Möbius Strip Riddle Solved

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."

Comments

What if I make an SLA (stereolithography)?

[sdo1]

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:What if I make an SLA (stereolithography)?

[Control Group]

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.

Re:What if I make an SLA (stereolithography)?

[Telvin_3d]

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:What if I make an SLA (stereolithography)?

[kebes]

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.

Re:What if I make an SLA (stereolithography)?

[jd]

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.

Re:What if I make an SLA (stereolithography)?

[XenoPhage]

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 of this formula. It seems to me that creating a mobius strip with equal densities throughout is somehow useful, but I can't determine why..

Re:What if I make an SLA (stereolithography)?

[name_already_taken]

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 bumpy at the atomic scale and not representative of the pure mathematical form.

Re:What if I make an SLA (stereolithography)?

[poopdeville]

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 terms of curvature, if I recall correctly).

Re:What if I make an SLA (stereolithography)?

[bpjk]

Speaking from complete ignorance here, but I doubt that difference is relevant. I don't think the definition of "Möbius strip" includes instructions on how to take a material and bend it in such-and-such a way to get this form. I rather suspect it's simply defined by it's shape and nothing else (same as with a circle, doughnut, pyramid and any other shape).

Even better: suppose I take a strip of perfect paper, i.e. a strip with perfect uniform density and internal stress. I then twist the strip to get a 180 degree turn at one end very carefully such that the twist is uniformly distributed along the entire length of the strip (i.e. exact same degree of twist for equal lengths of paper).

I then bend the strip very carefully to make the ends meet such that the bend is uniformly distributed along the entire length of the strip (i.e. the exact same degree of bending for equal lengths of paper).

This will give me a Möbius strip with uniform stresses throughout (except where the ends meet but that's left as an exercise for the reader). I mean, just because when turn a strip into a Möbius strip using our usual crude ways (i.e. hands) and get a localised bend-and-twist, doesn't mean it can't be done uniformly.

Ben.

Re:What if I make an SLA (stereolithography)?

[poopdeville]

This isn't insightful or informative. Please look up Model Theory. 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:What if I make an SLA (stereolithography)?

[ThinkingInBinary]

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.

Re:What if I make an SLA (stereolithography)?

[Hatta]

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

Begging the question does not mean raising the question.

Re:What if I make an SLA (stereolithography)?

[Hatta]

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?

Re:What if I make an SLA (stereolithography)?

[Sparr0]

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.

Re:What if I make an SLA (stereolithography)?

[be-fan]

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"

Re:What if I make an SLA (stereolithography)?

[Hatta]

Why? Isn't an extruded object just as one sided as a twisted strip?

Re:What if I make an SLA (stereolithography)?

[be-fan]

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:What if I make an SLA (stereolithography)?

[be-fan]

This will give me a Möbius strip with uniform stresses throughout

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 somewhere in the middle. This is a non-uniform stress distribution. The same thing will be true for adjacent slices of the flat sheet, even a mathematically perfect one.

Re:What if I make an SLA (stereolithography)?

[DigitAl56K]

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?

Re:What if I make an SLA (stereolithography)?

[kasparov]

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. 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 cutting you would just cut that part in half. You would end up with two separate pieces. See? It isn't a mobius strip without a real twist with a spring force.

Disclaimer: I am not a mathematician and I do not play one on TV. This is just the result of a thought experiment that I undertook after reading your question. I'd never really thought about it before. Answers from a real mathematician may vary. :-)

Re:What if I make an SLA (stereolithography)?

[Thrip]

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."

Re:What if I make an SLA (stereolithography)?

[SaXisT4LiF]

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 representations of a Mobius strip as the thickness approaches zero?"

Re:What if I make an SLA (stereolithography)?

[moderatorrater]

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.

Re:What if I make an SLA (stereolithography)?

[heinousjay]

Can I beg you to stop worrying about it?

Re:What if I make an SLA (stereolithography)?

[suv4x4]

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? Yes to both. So it's correct.

Which begs the question, why am I on my own turn wasting time explaining any of this.

Re:What if I make an SLA (stereolithography)?

