In many respects, the Möbus strip is rather odd and could seem entirely disjoint or unmotivated. Still, it also stands as an essential example within mathematics that is akin to the Weierstrass function, or Cantors set, etc. For example, if you wanted to construct a notion of integration of differential forms on some smooth manifold, then one of the critical technical components needed for it to be well defined is the need for the smooth manifolds' atlas to be orientable. The Möbius strip is a vital example in this regard, as it's not possible to integrate a differential form on a Möbius strip, when given a smooth differentiable structure, since it is non-orientable. So, unfortunately, we cannot always assume that we can integrate a differential form over just any smooth manifold. Another consequence is that the generalized Stokes theorem does not hold when integrating a differentia form on the Möbius strip. There are ways to integrate on non-oriented manifolds, but they require much more technical machinery. Love your content by the way, and the production quality is excellent!
Totally! 😅 There are many interesting things to think about with Möbius strip I just find the “let’s cut it in half” idea/problem to be unmotivated in comparison to what one could investigate (like the non-oriented integration problem and some of the things you have listed here for example). Anyhow, I’m glad you are enjoying my content! Thanks!!
I have a question: after cutting the strip into 8 pieces, what is the amount of linked bans any 1 linked ban will be linked to? Is this number inversely distributed between the links from one end to the other or is it equal throughout? And lastly, the coloring the strip 3 colors then cutting it, I would love to see a video on that😁
Mobius Strips have a physical use: the fabric ink ribbons in dot matrix printers. (Although in small-size applications like cash registers, these type of printers have recently fallen out of favor vs. the faster, quieter thermal printer.). The ribbon cartridge contains a long continuous loop of fabric stuffed random inside, that constantly advances in use. When manufactured, one end of the ribbon strip is flipped around before gluing to the other end, forming the Möbius. The ribbon is more than twice as wide as the printhead, as it has to be since a very narrow ribbon would be too fragile for the printer to handle.
Hahahahaha this reminded me of myself talking to my friends about random math things at university, it's great to have playdoh on hand, makes for all sorts of fun demonstrations! And I don't have the answer to that, but I have tried to find it before.
Oh!! Playdoh never crossed my mind as a viable demonstration tool! And also yes, I too have had these conversations with friends in college 😂. Glad that level of depth is coming across through the camera as well lol.
It turns out that the Möbius band isn’t a knot but you can force it to be a very simple case of one depending on how you think about defining knots. If you’re only interested in the knots themselves, then saying the Möbius band is a knot doesn’t totally make sense since knots do not have thickness so if you made the Möbius strip not have thickness you would get a circle or the unknot. If you really want the Möbius strip to be a knot you have to allow knots to have thickness and due to that thickness you also have to encode the number of twists in the knot, so in this case the Möbius strip is the unknot with one twist, so more on the trivial side of the knot spectrum but doable.
I was looking for a video that shows cutting a mobius strip into more than three "pieces" , what you ended up with an eight piece.... mess. How many separate loops are there? How many are linked? Guess I'll have to try cutting them myself. EDIT: Experimenting with physical models like this and observing patterns, can we come up with a formula to plug in "n" number of cut divisions of the strip to predict number/size/linkage of loops?
I definitely agree that most of the times I saw the mobius strip talked about it was just a cool party trick. 😅 Recently, though I learned a way of viewing the Porjective Plane as a Disk and a Mobius Band glued to each other along their respective boundaries. The easiest way to see this is to remove a disk from the Projective Plane and have a line someway away from disk. Then, shoot rays with arrowheads out from the disk they will hit the line normally or they will go around the points at infinity and come back around and hit the line. When the rays hit the line normally the arrowhead is normally oriented and when a ray has gone around the arrowhead is reverse oriented. So there is a half twist along the line at infinity and in fact all the lines going through the plane that don't intersect our disk are our lines going around a Mobius strip! It's kind of weird to see at first, but there's a great old video series M335 Geometric Topology here on RUclips that models it excellently.
CHALK kinda complicated to explain. They’re a K-pop girl group with a lot of lore, some of which says their members reside on a universe shaped like a Möbius strip... I didn’t know what this was so I came to RUclips for help and your video made me a bit less confused so thanks !
"POOF, Here's a cool thing kind of math stuff" Those kind of things that seem useless and just "cool" are gonna be the most important thing in math in the future.
In many respects, the Möbus strip is rather odd and could seem entirely disjoint or unmotivated. Still, it also stands as an essential example within mathematics that is akin to the Weierstrass function, or Cantors set, etc. For example, if you wanted to construct a notion of integration of differential forms on some smooth manifold, then one of the critical technical components needed for it to be well defined is the need for the smooth manifolds' atlas to be orientable. The Möbius strip is a vital example in this regard, as it's not possible to integrate a differential form on a Möbius strip, when given a smooth differentiable structure, since it is non-orientable. So, unfortunately, we cannot always assume that we can integrate a differential form over just any smooth manifold. Another consequence is that the generalized Stokes theorem does not hold when integrating a differentia form on the Möbius strip. There are ways to integrate on non-oriented manifolds, but they require much more technical machinery.
