Can a Klein Bottle be cut up into Mobius strips with no leftover pieces?

Question

Can a Klein Bottle be cut up into Mobius strips with no leftover pieces?

Answer 1

A Mobius strip is a surface homeomorphic to M={(s,t) in [0,1]^2;((2*t-1)*cos(pi*s),

(2*t-1)*sin(pi*s),s)}. A Klein Bottle is what results when two Mobius strips are connected at their edges.

Answer 2

If you can figure out how to sew two Mobius strips together, I can make you a pretty good imitation of a Klein bottle.



[I always get stuck when I try. I seem to need an extra dimension]



Edward: Been really happy with a bottle Boggle Old Vine Zinfandel recently, suggest you go buy a real bottle instead of salvating over something in an imaginary container ;-)



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Cat, I think you, me, Edward and Kittana would really enjoy one or several bottles of wine together. Should we invite our favorite bug-eyed, bug eating lizard friend? We could also invite Wombat, but I hear animals like him are often messy when they drink too much.



Anyhow, I think Edward has finally sobered up. He opened his bottle of wine, started drinking and now doesn't remember a thing. He claims it was a time warp. lol



Maybe it is just that a Klein bottle can hold more wine than a normal bottle? We will have to experiment to determine if that's true. Although if Edward is right about the time warp business (I've always heard that time is the fourth dimension therefore Edward had to violate the rules of time to crack open his bottle) we might even have more fun! All in the name of science!!!!

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I think we need to invite everyone who responded here. I just looked at Ben's little blurb on Y!A not only does he give good answers, but he makes wine and beer. All the more fun.

Answer 3

Great question!

BTW I got a Klein bottle of great wine for my birthday. ... yes... thank you.

We can share that wine if any one can explain how to get to the wine without damaging the bottle.

I would ask Felix K, but he is dead as a doornail. :-(

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Still, I would say we can do this, that is cut up that Klein Bottle into Mobius strips with no leftover pieces since the Klein Bottle can be constructed by folding a Mobius strip in half lengthwise and attaching the edge to itself. (Pst.. Felix K said that)





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Thank you (Ω) The cat ;-). Is the 'two thumbs' up a Canadian thing. LOL. Hugs to you funny gal..



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Yes Ω Remo Aviron you have a point and a good taste in inexpensive but good red wines. A bottle of Bogle Old Vine Zinfandel sound like a good idea. Come over and join me.



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EXTRA!

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Would you believe I have opened the Klein Bottle and had some of that wine. WOW! I think I have leaped forward in time. Has anything happened while I was gone? :-P

Answer 4

I just want to point out that the Klein bottle can also be cut so as to produce a single Mobius strip (and no leftovers).



Since imagery is impossible in Y!A (even more so than building myself a Klein bottle), I'll try to walk through a demonstration through wikipedia. Get yourself to the Klein bottle article there. If you happen to be gifted with spatial coordination (I'm not particularly good at it), the first image on the right may do. Imagine cutting along one of the "vertical" grid lines, following all the way until you get back to your starting position. Now open the bottle up along this new cut; you've got yourself a Mobius strip. [It might help to "open" the bottle along two of the vertical lines, then trace along those lines with two fingers; when you get back to your starting point you'll notice your fingers have interchanged positions, as in the Mobius strip.]



If you're not so good with spatial manipulation, we can use the "origami" patterns of the two objects. In the wiki's "construction" section they have a polygonal presentation for the Klein bottle. When they demonstrate the construction, they first paste along the red lines to form a cylinder; if we instead first paste along the blue lines we have ourselves a Mobius strip (this might be easier to see by elongating the square into a rectangle with long red sides; we're pasting the two blue ends together, but in opposite directions, which is exactly the "half-twist" of the Mobius strip). So now the red lines trace along half of the strip each, each one starting where the other ends. Now we need only paste the red lines so that their orientations match ("only" I say!) to get the Klein bottle.



If you still can't get a handle on this, I refer you instead to the "dissection" section on the wiki. No help for understanding it, but they do tell us that "In fact, it is also possible to cut the Klein bottle into a single Möbius strip."



