double-mouth with a nicely rounded, figure-8 shaped end cap. This results in this ” Inverted Double-sock” Klein bottle with a figure-8 profile.
On the right, you see the blue half of this bottle. This by itself forms a Moebius band – in this case, its edge forms a double loop that coincides with itself.
And when we compare the blue half and the yellow half, we find that they are both exactly the same.
So both are either right-twisting or left twisting Moebius bands. If we form the mirror image, we get from one type to the other!
These bottles are regular hoimotopically identical to the twisted figure-8 Klein bottles.

29. Klein Bottles Based on KOJ (in the same class as the “Inverted Sock”)
ISAMA 2004
Klein Bottles Based on KOJ (in the same class as the “Inverted Sock”)
Here a bunch of unusual knotted Klein bottles – all regular homotopically equivalent to the Inverted Sock.
Note, that all these knottles have an odd number of KB turn-back mouths where an inside-out surface reversal takes place.
Always an odd number of “turn-back mouths”!

30. A Gridded Model of Trefoil Knottle
ISAMA 2004
A Gridded Model of Trefoil Knottle
And here is another view and a physical model of one of those bottles. It forms a trefoil knot. So I call it a Klein knottle.
It has been fabricated on a FDM machine.

31. Klein Knottles with Fig.8 Crosssections
ISAMA 2004
Klein Knottles with Fig.8 Crosssections
Here are a bunch of weird and unusual Klein bottles with figure-8 cross sections. They are all chiral and thus homotopically equivalent to the simple twisted figure-8 Klein bottle. The first is a twisted figure-8 ring, but now entangled as a trefoil knot. ---
The next two sculptures are intriguing new structures – all making use of the figure-8 shaped “inverted Double Sock” turn-back mouth.
===
Interestingly, they have an even number of KB mouths! -- But they are still single-sided Klein bottles! – How can this be? -- Well the trick is the same that is used by the twisted figure-8 ring. This one has zero KB-mouths – which is an even number. But it gets its single-sidedness by introducing a flip through an odd multiple of 180° -- this bring one side of the original 2-sided tube surface into contact with the other side. This is what I am doing here: by judiciously managing the twisting of the various tube segments, I make sure that in the end I will have a net reversal of the surface when the loop closes. In the middle picture I twist the tube segments 6 times through 90°, and in the 3rd case: I twist it twice through 90° and twice trough 0° degrees.
Triply twisted trefoil knot
6-pointed fig.8 star
Twisted fig.8 zig-zag

32. A Gridded Model of Figure-8 Trefoil
ISAMA 2004
A Gridded Model of Figure-8 Trefoil
And here again is a gridded physical model of the triply-twisted, trefoil-knotted, figure-8 Klein bottle.

33. Decorated Klein Bottles: 4 TYPES !
ISAMA 2004
Decorated Klein Bottles: 4 TYPES !
The 4th type can only be distinguished through its surface decoration (parameterization)!
But this is only part of the story. Things get a bit more complicated if we not just consider color-less glass surfaces, but allow surfaces with some decoration on it or some explicit coordinate system. A paper by Hass and Hughes tells us that then we have to distinguish FOUR different types of Klein bottles!
For instance if we have a texture with these longitudinal arrows, then the bottle in which the arrows come out of the mouth is different from the one in which the arrows point into the mouth.
===
A regular homotopy operation cannot make a smooth transformation between these two states.
-- The way I found this distinction was by experimenting with a “graft” operation that added an extra collar around the mouth of the bottle. The extra self-intersection can then be removed by inflating the blue-to-purple branch of this bottle.
So, if we call the figure on the left the bottle type #1, then the one on the right would be bottle type #4.
Arrows come out of hole
Added collar on KB mouth
Arrows go into hole

34. Which Type of Klein Bottle Do We Get?
ISAMA 2004
Which Type of Klein Bottle Do We Get?
And which type of Klein bottle we get is determined when we decide which of the two ends of our circular tube we will narrow down!
In one case the upper half of this Klein bottle, marked with the white arrows, will form a left-twisting Moebius band; in the other case it will be right twisting.
The bottom half is the mirror image of the upper half.
It depends which of the two ends gets narrowed down.

35. Klein Bottle: Regular Homotopy Classes
ISAMA 2004
Klein Bottle: Regular Homotopy Classes
So, in conclusion, here is the complete map of the world of Klein bottles.
At the top is the classical inverted sock. At the bottom : the two twisted figure-8 rings.
There are three structural domains.
If we also consider surface decoration, then the top domain splits into two sub-domains where the bottles are mirror images of one another.
Without decoration they are indistinguishable, since the classical Inverted Sock Klein bottle is its own mirror image.

36. Q U E S T I O N S ?