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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."
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Möbius Strip Riddle Solved

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  • by sdo1 ( 213835 ) on Tuesday July 17, 2007 @04:32PM (#19892503) Journal
    If I make one from a 3-d printer or SLA, then what? That's a Mobius strip with no stresses and equal energy density throughout.

    Does throw out their math?

    -S

    • Re: (Score:1, Redundant)

      by PCM2 ( 4486 )

      That's a Mobius strip with no stresses and equal energy density throughout.

      I don't have an answer to your question, but your assumption certainly begs the question: Are you sure about that?

      • He can't be sure, I suppose, but it's a safe assumption - the article specifically indicates that the stresses from folding are what cause the higher energy density. If there's no folding involved, the implication is that the energy density isn't differentiated.

        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.
        • 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.

          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)

            by benhocking ( 724439 )

            If it were the bending, then wouldn't the energy densities depend on the material used to create the shape?
            That's probably the reason why it's an algebraic equation and not just a tensor. No, I agree with the other poster that it's probably a result of the bending. OTOH, IANAT.
          • Re: (Score:3, Informative)

            by poopdeville ( 841677 )
            The term 'shape' is being overloaded. There are two kinds of 'shape' in this context. There's the topology, and there's homotopies (continuous transformations) of the topology. As an example of this distinction, a mug and a donut have the same topological structure, but are "merely" homotopic. The topology is what characterizes an object as a Mobius strip.

            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
          • by be-fan ( 61476 )
            If it were the bending, then wouldn't the energy densities depend on the material used to create the shape?

            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.
      • Re: (Score:3, Insightful)

        by Hatta ( 162192 )
        I don't have an answer to your question, but your assumption certainly begs the question: Are you sure about that?

        Begging [begthequestion.info] the question does not mean raising the question.
        • I beg your pardon (Score:2, Insightful)

          by baxissimo ( 135512 )
          But if most everyone thinks it does, it might as well.
        • That's very true. If you are insistent in completely ignoring its modern, popular usage.

          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?
          • by Thrip ( 994947 ) on Tuesday July 17, 2007 @11:00PM (#19896527)

            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?
            What if 49% of the people started calling apples "oranges"? What if 10% did? What is the cut-off where something that started out as a misunderstanding becomes the new understanding? These days, if you can find a few other people who share your misapprehension, you can declare it "the new usage."
            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."
            • by gkhan1 ( 886823 )
              The meanings of words change, every day. Things that meant one thing yesterday means something else today. This is not something most people understand, but it is the truth. You cannot insist on a formal code of language which is absolute because a language is a living thing. Why don't you read this fine article [slate.com] written by an editor of the OED. It serves as a good example of ever-changing language.
              • "Pruning is the removal or reduction of certain plant parts that are not required, that are no longer effective, or that are of no use to the plant. It is done to supply additional energy for the development of flowers, fruits, and limbs that remain on the plant. Pruning essentially involves removing plant parts to improve the health, landscape effect, or value of the plant."

                Those of us concerned with the effectiveness and beauty of language realize that word meanings certainly can and do change. We feel r
        • He's pointing out that there's an unsupported statement that begs the questions. He then asks for some supporting evidence. Where's the improper usage?

          "Begging the question" is a form of logical fallacy in which an argument is assumed to be true without evidence other than the argument itself.
          Fits the definition to me.
        • by suv4x4 ( 956391 )
          Begging [begthequestion.info] the question does not mean raising the question.

          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
          • by Hatta ( 162192 )
            Languages evolve, and most of the interesting phrases in English (or any language) have origins that used to mean something else.

            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
        • by pjt33 ( 739471 )
          "To beg the question" as an intransitive verb has a particular meaning, but why does that necessarily preclude the transitive form from having its natural meaning?
        • In this case, the OP's assumption was that the equation might not apply to a Moebius strip produced with stereolithography because a Moebius strip produced using stereolithography wouldn't have the properties that the equation describes. That is begging the question.
    • by Telvin_3d ( 855514 ) on Tuesday July 17, 2007 @04:45PM (#19892739)
      I don't think so. i think the difference would be similar to the one between vector and raster graphics. If you have a vector circle and you print it out, it ceases to be a perfect, mathematically defined, circle. it is instead a picture that looks like a circle.

      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.
      • Re: (Score:3, Insightful)

        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.

