What is the surface area of a Mobius strip made from a strip of paper?

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Discussion Overview

The discussion revolves around the surface area of a Möbius strip created from a strip of paper, exploring theoretical implications, definitions of area for non-orientable surfaces, and the nature of dimensions involved in such constructs.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the area of the Möbius strip is equal to the area of the original strip (A), while others argue it could be double that (2A).
  • One participant questions how to define the area of a non-orientable surface, suggesting that there may be multiple definitions or no clear definition at all.
  • Another participant suggests that if the original strip's area includes both sides, then the Möbius strip retains that area (A), but if it only considers one side, it would be 2A.
  • A participant introduces a thought experiment involving a four-dimensional perspective, arguing that the concept of "sides" in a Möbius strip is misleading due to its topological properties.
  • Some participants assert that the Möbius strip is a three-dimensional object, with one claiming it has four dimensions, which is met with skepticism from others.
  • There are discussions about the Gaussian curvature of the Möbius strip, with conflicting claims about whether it is zero or not, indicating a lack of consensus on its geometric properties.
  • Several participants mention related concepts, such as the Kline bottle and other multi-dimensional constructs, but these are not universally accepted as relevant to the discussion of the Möbius strip.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the area of the Möbius strip, with no consensus reached on whether it is A, 2A, or something else entirely. The discussion also reveals disagreements about the nature of dimensions and curvature related to the strip.

Contextual Notes

There are unresolved questions regarding the definitions of area for non-orientable surfaces and the implications of Gaussian curvature, as well as the assumptions about dimensionality in relation to the Möbius strip.

kureta
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[SOLVED] Mobius Strip

we have a normal strip of paper with surface area=A. if we make a mobius strip with it what will be the area of the mobius strip? is it A or 2A?
 
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How do you define the areal of a non-orientable surface?
 
i don't know. are there several ways or no way to define that?
 
When you say a "normal strip of paper with area A" does that include both sides?

If so, then the area of the Moebius strip you get by twisting and glueing the ends also has area A. In not, then it has area 2A.
 
i have been thinking about my question. we are talking about a 2 dimensional surface. but talking about sides requires 3 dimensions. think about a 3 dimensional analogous. i am in a cube shaped room and a 4 dimensional guy is observing this room. the room lacks a length in 4th dimension just like the moebius strip lacks thickness. if this 4 dimensional guy bends my room to the 4th dimension, twists it and glues two opposite walls of the room which have doors, when i walk out one of these doors i will get in the same room with my orientation changed. if moebius strip's area was 2*one side of the initial strip's area then i would enter a newly created room or volume of the room would be doubled. so i think the source of misconception is this: we are intuitively thinking that lines and points are "on" the surface but in fact they are "in" the surface. we think of the other side as "the other side" but when we turn around the strip all we do is a transformation: a rotation in 3d = a reflection in 2d. a point traveling in a moebius strip therefore does not pass to the other side (in fact there is no other side) it simple can perform a reflection (without the need of a 3rd dimension) because of the topological properties of it's space. please let me know if i am wrong somewhere. thanks for your help.

Let's think about the 2d surface z=0 in 3d space. we'll call its side which faces towards z+ side-a and the other side side-b. let's choose an arbitrary point q, say x=1 and y=3. it is meaningless to ask whether q is on side-a or side-b and therefor it's meaningless thinking q as unable or able to go to the other side. so it's meaningless to add "the other side"s area when calculating the area of the moebius strip because there is no such thing.
 
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kureta said:
we have a normal strip of paper with surface area=A. if we make a mobius strip with it what will be the area of the mobius strip? is it A or 2A?




It is neither. Its Gaussian curvature is not zero, so it is not isometric to the plane.




How do you define the areal of a non-orientable surface?

Orientability is not required for defining local volume elements.
 


The mobius concept goes much much farther, there are 3D 2 sided mobius, propagating mobius, etc. This has nothing to do with the Kline bottle. I have provided many demonstrations with examples. http://www.youtube.com/patp2
 


Gasparri said:
The mobius concept goes much much farther, there are 3D 2 sided mobius, propagating mobius, etc. This has nothing to do with the Kline bottle. I have provided many demonstrations with examples. http://www.youtube.com/patp2

Lol, nice vids
 


I really don't see this having anything to do with differential geometry. Intuitively, for a physical piece of paper, the answer is simply 2A. Think of it this way, say you had a black marker and you started coloring/filling the whole of your paper from one edge of the strip to the other (assume no over lap in strokes and an equal amount of ink usage and such). Now before you made it a mobius strip if you just look at one side of your strip of paper and start marking and call the "area" you marked A then when you make it a mobius strip and do the same thing (start at one spot and just color until you reach an edge or come back to a spot you've already marked) you would color 2A and the whole thing would be black (instead of just the one side you colored before making it a mobius strip)
 
  • #10


Uh,... Ok, if you say so. However I contend the mobius strip is a 3d object. It's a perspective thing, when you look at it from a different perspective it changes nature like a photon on 'being observed'. Wave or particle? Depends on if you look I suppose. Likewise the mobius strip. I am making a video to demonstrate this and from that you can draw your own conclusions. 2A? My discovery is that the mobius has 4 dimensions in reality and now that I have this I see it everywhere in nature. All of us have many many built in mobius functions, on fact it's what makes us what we are.
 
