I know what happens if I dissect a Mobius strip with a cut along the centerline and also with a cut one-third of the way across the strip. What happens with the dissections one fourth, one fifth, one sixt, etc. all the way up to, say, one hundredth of the way across? Also, what’s the highest denominator known for which the dissection has been accomplished and what was the result?
Soon as I find the tape I’ll do some tests… but now I want to know what happens with a Mobius strip made of a thick material that gets divided across the thickness dimension. [I suspect a canonical MS has ‘no thickness’ as well as only one side, but those are hard to work with]
I just had to look it up and this seems to be the definitive answer:
When you cut the Mobius strip in half you made a strip twice as long with 4 half twists.
Cut the half-Mobius in half again and you get two linked strips, both with 4 half twists.
Cut the Mobius strip in thirds and you make two linked loops one with one half twist and one with 4 half twists.
Mark a line 1/n from one side of a Mobius band of width W, then cut along this line.
There will be a strip left in the middle of width W - W/n-W/n.
A strip will be left in the middle for any value of n except n=2.This is why marking a line down the middle of the Mobius strip, 1/2 way from one edge, produces a different result from all other fractions.
The rule
If you bisect a strip with an even number, n, of half twists you get two loops each with n half-twists. So a loop with 2 half-twists splits into two loops each with 2 half-twists.
If n is odd, you get one loop with 2n + 2 half-twists. So a Mobius strip with 1 half-twist becomes a loop with 2+2 = 4 half-twists.
http://isaac.exploratorium.edu/~pauld/activities/mobius/mobiusdissection.html
To sum up: All cuts anywhere but along the centerline are equivalent, and so you end up with the same result as the case where you cut at exactly 1/3, just with different widths for the resulting strips.
And if you actually do the dissection, you’ll easily see why: if you cut along the centerline, once you make one loop you’ll meet back with the starting cut. But with any other position, once you make one loop you’ll see your starting cut, but it will be on the other half of the strip and therefore you keep going. Only once you go around again, swapping halves again, do you meet back with the starting cut.
Thanks everyone! I was a bit confused with the first answers, although that was due to my parsing so not the posters’ fault. All of the answers together really make it clear.
Too bad we can’t physically do dissections of a Klein bottle.
?
Why not? A Klein bottle is just two Mobius bands glued together, so, whatever you were doing to diagram the bands, you can do it with a Klein bottle. The simplest way to unfold it gives a square with opposite sides identified (with one twist).
AFAIK you cannot get make an actual Klein bottle.
They also have knit ones (Klein bottle hats and scarves…)
Those are novelty items and are not actual Klein bottles, regardless of the name under which they’re being sold.
Yes, those are… silicate snake oil.
I adore alliteration!
Okay, can we digress into side-topics now? Here’s a cute video of superconducting quantum levitation on a 3π (three half-twists) Möbius strip. (Video includes links at the end to several other similar.)
Sorry about the paywall, but I couldn’t find a non-pay site with this short story.
It’s the second one I’ve seen on the theme of interacting physically with an actual Klein bottle. Sadly, I cannot recall the name of the first one. The characters in the first one, IIRC, were college chums who had met for a reunion.