4 ms·

Möbius strips and differential equations

86 points by mbustamanter 2mo ago

1970-01-01 2mo ago
Here's the reason you need enginners in the universe. Yes, the strip has non-orientability. Howwever, just introduce spin about the center, as you would a wheel, and you get all the same benefits of orientation as long as you're not in a completely empty void in space. Upside sees constantly changing, inside sees the other side of the strip.
- srean 2mo ago
  I think you are saying something interesting. Could you elaborate a little though because I am not quite getting you yet hooked to understand what you are saying.
  - 1970-01-01 2mo ago
    Spin gets you centripetal force. You know where inside and outside are globally because the forces will be opposite depending on which "side" you're standing, which directly contradicts their statement "..globally you cannot label one side as up and one side as down. If you try to label one side as up, and then you move continuously around the Möbius strip, you'll end up labelling that same side as down!"
    - srean 2mo ago
      Got you.
    - karmakurtisaani 2mo ago
      I get what you're saying, but with the spin you need to introduce the notion of movement, mass and force. It adds complexity you might not be able to afford, especially since typically you want to investigate a whole bunch of shapes, not just one strip. Also, since everything on your model is continuous, there must be a point on the strip where the centrifugal force is 0. Is that point inside or outside?
      - 1970-01-01 2mo ago
        >introduce the notion of movement, mass and force Hence my statement about not being in a void. >Is that point inside or outside? Depends on which foot is inside and which is not. That's the beauty of reality. It's fully deterministic at the marco level.
        karmakurtisaani 2mo ago
        Ah. I think it's better we don't discuss this topic from a mathematical perspective and agree you can indeed do a lot of different things with a lot of different outcomes.
      - srean 2mo ago
        Note that for a particle there is no discrepancy.
    - pdonis 2mo ago
      > If you try to label one side as up, and then you move continuously around the Möbius strip, you'll end up labelling that same side as down!" This statement is true (though it is using a different definition of "up" and "down" than you are), and your claim that spinning the strip will show a contradiction with it is false. See my post downthread in response to your "spin the strip as you would a wheel". I phrased it there in terms of which edge is the "outside" and which is the "inside" edge, but the same is true of "up" vs. "down" sides relative to the strip (note that I emphasized that phrase in my other post). Note that the issue is not that spinning the strip does not define an "inside" (towards the axis) vs. "outside" (away from the axis) in the 3-space in which the strip is embedded; of course it does (and similarly for "up" and "down", if you want to use those terms instead, in the embedding 3-space). The issue is that there is no way to use this to construct a continuous orientation on the strip itself, the way you could on an orientable surface, such as an ordinary ring (a disk with a hole cut in the center).
- pdonis 2mo ago
  What are you saying should spin? The particle moving along the Mobius strip, or the Mobius strip itself?
  - 1970-01-01 2mo ago
    Spin the strip as you would a wheel. The forces tell you where you're standing on it; Just measure them: If you painted a line down the center, the paint would fly-out from half of the surface (at high RPM) before it could fully adhere to the surface. The no-orientation topology only makes sense if you're concerned with topology, the universe has other ways of showing you real orientation.
    - pdonis 2mo ago
      > Spin the strip as you would a wheel. Meaning, about an axis perpendicular to the strip (or at least it's like that at one point--see below) and running through the center of the "hole" in the middle? You can do this, sure. But because of the shape of the strip, it won't be perpendicular to the rotation axis everywhere. Try this experiment: make a Mobius strip and pick a small area of it that you will define to be in the plane of rotation. Draw a mark on each side of the strip pointing "outward", i.e., away from the rotation axis. Now take a small arrow and start it on one side of the strip pointing in the same direction as the little mark. Move the arrow around the strip. You will find that it will reach a point on the other side of the strip, where the other mark you drew is--and when it's there, the small arrow will be pointing in the opposite directon to the other mark. In other words, the small arrow is now pointing at the "inside" edge of the strip, not the "outside". (Note that on an orientable surface, you would not be able to move continuously along the surface from one mark to the other at all without crossing an edge.) In other words, while yes, there is a well-defined "direction of centrifugal force" arrow everywhere in the space the strip is embedded in, it cannot always point in the same direction relative to the strip. In some places it points "horizontally" relative to the strip, towards one edge (but it's impossible to continuously pick "the same edge" everywhere on the strip relative to this). In other places it points "vertically", out of the strip. So the "direction of centrifugal force" does not give you a well defined orientation everywhere relative to the strip. You can't use it to say, for example, "this is the outer edge" and have that work continuously everywhere.
- deleted 2mo ago
  [deleted]
azeemba 2mo ago
I enjoyed reading this. I have been making my way through Needham's Visual Complex Analysis (https://global.oup.com/academic/product/visual-complex-analysis-9780192868923?cc=us&lang=en& https://global.oup.com/academic/product/visual-complex-analy...) book which takes a similar visual-focused view of teaching all of complex analysis. Chapter 2 in particular covers multivalue functions and branch points to make a similar point about how some paths will yield different values at same points.
- Ydarbleoj 2mo ago
  Needham is great and his Visual Differential Geometry and Forms is excellent.
  - Agingcoder 2mo ago
    Yes im reading it now. I find that the total lack of formalism makes things difficult from time to time just like his insistence on using basic geometry when there are easier ways. That being said, its an excellent book.
jesuslop 2mo ago
Thanks to the authors, super-enjoyable and bookmarked. Part of plenty of good stuff.
TwelveEyes 2mo ago
You know what, that was really cool. I’ve never seen anything like that.
somat 2mo ago
Probably junk science, But I like this attempt to describe the electron via möbius twisted em fields. https://quicycle.com/understanding-electrons/ https://quicycle.com/understanding-electrons/

