What is the equation for a möbius strip with a constant bend?

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In summary, the conversation is about finding an equation that describes the edge of a Möbius strip, which has been a challenging problem for experts in mechanics. The questioner is interested in a specific case where the center circle has a radius of 1 and the width of the strip is 0.5. They have tried various parametric equations and models, but are still struggling to visualize the relationship between a torus and the Möbius edge. Another equation is suggested and the conversation ends with the hope that it will provide a better understanding of the desired curve.
  • #1
Alfi
The Mobius edge.
Not the two dimensional object, the one dimensional curve.

btw - I am just a re-awakening old guy trying to relearn math from the ground up. :) just so you know your questioner. I think I have worked out the parametric for a mobius strip. I get the classic x,y,z, coordinates for a strip with two variables. R constant at 1.0 for me.
x = (R + L*Cos(A/2)) * Cos(A)
y = (R + L*Cos(A/2)) * Sin(A)
z = L*Sin(A/2)
A, ranging from 0 to 360 degrees, L ranging from -Lmax to +Lmax,
I use an R of 1 and a L approaching +/- 0.50

My problem, is trying to restrict the L quantity till it is just a single curved line. Eventually I hit a division by zero error in the graphing programs.

I can make it draw up to some point (?) , like L= 0.5 +- .00001 and then I end up with just a very narrow two dimensional strip, not a line.
Is there a simple equation to define the line or am I chasing a wrong path?
Thanks.
 
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  • #2
Alfi said:
The Mobius edge.
Not the two dimensional object, the one dimensional curve.
You mean -- you are trying to draw the curve that represents the edge of a Möbius strip?

Wouldn't that just be the curve you get from plotting for each A the two L-values Lmin and Lmax? If you want to get a single parametrization for the entire curve (rather than drawing the two curves I suggested), do you see why it's not enough to just the polar angle (A) range from 0 to 360?
 
  • #3
If you want to get a single parametrization for the entire curve

I do
 
  • #4
I'll try to figure out another way to ask the question.
Thanks.
 
  • #5
http://www.cosmosmagazine.com/news/1457/moebius-strip-riddle-solved-last
( not exactly a good source but it's all I got *shrug* Sorry.

Since 1930, the Möbius strip has been a classic poser for experts in mechanics. The teaser is to resolve the strip algebraically – to explain its unusual shape in the form of an equation.

I am interested in the equation that describes the edge only.
I don't have the training or the knowledge to figure it out myself ( I've tried .. I'm out of my league and I know it )

Not a general equation! , Just in the specific case of the center circle radius being unity ( x^2 + Y^2 = 1 ) and the width or the separation between the curve and the center circle at 0.5 of the radius.
Think of a torus with a curved line (mobius edge) drawn on the outer surface.

I just don't understand why the stress or curve or whatever would not be constant through out the line. The picture they show in the link is not the idealized torus or mobius strip that i envision.

I have built dozens of models in various forms to try and visualize how a torus and the mobius edge are related. My work room is getting littered with the damn things. :)

To me, in my very limited understanding, the edge ( the drawn curved line on the surface of the torus) must be a subset of the torus.
If I cut the torus along the edge ( top side to bottom side ... and once again I prove that I don't know the correct terminology ) for 360 degrees, the torus remains intact. It does not fall into two pieces. The part that gets cut out ( width of the blade ) is a mobius strip. If I had a very thin blade the area cut out is two dimensional, length and width only, but needs to be three dimensional in it's description.

I have probably made a dozen errors in my questions. Please forgive me for my ignorance . *crosses fingers that I explained my desire in some kind of meaningful way *
 
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  • #6
wait a min
 
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  • #7
Try this parametric equation with Matlab or something similar with 0<=t<2pi

[ (L./(2.*pi)+W.*cos(t./2)./2).*cos( t ) , (L./(2.*pi)+W.*cos(t./2)./2).*sin( t ) , W.*sin( t./2 )./2 ]

L is the length of the paper and W is the width. Remember that 2*pi*W<L

This möbious band is twisted with a constant "bend" and the centre line of the paper strip is located on a circle of radius L/(2pi) on the xy plane.

The equation is this:

[tex]\vec{r}(\theta)=[ (\frac{L}{2\pi}+\frac{W}{2}\cos(\frac{\theta}{2}))\cos( \theta ) , (\frac{L}{2\pi}+\frac{W}{2}\cos(\frac{\theta}{2}))\sin( \theta ) , \frac{W}{2}\sin( \frac{\theta}{2} )][/tex]
 
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1. What is the Mobius edge?

The Mobius edge is a concept in mathematics and topology, named after mathematician August Ferdinand Möbius. It is a one-sided surface with only one boundary, formed by taking a strip of paper, giving it a half-twist, and then connecting the ends. This results in a shape that has only one surface and one edge.

2. How is the Mobius edge different from a regular edge?

The Mobius edge is different from a regular edge because it has only one side, while a regular edge has two distinct sides. Additionally, a Mobius edge has only one boundary, while a regular edge has two distinct boundaries.

3. What are some real-life applications of the Mobius edge?

The Mobius edge has been used in various fields, such as architecture, engineering, and art. In architecture, it has been used to design unique structures with only one surface, such as the Mobius House in the Netherlands. In engineering, it has been used to create conveyor belts that have a longer lifespan due to the even wear on both sides. In art, the Mobius edge has been utilized in sculptures and jewelry designs.

4. Can the Mobius edge exist in higher dimensions?

Yes, the concept of the Mobius edge can be extended to higher dimensions. In four dimensions, it is called a "Mobius strip" and has two boundaries, while in five or more dimensions, it is called a "hypersurface" with no boundaries.

5. What are some interesting properties of the Mobius edge?

One interesting property of the Mobius edge is that it has only one side and one edge, making it a non-orientable surface. This means that if you were to walk along the edge of a Mobius strip, you would end up on the opposite side without crossing an edge. Another property is that it has a constant width, meaning that no matter where you measure its width, it will always be the same.

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