Möbius Strip Riddle Solved 184
BigLug writes with news that two experts in non-linear dynamics, Gert van der Heijden and Eugene Starostin of University College London, have developed an algebraic equation that describes the Möbius strip — something that, you may be surprised to learn, had never been done since the form's discovery in 1858. ABC.net.au has an accessible short summary: "What determines the strip's shape is its differing areas of 'energy density,' they say. 'Energy density' means the stored, elastic energy that is contained in the strip as a result of the folding. Places where the strip is most bent have the highest energy density; conversely, places that are flat and unstressed by a fold have the least energy density."
What if I make an SLA (stereolithography)? (Score:4, Interesting)
Does throw out their math?
-S
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I don't have an answer to your question, but your assumption certainly begs the question: Are you sure about that?
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Of course, it's possible that it's poorly worded on their part or poorly interpreted on mine, and the differing energy densities are, in fact, a property of the shape rather than the process used to create it - but that's not the way I read it.
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I read it the opposite way.. That the energy densities are a a property of the shape, rather than the bending involved. If it were the bending, then wouldn't the energy densities depend on the material used to create the shape?
I'm curious as to the practical aspects
Algebraic equation (Score:3, Insightful)
That, too, of course (Score:2)
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The problem solved is finding a surface homotopic with a Mobius strip with the lowest global energy density (which can be defined as an integral in te
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For most simple, homogenous materials, you can factor the material properties out of the equations describing the strain distribution. Eg: the equations describing the deformation of a brick with a load on top are the same whether the brick is aluminum or steel, they are just parameterized by the Young's modulus and Poisson's ratio of the material used.
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Begging [begthequestion.info] the question does not mean raising the question.
I beg your pardon (Score:2, Insightful)
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If the rest of the world decided to start calling apples "oranges" tomorrow and you decided to go about correcting them, who in fact would be more wrong?
Re:What if I make an SLA (stereolithography)? (Score:5, Insightful)
When I hear someone trot out the "modern, popular usage" of "beg the question" or, say, "enormity" or "irregardless," well, I know those things are sanctioned by more populist dictionaries, but I pretty much assume the person is just using words they don't understand, which gives me a negative impression of them. And when people defend those usages, I think "here is someone who can't stand to find out they were wrong about something."
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Those of us concerned with the effectiveness and beauty of language realize that word meanings certainly can and do change. We feel r
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The site explains how people use "begs the question" as a variant of "raises he question", and that's wrong.
Know what else is wrong? Registering a domain "begthequestion" and dedicating it to tutoring people how to talk. Languages evolve, and most of the interesting phrases in English (or any language) have origins that used to mean something else.
Did you understand what he meant to say? Do a lot of people use the phrase as he did
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Begging the question is a technical phrase with a specific definition. If we let people use it to mean something else, we won't have a term for begging the question anymore.
Millions of ignorant computer users call their monitor their "computer" and their tower their "hard drive". By your logic we should just shut up and let them, since we understand what they meant anyway.
Besid
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I stand by the usage (Score:2)
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Yes, I'd say that hidden in that response is an assumption that the principle in question is true. The circular reasoning is the important feature, and you're quite right about answering questions in a tautologous way.
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"..your assumption begs the question: Would the energy density actually be equal?"
"..your assumption begs the question. Are you sure about that?"
Re:What if I make an SLA (stereolithography)? (Score:5, Informative)
In a similar way, if you used this formula to generate a mobius strip in the 3D program of your choice and then print it out on a 3D printer, it ceases to be a true mobius strip and becomes an object that is shaped like a mobius strip. it is a subtle, but definable, difference.
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Wouldn't that apply to anything made of atoms regardless of whether it's produced on a 3D printer, carved from stone, or whatever? I'm thinking of the atoms as similar to 3D pixels - even a mobius strip assembled atom by atom is bump
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[1] Topologically, the Möbius strip can be defined as the square [0,1] × [0,1] with its top and bottom sides identified by the relation (x,0) ~ (1-x,1) for 0 ? x ? 1.
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Uh, no. Even if you just take your perfect piece of paper and twist it slightly, you'll get a non-uniform stress distribution.
