Bay Area Science Festival, 2013 ISAMA 2004 Bay Area Science Festival, 2013 Magic of Klein Bottles Klein bottles by Cliff Stoll Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

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 isthe 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.

Several Fancy Klein Bottles ISAMA 2004 Several Fancy Klein Bottles The geometry of this bottle can change drastically; but these are all honest Klein bottles! Cliff Stoll Klein bottles by Alan Bennett in the Science Museum in South Kensington, UK

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 ! 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 …

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!

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).

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”

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:

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 . . .

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.

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

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. 12

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.

(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. 14

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!

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?

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

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:

ISAMA 2004 A Twisted Klein Bottle We can also make a twisted cut . . . Split it along a twisted longitudinal grid line . . .

Split Klein Bottle  Two Moebius Bands ISAMA 2004 Split Klein Bottle  Two Moebius Bands It still results in two Moebius bands, but now they are interlinked, and one of them is triply twisted!

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 . . .

“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.

“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.

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. http://netcyborg.free.fr/

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”!

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.