Oddities of the Möbius strip (high level math warning)?

Question

Hi there Yahoo! Answers people. I once again turn to your wisdom with a problem that has completely gotten me stumped and had my mind tied up for nearly a week now.

So, we're studying developable surfaces and we're trying to prove that for a Möbius strip to be constructible out of paper (non-compressing, non-stretching, bounded) if it has width 1 it must have length between π/2 and √3. At the end of the proof we see that for it to be π/2 the paper must past through itself, but that's not my main problem. Our book is horrible at doing proofs in a way people can actually understand them. It first goes over rulings, straight intervals on the surface that pass through a given point, and says that if a surface is developable these rulings have parallel tangent planes at each point.

So here's my question: when concerning a Möbius strip given by the parametrization

M(u,v) = <cos(u), sin(u), 0> + v <cos(u/2) cos(u), cos(u/2) sin(u), sin(u/2)>

it is a ruled surface as it satisfies the form

x(u,v) = b(u) + vδ(u)

but is that ruling, i.e. x(u₀,v) for some fixed u₀, equivalent to the developable ruling above?

Our book shows a flat representation of the Möbius strip which has rulings that are trapezoidal in shape, but the rulings given by the parameterization are square, each line is perpendicular to the 3-d surface, even if it was cut and unrolled it would be perpendicular, wouldn't it? So are the trapezoids projections of the parametric rulings onto the plane? Or are the parameterization and making the Möbius strip out of paper (from what I understand is an embedding of the plane into 3-space) completely different?

If someone could shed some light onto this that'd be fantastic.

http://www.ucl.ac.uk/~ucesgvd/moebius.pdf

That link has pictures of what I'm talking about as far as the trapezoidal ruling. From what I can tell, in the context of this paper, those lines are "straight generators" for the shape, but I have no idea what that means as I can't find a definition of a "straight generator" anywhere.

As far as my math background (so you can make your explanation more understandable to me): I am currently taking Topology and we're studying homeomorphisms now; I've taken up through Vector Calculus (not analysis) and have also taken abstract algebra.

Thanks for any help you could give.

Update:

I know it's long already, but 1 more thing. I found that half the edge length (0 to π) in the parameterization is more than the mid-line circumference. Should that happen when you make a Möbius strip out of paper in the real world, and is that where the discrepancy between the rulings come from?

Thanks again.

Morewood · Accepted Answer

I'm sorry that I don't have time to deal with this properly. It might be better directed to your professor and your classmates.

I did notice that your reference, in first full paragraph on page 2, says of your parametrization:

M(u,v) = <cos(u), sin(u), 0> + v <cos(u/2) cos(u), cos(u/2) sin(u), sin(u/2)>

"A common paper Mobius strip is not well described by this model", for precisely the reason you note in the "additional details": distances are not preserved.

Further down that same page, your reference does give a parametrization for a developable Mobius strip {formula (1)}:

M(u, v) = R(u) + v [ B(u) + (τ(u)/κ(u))T(u)]

where R(u) is the centerline, with unit tangent T(u), unit binormal B(u), curvature κ(u), and torsion τ(u). I leave it up to you to verify that this is developable and to find and verify the isometry with a paper strip. (That might also be in your reference. At very least there are strong hints.)

Trending News

Oddities of the Möbius strip (high level math warning)?

1 Answer