        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
      • Re: (Score:3, Informative)

        by poopdeville ( 841677 )
        This isn't insightful or informative. Please look up Model Theory [wikipedia.org]. Physical objects can be and often are models of abstract languages. A paper Mobius strip satisfies the topological definition of a Mobius strip[1] under a suitable homotopy, and is thus a model of the language defining the Mobius strip.

        [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.
        • Re: (Score:3, Interesting)

          by SaXisT4LiF ( 120908 )
          I agree with the modding on Telvin's post because it points out the subtle line between the mathematical definition of a Mobius strip and the study of the physical properties of objects that are similar in appearance to a Mobius strip. The mathematical definition of a Mobius strip calls for a surface with zero thickness to it, while physical reproductions of its likeness inherently have some non-zero thickness. The research referred to in the article seems be asking the question: "What happens to physical
    • by kebes ( 861706 ) on Tuesday July 17, 2007 @04:54PM (#19892855) Journal

      If I make one from a 3-d printer or SLA, then what? That's a Mobius strip with no stresses and equal energy density throughout.
      Sure. In principle you can generate an arbitrary shape with an arbitrary internal stress distribution (including no stress distribution).

      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.
    • by jd ( 1658 )
      I wouldn't have thought so - the gradient is non-uniform and there's a point of inflection. Ergo, the resultant force at any two points along the shape must be different. Since energy is force times time, the energy can't possibly be uniformly distributed. This would be true however the shape was created, provided there was some interaction between any given point and neighboring points.
      • Correction (Score:3, Informative)

        by benhocking ( 724439 )
        Energy is power times time, or force times distance.
        • by jd ( 1658 )
          Gaah. You are correct. (My excuse is that I'd failed to update my brain to the latest Linux kernel, causing an unexpected error.)
    • 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.

      • by Hatta ( 162192 )
        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.

        What's the difference?
        • by Sparr0 ( 451780 )
          The twist produces a spring force towards un-twisting. It might be a quite small force, for a tissue paper strip, or a quite large force, for a rubber strip, but that force is an important part of the definition.
          • by Hatta ( 162192 )
            Why? Isn't an extruded object just as one sided as a twisted strip?
            • Imagine the difference between a flexible medium with a twist, such as a piece of paper. Now cut it like he does in this video [krampf.com]. Notice how when he cuts it, the twist moves. He never cuts the twist in half and the result is a single large twisted ring of paper. Now imagine something shaped like a mobius strip in a fixed medium, such as wood, that has no spring force that keeps the twist moving away from your scissors. You would eventually get to the stationary part that looked like a twist and when cut
    • by be-fan ( 61476 )
      The analysis is for what happens if you take a flat sheet and bend it into a mobius strip. Hence the line in the article: "wide developable strip undergoing large deformations"
  • by CastrTroy ( 595695 ) on Tuesday July 17, 2007 @04:33PM (#19892511)
    Looking at all the linked articles, I wasn't actually able to find the equation. Does anybody have the equation?
  • If only... (Score:5, Funny)

    by InvisblePinkUnicorn ( 1126837 ) on Tuesday July 17, 2007 @04:34PM (#19892515)
    Now if only they could build a little bridge out of matchsticks so those poor ants can get off that damn endless path.
  • "Here we use the invariant variational bicomplex formalism...." I can comprehend the first four words.

    Who the hell talks like that?
  • This is an integral (hence analytic) equation if you read the article. An algebraic equation would be much more interesting as it would be a lot easier to study and maybe gain geometric insight from.
    • by coult ( 200316 )
      If you look at the wikipedia article on Mobius strips (linked by the poster, actually), they show a fairly simple example of algebraic equations which define a Mobius strip. The article in question here is not about that - its about the physics of forming Mobius strips from other shapes. No one seems to get that.
  • strange feeling (Score:3, Insightful)

    by mapkinase ( 958129 ) on Tuesday July 17, 2007 @04:38PM (#19892597) Homepage Journal
    First I got slightly excited, then I realized that people are talking about Moebius strip as a physical object rather than mathematical.