86 points by mbustamanter 2mo ago

1970-01-01 2mo ago

Here's the reason you need enginners in the universe. Yes, the strip has non-orientability. Howwever, just introduce spin about the center, as you would a wheel, and you get all the same benefits of orientation as long as you're not in a completely empty void in space. Upside sees constantly changing, inside sees the other side of the strip.

srean 2mo ago
I think you are saying something interesting. Could you elaborate a little though because I am not quite getting you yet hooked to understand what you are saying.
- 1970-01-01 2mo ago
  Spin gets you centripetal force. You know where inside and outside are globally because the forces will be opposite depending on which "side" you're standing, which directly contradicts their statement "..globally you cannot label one side as up and one side as down. If you try to label one side as up, and then you move continuously around the Möbius strip, you'll end up labelling that same side as down!"
  - srean 2mo ago
    Got you.
  - karmakurtisaani 2mo ago
    I get what you're saying, but with the spin you need to introduce the notion of movement, mass and force. It adds complexity you might not be able to afford, especially since typically you want to investigate a whole bunch of shapes, not just one strip. Also, since everything on your model is continuous, there must be a point on the strip where the centrifugal force is 0. Is that point inside or outside?
    - 1970-01-01 2mo ago
      >introduce the notion of movement, mass and force Hence my statement about not being in a void. >Is that point inside or outside? Depends on which foot is inside and which is not. That's the beauty of reality. It's fully deterministic at the marco level.
      - karmakurtisaani 2mo ago
        Ah. I think it's better we don't discuss this topic from a mathematical perspective and agree you can indeed do a lot of different things with a lot of different outcomes.
    - srean 2mo ago
      Note that for a particle there is no discrepancy.
  - pdonis 2mo ago
    > If you try to label one side as up, and then you move continuously around the Möbius strip, you'll end up labelling that same side as down!" This statement is true (though it is using a different definition of "up" and "down" than you are), and your claim that spinning the strip will show a contradiction with it is false. See my post downthread in response to your "spin the strip as you would a wheel". I phrased it there in terms of which edge is the "outside" and which is the "inside" edge, but the same is true of "up" vs. "down" sides relative to the strip (note that I emphasized that phrase in my other post). Note that the issue is not that spinning the strip does not define an "inside" (towards the axis) vs. "outside" (away from the axis) in the 3-space in which the strip is embedded; of course it does (and similarly for "up" and "down", if you want to use those terms instead, in the embedding 3-space). The issue is that there is no way to use this to construct a continuous orientation on the strip itself, the way you could on an orientable surface, such as an ordinary ring (a disk with a hole cut in the center).
pdonis 2mo ago
What are you saying should spin? The particle moving along the Mobius strip, or the Mobius strip itself?
- 1970-01-01 2mo ago
  Spin the strip as you would a wheel. The forces tell you where you're standing on it; Just measure them: If you painted a line down the center, the paint would fly-out from half of the surface (at high RPM) before it could fully adhere to the surface. The no-orientation topology only makes sense if you're concerned with topology, the universe has other ways of showing you real orientation.
  - pdonis 2mo ago
    > Spin the strip as you would a wheel. Meaning, about an axis perpendicular to the strip (or at least it's like that at one point--see below) and running through the center of the "hole" in the middle? You can do this, sure. But because of the shape of the strip, it won't be perpendicular to the rotation axis everywhere. Try this experiment: make a Mobius strip and pick a small area of it that you will define to be in the plane of rotation. Draw a mark on each side of the strip pointing "outward", i.e., away from the rotation axis. Now take a small arrow and start it on one side of the strip pointing in the same direction as the little mark. Move the arrow around the strip. You will find that it will reach a point on the other side of the strip, where the other mark you drew is--and when it's there, the small arrow will be pointing in the opposite directon to the other mark. In other words, the small arrow is now pointing at the "inside" edge of the strip, not the "outside". (Note that on an orientable surface, you would not be able to move continuously along the surface from one mark to the other at all without crossing an edge.) In other words, while yes, there is a well-defined "direction of centrifugal force" arrow everywhere in the space the strip is embedded in, it cannot always point in the same direction relative to the strip. In some places it points "horizontally" relative to the strip, towards one edge (but it's impossible to continuously pick "the same edge" everywhere on the strip relative to this). In other places it points "vertically", out of the strip. So the "direction of centrifugal force" does not give you a well defined orientation everywhere relative to the strip. You can't use it to say, for example, "this is the outer edge" and have that work continuously everywhere.
deleted 2mo ago
[deleted]

azeemba 2mo ago

I enjoyed reading this. I have been making my way through Needham's Visual Complex Analysis (https://global.oup.com/academic/product/visual-complex-analysis-9780192868923?cc=us&lang=en& https://global.oup.com/academic/product/visual-complex-analy...) book which takes a similar visual-focused view of teaching all of complex analysis. Chapter 2 in particular covers multivalue functions and branch points to make a similar point about how some paths will yield different values at same points.

Ydarbleoj 2mo ago
Needham is great and his Visual Differential Geometry and Forms is excellent.
- Agingcoder 2mo ago
  Yes im reading it now. I find that the total lack of formalism makes things difficult from time to time just like his insistence on using basic geometry when there are easier ways. That being said, its an excellent book.

jesuslop 2mo ago

Thanks to the authors, super-enjoyable and bookmarked. Part of plenty of good stuff.

TwelveEyes 2mo ago

You know what, that was really cool. I’ve never seen anything like that.

somat 2mo ago

Probably junk science, But I like this attempt to describe the electron via möbius twisted em fields. https://quicycle.com/understanding-electrons/ https://quicycle.com/understanding-electrons/