  • #11


Gasparri said:
Uh,... Ok, if you say so. However I contend the mobius strip is a 3d object. It's a perspective thing, when you look at it from a different perspective it changes nature like a photon on 'being observed'. Wave or particle? Depends on if you look I suppose. Likewise the mobius strip. I am making a video to demonstrate this and from that you can draw your own conclusions. 2A? My discovery is that the mobius has 4 dimensions in reality and now that I have this I see it everywhere in nature. All of us have many many built in mobius functions, on fact it's what makes us what we are.

What? Your mobius strip has 4 dimensions? I think you're doing it wrong dude. Maybe you're thinking Klein bottles or Tesseract or something? Or I vaguely remember that there's a 3 dimensional analogue or some such of a Mobius strip
 
  • #12


maverick_starstrider said:
What? Your mobius strip has 4 dimensions? I think you're doing it wrong dude. Maybe you're thinking Klein bottles or Tesseract or something? Or I vaguely remember that there's a 3 dimensional analogue or some such of a Mobius strip

No, no Klein bottles here. I don't drink either. The KB has a fatal flaw in that the puncture violates the rules of the rubber sheet; the instant a hole is made 'POOF'! the bottle vanishes. When you get the 'true' perspective of a mobius strip as having x,y,z axis, the next dimensionality that arises is the 'S'phere from the noted center of the mobius strip.
The surface of the sphere is attached to the center line of the mobius at the apex of it's y axis. Then for any mobius strip of N length the volume of S , Sv, is directly related. For any true mobius strip; It only has one side. It only has one edge. The side is out of phase to the edge by 90 degrees. When you separate the phases (x,z) from Y you may assemble the mobius in one of two ways; as 2 sided mobius, or as a 1 sided mobius. To make a 2 sided mobius you use the same (x,z) plane common twice. This 2 sided mobius can ony be 'mobius' with odd numbers of elements such as 1,3,5,7,9... The 1 sided mobius can have any number of elements and it will always have 1 side, 1 edge.

If this sounds confusing seeing the models should clear things up.
 
  • #13


All of us have many many built in mobius functions, on fact it's what makes us what we are.



...Twisted?
 
  • #14


Perhaps that is not so far off the mark. In fact liner vectors can be so arranged
that the polygon of forces result is a stable dynamic one by three vortex system.
Multiple geometrically arranged vortex systems occur in nature.
 
  • #15


Multiple geometrically arranged vortex systems occur in nature.


Yeah, but they also don't.


Anyway, I'm not a vector, I'm a human being (dammit).


I'm not linear,
and I'm surely not stable.
 
  • #16


You may not be a vector but they own you.
See here, I explain why they got that mobius strip thing all twisted up.
 
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  • #17


Reb said:
It is neither. Its Gaussian curvature is not zero, so it is not isometric to the plane.






Orientability is not required for defining local volume elements.

the Gaussian curvature is zero
 
  • #18


wofsy said:
the Gaussian curvature is zero



I'd be surprised. http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/moebius.html" page calculates for us that, for the standard parametrization,


moebiusgauss.gif
 
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  • #19


Reb said:
I'd be surprised. http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/moebius.html" page calculates for us that, for the standard parametrization,


moebiusgauss.gif

There is no doubt that the Mobius strip obtained from a piece of paper has Gaussian curvature zero.
 
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  • #20


Reb said:
I'd be surprised. http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/moebius.html" page calculates for us that, for the standard parametrization,


moebiusgauss.gif

Wikipedia weighs in:

"A model can easily be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip together to form a loop. In Euclidean space there are in fact two types of Möbius strips depending on the direction of the half-twist: clockwise and counterclockwise. The Möbius strip is therefore chiral, which is to say that it has "handedness" (as in right-handed or left-handed).

It is straightforward to find algebraic equations the solutions of which have the topology of a Möbius strip, but in general these equations do not describe the same geometric shape that one gets from the twisted paper model described above. In particular, the twisted paper model is a developable surface (it has zero Gaussian curvature)."

http://en.wikipedia.org/wiki/Mobius_strip
 
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  • #21


The process of making a Mobius band from a strip of paper preserves the flat geometry of the piece of paper. There is no stretching - All distances and angles are preserved.
The same thing is true of making a cylinder or a cone out of a piece of paper.

The Klein bottle which is two Mobius bands attached together can also be given a metric of zero Gaussian curvature.
 

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