Think about it this way. Take your hands and put them together firmly. Slightly twist your left hand, trying to move your left thumb upwards and away from you. What's the sheer stress on your right hand due to the torque? It's upward near your palm, and downward towards your finger tips. That means its zero somewhe
Re:What if I make an SLA (stereolithography)? (Score:5, Insightful)
The paper in question, however, was modeling the minimum-energy state that a Möbius strip would adopt assuming that the local energy on the strip is based on local curvature (and that stretching energies can be neglected). As they point out, this is a very good approximation for building a Möbius strip by bending common thin materials (e.g. a sheet of paper or plastic). Knowing stress distributions is of course important for things like failure mechanics.
They also note that in the field of synthesizing nano-ribbons and nano-Möbius strips (yes, it's been done!), this bending energy can be critical to understanding the behavior of the final object, and is also important in understanding how such objects can be synthesized. (The growth of anisotropic nano-crystals, including nano-ribbons, is strongly dependent on the relative energies of the various growing surfaces.)
Having said all that, I think it's pretty clear that the authors tackled this particular mathematical problem because it was fun, and because of the notoriety of the Möbius strip. Ultimately it's a neat piece of mathematics and makes for some cool-looking graphs.
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Correction (Score:3, Informative)
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If you make one from a 3-D printer, then it's not a strip anymore. The Mobius strip is defined as a strip with one twist in it, not a thin toroidal volume of mass shaped such that the normal vector of the surface rotates 180 degrees when you travel around it once. The twist is essential.
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What's the difference?
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I Can't find It. (Score:3)
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Re:I Can't find It. (Score:5, Informative)
Layne
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If only... (Score:5, Funny)
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Well... now I can sleep tonight (Score:1, Funny)
Who the hell talks like that?
Mathematicians (Score:3, Funny)
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Not an algebraic equation (Score:2, Informative)
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You cannot embed RP^3 in R^3 at all, not even continuously. This follows for example from the Alexander duality theorem, which has as an easy corollary that a non orientable compact n-manifold cannot be embedded in R^{n+1}.
As for your other reply: indeed, real algebraic varieties are not in general orientable (I do not know of a non-affine example, though). The fun starts when you want them inside R^3 ;-)
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That you can emdeb the band smoothly in R^3 is no news: I can do it with some paper in a few seconds ;-)
If f is a real polinomial function on R^n, then its gradient provides a non vanishing normal vector field on any connected subset of the set of smooth points of its set of zeros. So no such subset can be non-orientable. Thus, connected smooth subsets of real algebraic hypersurfaces are orientable. That is why I said that I doubt you can present the Möbius band as a semi algebraic subset of R^3. To g
strange feeling (Score:3, Insightful)
And I lost interest. Does it qualify for "inaccurate"? I do not know.
Heavy Mettle (Score:1, Offtopic)
Interesting (Score:5, Funny)
Um... (Score:1)
What are the implicaions of this riddle being solved, if any?
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The discoverers got an article written about their paper, and it was linked to by Slashdot.
(Was that too subtle? I half expect "Offtopic" and "Troll" mods instead of the "Funny" I was going for.)
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x = [R+scos(1/2t)]cost
y = [R+scos(1/2t)]sint
z = ssin(1/2t)
for s in [-w,w] and t in [0,2pi).
With interactive pictures! [wolfram.com]
too bad though... (Score:2)
Obligatory link (Score:4, Interesting)
Why did the Chicken Cross the Mobius Strip?.... (Score:5, Funny)
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The scientific principle of Möbius strippers (Score:5, Funny)
Re:The scientific principle of Möbius strippe (Score:2)
algebraic equations for Mobius strips are not new (Score:5, Interesting)
Möbius trick (Score:3, Interesting)
As a kid, I useeed to play with Möbius strips made out of paper, here is a really good trick for kids.
1) Build 2 Möbius strips out of paper.
2) Cut one in the middle of the strip -> gives a longer Möbius strip ( not two smaller one )
3) Cut the other at one third of its width and continue all around the strip -> gives a 2 Möbius strips, one shorter than the other.
Funny, I still remember this after so many years.