Sorry, I feel like I've disappointed everyone by not using enough 4-D jokes...oh!



You know you're a math major iff:

you've used the excuse "I did my homework, and locked it up, but a four-dimensional dog got in and ate it."



And one other thing: KLEIN STEIN!

http://www.kleinbottle.com/



EDIT: We really need a discussion board for us math answerers.



I would love some wine. The real problem would seem to be keeping the wine in the bottle, not getting it out...having no interior and all. Then again, if we do regard the fourth dimension as time, perhaps some of the wine only exists in the future! Something to look forward to!



[I love having a topology class at the same time as this discussion :P ].

Answer 5

One of the fun properties of the Klein bottle is that,( if you imagine yourself an ant crawling around on the surface,) if you travel horizontally, you eventually end up where you started. Whereas, if you travel vertically, you end up on the *opposite side* of where you started...



What does that have to do with it, you ask? well, That means if you drew a set of closely spaced curves, reconnected on the ends as you came around, they would obviously trace out a Mobius strip, (and you would end up with just a single curve, not two; Mobius strips have only one edge.) Then if you widened the strip, by "pulling the curtain around the bottle," as it were, you could eventually narrow the region not included in the strip to a single slit.



Hence, if that ant were dragging a pair of box cutters, or "Klein bottle cutters" around it's vertical, "round the bottle journey," you would end up with a single Mobius strip.



This leads to a fun collorary.



Since the bottle, technically has only one side, the ant would actually be cutting at two points on the bottle at once.



~WOMBAT

Answer 6

I am thinking of some sort of parametric curve. Imagine the initial point as being at (0, 1/2, 0) and it ultimately winds up at (x, -1/2, 0). The angle through which the point is being rotates starts of at 0 and ends when the angle = π. When the angle is at π/2, the point will be at (x/2, 0, 1/2). The trick will be to find the arc length of that curve. Currently working on it; will see what I can come up with. edit: parametrize the curve. x = t, y = cos(t), z = sin(t) Now let's get the length on the arc 0 <= t <= π/2. Double this length will be the minimum length of the strip needed to create a Mobius.

Answer 7

That is a rather twisted question!

I hope you will get a sharp answer.

A star for you:D



Love your added details. Sounds like fun.

Can you direct me to a 4 dimensional space insertable brain software that would fit in our 3 dimentional brains before we start?



Lets start with 4 dimensional scissors. Be right back;)



How many surfaces do we need to cut and how many surfaces does these sissors have?



Edward, I wish I could give you another thumb up!





Edit:



Mr Aviron, I am sorry your answer went over my head, (the sewing part created diversion)... I would give you an extra thumb up also!



Could my pal Kittana and I join you two gentlemen for a bottle or two?



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WΩMBAT you say: "Since the bottle, technically has only one side, the ant would actually be cutting at two points on the bottle at once."

I think this is it!



Scythian, is that what you mean by:"that's how I was looking at this problem." ?

Does the ant cut at two points at once?



***Perhaps poking a hole in the Universe would solve the Israeli-palestinian "conflict";)***

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Yipee, Edward has opened the bottle!

Edward, did a genie from another dimention come with the wine?





Time warp, spatial short cut, whatever, we can celebrate now!

All in the name of science of course, Mr Aviron;)



Hey what about Scythian, he is the one that has asked the question after all.

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Last edit: I promise.



Ben, you deserve 2 additional thumbs up and some wine!

I especially liked the last sentence to your second paragraph.



And Scythian, thank you for your amazing questions, and your patience.

Answer 8

Ahhh i sure wud have loved to try cutting it but then i don't want to waste money u sure must have tried it right?







EDIT : Whoooa dudes i think i seriously drank so much that i just didn't realise i'd missed out on so much with u guys trying to open that Klein bottle up lol seems like it indeed just emptied out only for me lol while all u just planned on opening it lol...awwwwwwwwww CAT i was way too thirsy i guess :D

Answer 9

No. The Klein bottle only exists theoretically. It passes through itself without making a hole, which is a physical impossibility.

Answer 10

yes, i guess.