OK, serious question, and I'm looking for real answers. If you reply, please say whether you looked it up, calculated it for yourself, or did the physical experiment. All approaches valid, I just need to correlate the sources.
Take a one-half-twist Moebius strip & cut it in half. You get one big twisted loop. How many half-twists does it have?
Take a 3-half-twist Moebius strip & cut it in half. You get one big twisted loop, tied in a trefoil. How many half-twists does it have?
The number of twists is the property of embedding, not of the internal geometry of the surface. Internally, it's the same rectangle with a pair of opposite sides "glued together" in either direction.
Cutting MS along the central line gives the cylinder S×I. How the number of twists changes may be understood by modeling MS as the surface generated by the rotating chord between points on the torus.
@amiloradovsky That's all true, and I already know that. So I'm asking about the embedding, and the number of twists. I'm looking for actual numbers here, either from calculations, or from physical experiments. You may deduce that there are reasons for me asking this question in this way.
@ColinTheMathmo The number of half-twists should double after the cutting. So the answers are 2 and 6 respectively. Is there a catch?
@amiloradovsky Yes. Try this.
Make a one-half-twist Moebius Strip, cut in half, put to one side.
Take a strip and try to replicate what you have. How many half twists do you have to insert to make that happen?
@ColinTheMathmo Tried for one-half-twist MS: got 4… not sure why.
@amiloradovsky ... and there is my reason for asking. With one half-twist you end up with 4. How many do you end up with if you cut a three-half-twist Moebius strip?
@ColinTheMathmo I guess, the additional half-twists are due to the wrapping around the torus' axis. So the answer may be 6 + 2 = 8. — Let's see if I'm wrong again…
@amiloradovsky It's a good question.
@ColinTheMathmo Nope, 8 is right (unless I messed up the counting…).
Generally, I hypothesize, the answer is
m = 2 × (n + 1)
where n — the (odd) number of half-twists in the original, m — after the cutting.
m = 0 (mod 4)
Although it may be somewhere on Wikipedia…
@amiloradovsky My early experiments agree with your formula. So now the question is:
Why is the answer on so many pages on the internet different?
This page, for example, claims that with the one-half-twist Moebius Strip one gets only 2 half twists in the result:
@amiloradovsky I've not been able to find this in a comprehensive form, and getting the same answers that I do.
@ColinTheMathmo The formula now seems more or less obvious to me. But I'm not sure what subfield of topology is needed to prove it (to express the (half)twists), perhaps the theory of links and braids.
@amiloradovsky It is an interesting question. Most interesting to me, though, is that lots of pages on t'internet give an answer different from the one I believe to be true.
@ColinTheMathmo One half twist, cut in half: 4 half twists.
3 half twists... I also count 4, but counting half twists on a trefoil is really hard.
Done experimentally with a strip of paper.
Off to look up an answer now...
@derwinmcgeary Cool - thanks. Good luck at finding an answer - please let me know what you find.
@derwinmcgeary You should unknot it first.
@ColinTheMathmo Is the number of half twists in a trefoil Möbius band well defined?
@henryseg I have a definition that I think is clean and clear - allow one part of the strip/loop to pass through another. This allows "unknotting" in a manner that can be argued is independent of the twists.
@ColinTheMathmo That depends on a choice of unknotting move - how do you know that it doesn’t matter where you do it?
@henryseg Short answer is that I don't. I'm exploring some of these questions, and looking at the different answers people are getting. And there are different answers, and I don't understand why.
But just taking the single-half-twist Moebius band, some people say the result of cutting has two half twists, others (including me) say it has four, so I'm concluding that I don't actually understand anything about this at all.
@ColinTheMathmo You can define the twist number for an unknotted loop by considering the disk in space that the loop at the core of the Möbius strip bounds. This is unique up to deformation. Then see how many times the boundary of the Möbius strip crosses that disk.
@henryseg And for the cut one-half-twist Moebius Strip I get an answer of 4. So I don't understand why there are place on t'internet and people IRL who say the answer is two.
@ColinTheMathmo Maybe they are counting full twists rather than half twists.
@henryseg I've asked when I can, and they're definite that they are counting half twists.
@ColinTheMathmo I mean the next obvious step is to get them to post a picture.
@henryseg Indeed. Problem I'm finding now is that when I ask, people are effectively saying:
I did what you ask, I'm not going to waste any more time on this.
... and going away. I'm being as encouraging and positive as I can, but I'm not getting anywhere.
@henryseg I am still engaging with one person who seems very confused in their descriptions.
I'd love to send you a link to the discussion on Facebook <spit> but I can't figure out how to do that. It's a deliberate walled garden, but I do get responses there, so don't see how to leave and retain the value.
@ColinTheMathmo Sometimes it is ok for people to be wrong on the internet...
@henryseg This was prompted by a working mathematician who is getting into outreach finding a well-known, well-regarded website saying the answer was two. I'll see if I can find it for you ... stand by ...
@ColinTheMathmo Is it in fact a wiki? This bit is also dubious: “The topology of Möbius strips make it a rare Euclidean representation of the infinite”
@henryseg It's a wiki, but I see *many* people claiming how wonderful it is, and sending students to it. It's only one example though, and the one my correspondent sent me to. It had been held up to them as a shining example of fantastic material
@ColinTheMathmo if it’s a wiki, fix it?
@henryseg The problem is that it's then yet another site that I have to register with, learn their systems, click on a confirmation link, remember a password, cope with reminder emails, and then get into edit battles with people who disagree.
My experiences aren't uniformly positive with things like this ...
@henryseg Just now I'd rather understand what's actually going on, and then if people ask, I have a definitive answer. I may yet try to hunt down everyone on t'internet that gets it wrong and try to fix it, starting with them, but not now, and not soon.
@ColinTheMathmo That is the price to pay for correcting wrongness. I’m not willing to pay that price either right now!
@henryseg Quite - but there is value in writing up something that's actually correct, and perhaps pointing at the existing site, quoting it, and pointing out that it's (currently) wrong.
@henryseg It's certainly true that parts of the site are ... yes, dubious is a good word.
@henryseg I think the mystery has been solved as to why people are getting a different answer - they're counting "half-twists" without opening up the loop into a loop - they are leaving it in a figure-of-eight.
@henryseg See here:
@henryseg Running around that the top edge goes to the bottom once, then returns to the top, giving one full twist.
@ColinTheMathmo There’s the importance of the surface (disk here) to get a well defined answer!
@henryseg So eventually we will return to the question of the three-half-twist Moebius Strip being cut in half, but for now I need to return to the company accounts.
Thanks for your feedback/input/interaction/time.
@ColinTheMathmo Does it also depend on a choice of knot diagram? What if you mess up the trefoil somehow first, and then make a crossing change (that unknots the trefoil)?
@henryseg As I say, I don't know. I've tried a few things that look like they might be different, and each gave the same answer. That's not the answer, that's just the starting point of the question.
But the question here (and in other places) was to get people thinking. I think I need first to understand the one-half-twist (and therefore unknotted) version. That will help clarify the definitions.
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