Love your content by the way, and the production quality is excellent!
Totally! 😅 There are many interesting things to think about with Möbius strip I just find the “let’s cut it in half” idea/problem to be unmotivated in comparison to what one could investigate (like the non-oriented integration problem and some of the things you have listed here for example). Anyhow, I’m glad you are enjoying my content! Thanks!!
(Also pinning this because you bring up a lot of cool stuff here that I didn’t focus on in the video. 😄)
I have a question: after cutting the strip into 8 pieces, what is the amount of linked bans any 1 linked ban will be linked to? Is this number inversely distributed between the links from one end to the other or is it equal throughout? And lastly, the coloring the strip 3 colors then cutting it, I would love to see a video on that😁
A perfect counter to an ability that turns you inside out.
Between this and crash course im learning more from RUclips then I ever then in school.
Mobius Strips have a physical use: the fabric ink ribbons in dot matrix printers. (Although in small-size applications like cash registers, these type of printers have recently fallen out of favor vs. the faster, quieter thermal printer.). The ribbon cartridge contains a long continuous loop of fabric stuffed random inside, that constantly advances in use. When manufactured, one end of the ribbon strip is flipped around before gluing to the other end, forming the Möbius. The ribbon is more than twice as wide as the printhead, as it has to be since a very narrow ribbon would be too fragile for the printer to handle.
Hahahahaha this reminded me of myself talking to my friends about random math things at university, it's great to have playdoh on hand, makes for all sorts of fun demonstrations! And I don't have the answer to that, but I have tried to find it before.
Oh!! Playdoh never crossed my mind as a viable demonstration tool! And also yes, I too have had these conversations with friends in college 😂. Glad that level of depth is coming across through the camera as well lol.
The mobus strip is a knot i think. some knots can only be untangled in the 4th or higher dimension
It turns out that the Möbius band isn’t a knot but you can force it to be a very simple case of one depending on how you think about defining knots. If you’re only interested in the knots themselves, then saying the Möbius band is a knot doesn’t totally make sense since knots do not have thickness so if you made the Möbius strip not have thickness you would get a circle or the unknot. If you really want the Möbius strip to be a knot you have to allow knots to have thickness and due to that thickness you also have to encode the number of twists in the knot, so in this case the Möbius strip is the unknot with one twist, so more on the trivial side of the knot spectrum but doable.
I was looking for a video that shows cutting a mobius strip into more than three "pieces" , what you ended up with an eight piece.... mess. How many separate loops are there? How many are linked? Guess I'll have to try cutting them myself.
EDIT: Experimenting with physical models like this and observing patterns, can we come up with a formula to plug in "n" number of cut divisions of the strip to predict number/size/linkage of loops?
Thanks for the great Video! Keep going! :)
That is the plan!! 😅😀 I'm glad you enjoyed the video!
Can you make a higher dimension structure form cutting it ? Like an E8 lattice
I definitely agree that most of the times I saw the mobius strip talked about it was just a cool party trick. 😅 Recently, though I learned a way of viewing the Porjective Plane as a Disk and a Mobius Band glued to each other along their respective boundaries. The easiest way to see this is to remove a disk from the Projective Plane and have a line someway away from disk. Then, shoot rays with arrowheads out from the disk they will hit the line normally or they will go around the points at infinity and come back around and hit the line. When the rays hit the line normally the arrowhead is normally oriented and when a ray has gone around the arrowhead is reverse oriented. So there is a half twist along the line at infinity and in fact all the lines going through the plane that don't intersect our disk are our lines going around a Mobius strip! It's kind of weird to see at first, but there's a great old video series M335 Geometric Topology here on RUclips that models it excellently.
Loona brought me here lol
Loona?
CHALK kinda complicated to explain. They’re a K-pop girl group with a lot of lore, some of which says their members reside on a universe shaped like a Möbius strip... I didn’t know what this was so I came to RUclips for help and your video made me a bit less confused so thanks !
Haha 😂 glad this helped then!!
hahaahahahahah the comment I was looking for
LMAO SAME
"POOF, Here's a cool thing kind of math stuff"
Those kind of things that seem useless and just "cool" are gonna be the most important thing in math in the future.
Try making Möbius Strips with multiple layers!
"This is going to be a thing".
Thank you.
Endgame brought me here.
Notifications brought me here.
@@CHALKND thank you
You're 9 second intro period is a joke that's what he said