Whats up with Mobius strips. How can a two-planed object exist in a 3-dimensional world?
Just make one and see how it works for yourself, it’s not that hard to do. Take a strip of paper and bring the two ends together, like you’re gonna make a normal loop of paper. Then give one of the ends a half twist, tape the ends together, and now you’ve got a Moebius strip–only one edge, only one side.
“A mathematician confided
A Moebius Strip is one-sided
But you’ll get quite a laugh
If you cut one in half
For it stays in one piece when divided.”
(I don’t remember who wrote that.)
**(I don’t remember who wrote that.) **
Probably some guy with a one-track mind.
For even more fun, try drinking your next beverage out of a zero-volume one-sided Klein Bottle, the 3-dimensional analog of a Mobius strip.
According to the website-
Only if you cut down the middle. Start the cut 1/3 of the way in from an edge and you end up with 2 linked strips.
How do you cut it in half by cutting 1/3 of the way in from the edge? This is a weird fourth dimensional effect, right?
It’s been a while since I’ve done this, but as I recall, cutting a standard one-twist Mobius strip down the middle results in one large strip with two twists. Cutting that strip down the middle (it helps to start with a very wide strip) results in two linked strips, each with one twist. Might be remembering wrong, though.
I love the Klein bottles! The site is so funny, I thought it was all a joke. Not only can you drink out of it - it’s a barometer! You can use it to measure solar luminosity to determine if it’s day or night!
Now I have to find someone who’ll buy me a $50 breakable object that will no doubt spew colored water everywhere when I try to demonstrate it…
Here are some interesting Mobius strip sites.
http://www.rose-hulman.edu/~berglunb/Mobius.html - weird animation!
http://www.quantonics.com/Level_4_QTO_Mobius_Strip_Quanton_Latched_Right.html - I love that they compare a Mobius strip to a bow tie.
<blatent, yet appropriate showing off of sig line>
Try doing a websearch on hexahexaflexagons sometime. These…things…are fascinating. I’m going to make one out of fabric, instead of paper.
And I love the Klein bottle site, particularly the cutting edge jigsaws.
Yes indeed. Just cut along the length of the loop, 1/3rd of the way in, start cutting anywhere on the strip, keep cutting and see what happens. Also try cutting a strip into 1/4ths. I once cut a strip into 1/5ths, it took some planning to get that done. Try cutting a strip in half/thirds etc, then recutting one of the newly cut loops a second time. Hmm. this is easier to show people than explain in text. There is a load of topological goodness in moebius strips.
I don’t understand Klein Bottles. I think I want one, though. But what is the big deal? As far as I can tell from the picture, it’s just a bottle whose top or handle sticks inside of it.
How is this considered zero volume?
How do you put anything into it? If you can’t, why is it any better than a mishaped bubble?
If there is a whole in the bottom where the top meets the base, then I assume you could fill it up through there. But then, how is it still considered zero volume???
Here’s how to make a Klein bottle. Take a square sheet of paper, attach the left and right edges with no twisting, to make a cylinder, and attach the bottom and top edges with a half twist, just as if you were making a Moebius strip.
Don’t actually try that, it will only frustrate you. Klein bottles can’t exist in three dimensions, which is why the “Klein bottles” on that page are self-intersecting. I’ve always thought of Klein bottles as bottles that don’t have an “inside” and “outside”, there’s no distinction between the two. Here’s a Klein bottle made of Legos:
I HAVE a klein bottle from that site! I got it for Christmas this year. I’m going to fill it with jellybeans or something and display it in my office, just to confuse people.
I also got my sig from that site.
I don’t get it, either; they sure look like they have an inside to me.
Okay, so now I have an ordinary cilinder, right?
Hard to do with paper, I agree. But let me use something flexible like a flat sheet of fabric. After making the cylinder, bring the two open ends together and give one a half-twist, right? So now I have something like a hollow donut with a kink in it.
Or why not skip a step & use a rubber hose, since you’re going to wind up with a cylinder after step 1 anyway. Then bring the two open ends of the hose together. Another hollow donut.
I don’t like this game any more.
“The result is not a true picture of the Klein bottle, since it depicts a self-intersection which isn’t really there. The Klein bottle can be realized in 4-dimensional space: one lifts up the narrow part of the tube in the direction of the 4-th coordinate axis just as it is about to pass through the
thick part of the tube, then drops it back down into 3-dimensional space inside the thick part of the tube.”
Yep. That’s the problem; no kinks allowed.
Another way to construct a Klein bottle is to join together the edges of 2 Mobius bands. I’m afraid that’s not any easier for me to visualize but accounts for this:
A German topologist named Klein
Thought the Mobius Loop was divine
Said he, "If you glue
The edges of two
You get a weird bottle like mine.
Thanx for the site, Arjuna, I have been looking for them (admittedly not very hard) since I saw one in the UCLA Math Department 30 years ago.
Now I get it. Except, I tried putting two Moebius strips together, and I managed to do it. Unfortunately, in so doing I seem to have ripped a hole in space-time. It’s a big black void gurgling in the corner of my office, and it looks unstable and could tear apart any second now, so if it explodes outward at the speed of light and destroys the universe, I’d just like to apolog