    And I lost interest. Does it qualify for "inaccurate"? I do not know.
  • Heavy Mettle (Score:1, Offtopic)

    by Doc Ruby ( 173196 )
    So now can my graphics coprocessor render moebius strips [wikimedia.org] on demand?
  • Interesting (Score:5, Funny)

    by Anonymous Coward on Tuesday July 17, 2007 @04:40PM (#19892647)
    Interesting idea, but I'm having trouble seeing both sides of their argument...
  • I tried to RTFA, and I'd really like to understand what they did, but reading the abstract warped my mind into it's own Möbius strip...

    What are the implicaions of this riddle being solved, if any?
    • Re: (Score:3, Insightful)

      by ivan256 ( 17499 )

      What are the implicaions of this riddle being solved, if any?


      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.)
  • this kinda take s the *magic* out of my opengl mobias screen saver. :(
  • Obligatory link (Score:4, Interesting)

    by amstrad ( 60839 ) on Tuesday July 17, 2007 @04:53PM (#19892847)
    Obligatory link to Cliff Stoll's [wikipedia.org] Klein Bottle site: http://www.kleinbottle.com/ [kleinbottle.com]
  • by CaffeineJedi ( 643314 ) on Tuesday July 17, 2007 @04:59PM (#19892907)
    To get to the other
  • by jollyreaper ( 513215 ) on Tuesday July 17, 2007 @05:05PM (#19892985)
    Möbius strippers never show you their backsides.
  • by coult ( 200316 ) on Tuesday July 17, 2007 @05:06PM (#19893001)
    TFA doesn't say what the poster says it does. The article is really about the physics of actually making Mobius strips out of various materials. The equations which parameterize a mobius strip are not complicated and can take many forms (a good math undergrad should be able to put it together with some help from Mathematica, for example).
  • Möbius trick (Score:3, Interesting)

    by ls671 ( 1122017 ) on Tuesday July 17, 2007 @05:14PM (#19893079) Homepage

    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.

    • The other poster at this level is correct. The strip obtained from cutting the Möbius strip in "half" is simply a full-twist piece of paper (orientable, having two sides). The two strips obtained from cutting the Möbius strip in "thirds" are another Möbius strip, composed of the "center third" of the parent strip, and another orientable, full-twist strip, composed of the "outer thirds" of the parent strip.

      Once you see where the individual strips come from, it's not too h
  • ..a stupid article. No just playing. I'm confused because the article didn't seem to present a case for what problems existed and exactly what they did to solve those problems. Oh a couple side notes for the publisher. First please let us know when the full details of the article require a paid subscription. Second, please make links with a target of _blank so that we don't get taken away from our beloved /.
    • by shish ( 588640 )

      please make links with a target of _blank so that we don't get taken away from our beloved /.
      Middle click.
  • by idontgno ( 624372 ) on Tuesday July 17, 2007 @05:30PM (#19893357) Journal

    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.

  • Sweden just figured out the differential equations governing a noose.
  • No one will ever solve my riddle!

    MUHAHAHAHAHAHA
  • We did this in a 2nd year Calculus course, Vector Analysis, when we were on the subject of non-orientable surfaces. That was in 2002.

    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

  • WHY? (Score:2, Funny)

    by yusing ( 216625 )
    I made many mobius strips when I was young. It puzzled me where the "other side" went when I taped the two ends together, and *really* frustrated me when, despite *self-evident* demonstrations, "other people" (stubbornly less mathematically-inclined) insisted that there were still two sides!

    It ... the other side ... was there before the taping, *not* there after the taping. Where does it go? Clearly it must go into AN INVISIBLE DIMENSION. Is it a dimension of sound? of sight? of mind? Is it vast as space, t
  • by kgp ( 172015 ) on Tuesday July 17, 2007 @09:44PM (#19895923)
    Two easier to read commentaries in Nature [nature.com] and Science [sciencemag.org]
  • Misleading (Score:2, Informative)

    by joeyblades ( 785896 )
    The Slashdot blurb and the ABC article are misleading. They claim that the algebraic description of a Mobius strip has escaped algebraic description for 8 decades. Nothing could be further from the truth. Mathematically and algebraically, the Mobius strip has been adequately comprehended from the beginning. In fact, this understanding has been fundamental to the work of Roger Penrose and Wolfgang Rindler in their development of spinors and twistor theory (one of the leading approaches to merging Relativity
  • Just the other day I was thinking; how possible would it be to take a punched-tape computer and give it a mobius strip as input, and have it perform valid instructions all the way round?

Fast, cheap, good: pick two.

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