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Once you see where the individual strips come from, it's not too h
My lack fo understanding denotes.. (Score:2, Insightful)
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I am not a topologist (Score:3, Informative)
but the energy they speak of might be related to Willmore energy [wikipedia.org]. I gather from the Wiki writeup and assorted Google-gleanings that Willmore energy is a mathematical expression of what we consider in the real world as distortion tension. The more you have to bend a shape the more localized Willmore energy density you have. A good clue to me is the line in the Wiki article: "A sphere has zero Willmore energy." The curvature of a sphere is constant, with no localized puckers or distortion. Hence, zero Willmore energy. An untwisted flat strip would also have zero Willmore energy, but twist it and curve around to join up into a Mobius, and it gains significant distortion; hence, increased energy.
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In other news, a team from.. (Score:2)
Re:In other news, a team from.. (Score:4, Funny)
Sweden just figured out the differential equations governing a noose.
A Nøøse once bit my sister ... No realli!
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No one will ever solve my riddle! (Score:2)
MUHAHAHAHAHAHA
Never been done.... NOT! (Score:2, Informative)
My memory is a bit fuzzy, and I don't have my notes, but I _think_ it was this:
x=1/2*(2*r+w-cos(theta)^2*w)*(2*cos(theta)^2-1)
y=sin(theta)*cos(theta)*(2*r+w-cos(theta)^2*w)
z=1/2*sin(theta)*cos(theta)*w
For all real values of theta, and a constant r and w for any particular Möbius strip. As I recall, the function was derived by taking a point a distance of w/2 f
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Yeah... sorry. After I had posted, I kinda thought I might have gotten it wrong, so I dug up my old Cal4 notes.
Here's the actual set of equations we used:
x = 1/2*(2*cos(theta)^2-1)*(sin(theta)*w+2*r)
y = (-cos(theta)^2*w+2*sin(theta)*r+w)*cos(theta)
z = 1/2*cos(theta)*w
As I mentioned before, w and r are held constant, r being the major radius of the loop and w the width of the strip. theta is allowed to vary freely, and can be any real value. x,y,z will be a point on the edge of the strip. This
WHY? (Score:2, Funny)
It
Intelligent General Reader write ups (Score:4, Informative)
Misleading (Score:2, Informative)
Tape reels (Score:2)
Re:Mobius strip (Score:5, Funny)
Now leave me alone while I figure out how to get to the top of these stupid MC Escher stairs.
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Re:Mobius strip (Score:4, Insightful)
Double-edged sword (Score:4, Insightful)
And sadly, the work of many generations of mathematicians is utilized by idiots so that they can drive their SUV, eat a fast-food hamburger, and talk on the cell phone all at the same time.
(As for me, I'm an EE. Sometimes I think about others I knew who were working several years toward their PhD. It's actually quite (morbidly) funny...)
Personally, I have renewed respect for janitors and garbage collectors. Without R&D folks, *technology* would no longer advance. Without janitors/garbage people, *populations* would cease to exist.
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Re:Mobius strip (Score:5, Interesting)
What this work did was use a new mathematical technique to analyze strain energy within a mobius strip. Computation of the strain energy (potential energy function) of various geometries is an important part of the finite element formulation used to analyze real mechanical structures. The fact that the geometry is so simple doesn't mean the work is useless. Finite element methods are formulated on very simple geometries. For example, you can do very precise analysis of something like an airplane skin using a fundamental element as simple as an isotropic 2D rectangular sheet.
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Re:Mobius strip (Score:4, Funny)
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Uh, yeah, people know about elastic energy. Nobody is claiming that elastic energy has just been discovered. What's new, according to the article, is applying that concept to determine a formula for the shape of a Moebius strip.
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Same way it's easy to say 'a quantum electromagnetic effect exists' but it is much harder to go on and use the basic equations to describe the operation of an avalanche diode.
My Very Own Summary (Score:2)
A Möbius strip is a developable surface, that is, a surface that can be created by bending a flat surface without stretching it. For example, a cylinder is developable but a sphere is not.
When the summary articles refer to "solving" the Möbius strip, they are talking about a general solution to the problem of predicting what physically happens to a flat sheet when it is deformed into a Möbius strip. So this work is in the domain of mechanics as opposed to pure geometry.
With a general solu