[fatphil]

Surely that can't be correct, can it? The energy density will not be equal at all points on the surface, which means that you can skew the mean energy density by making the strip wider at the low-energy-density places, and narrower at the high-energy-density places. Of course, the *strip* nature excludes these homotopies, which makes me think that a purely topological view cannot be appropriate to this problem. Surely this is just another minimisation problem like splining, well into the applied maths domain rather than pure maths?

I'm also sure the 'answer' depends on more than just the relative width, it must surely depend on the material too, as different materials respond differently to torsion than tension, and forming a mobius strip requires both.

Note that in the above "surely" means nothing more than "I, who pretty much flunked applied maths, would like to believe".

Re:What if I make an SLA (stereolithography)?

[gkhan1]

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 written by an editor of the OED. It serves as a good example of ever-changing language.

Re:What if I make an SLA (stereolithography)?

[fatphil]

I think you're missing his point. His point is that an extruded mobius strip is as much a mobius strip as one made from deformation and glueing of a strip. The topological property, which is the only thing previously of interest, is there in both of them, so they're both mobius strips, even if you made them a different way. Of course, his construction is utterly uninteresting and irrelevant to the question in hand, which is not one of topology, but one of mechanics.

Re:What if I make an SLA (stereolithography)?

[Thrip]

Well, I read the article, but I have heard the same argument a million times, and it's irrelevant to my point, which is that even if some authority backs you up, employing usages that are generally understood to be ignorant marks you as ignorant.

Yes, the language changes. I happily use a lot of words and phrases that didn't exist a few years ago. Some changes add precision and color to the language. But others detract from it.

"Beg the question," "literally," and "enormity" are essentially useless now, because if you employ them with their older meanings, you will likely be misunderstood, while if you use them in the newer senses, the better educated portion of your audience will think you're an idiot. Worse yet, there are no good synonyms for the first two terms, so the language has just lost any good way to express those concepts. Meanwhile, the newer meanings were already well served by existing terms, so these changes add nothing.

Yes, change happens. And just as I'm not thrilled about changes like deforestation, the rise of religious fundamentalism, and Windows Vista, I don't get behind laisez-faire linguists who encourage people to embrace usages that arise from ignorance and muddy the language.

I'm not one of these people who runs around correcting people every time they open their mouths. I'm really not. I just comment on it since the subject came up.

Re:What if I make an SLA (stereolithography)?

[Hatta]

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.

Besides you didn't read the FAQ:

But language is constantly evolving.
        That's great to know! Descriptivist linguists, whom we do not fault for their stand, are quite free to watch as we bring about an evolution in the vernacular understanding of "begging the question."

Languages evolve, and I'd like to see ours evolve towards more precision and expressive power. Not sloppiness.

Re:What if I make an SLA (stereolithography)?

[Hatta]

But "Are you sure about that?" isn't the principle under question here. The original assertion had to do with the stresses in an extruded mobius strip. These would have been acceptable:

"..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)?

[ioshhdflwuegfh]

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. Which begs the question: if the meaning changes so fast, and if that is the truth, and if you cannot insist on a formal code of language which is absolute, how come that what you are saying is truth? In which relative formal code of language you can assert what you assert? Isn't it more like that language, especially if we take it to be a living thing, does not change that fast?

Re:What if I make an SLA (stereolithography)?

[FiloEleven]

"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 responsible, however, for ensuring that the changes are beneficial in some way, be it more concise communication or a more beautiful expressiveness (which may sometimes be at odds with each other, but that's what keeps things interesting).

In this case, "begs the question" is a useful phrase with a specific meaning pertaining to logic. If it becomes co-opted to mean "raises the question", then we will have lost a useful part of our language. Granted, it is not a huge loss and the use of this phrase in its original meaning is relatively rare (save between philosophers and logicians), but its preservation is not something to be derided. Don't act like we're ruining your fun - you certainly don't have to listen to us, though we believe it would be to your benefit to do so. The meanings are in question; we are simply pushing for retaining the old while you are opting for accepting the new...just another in the long series of human disagreements.

(To be clear, 'you' in this post refers not specifically to the parent poster but to those opposed to the efforts of language caretakers.)

Re:What if I make an SLA (stereolithography)?

[ioshhdflwuegfh]

He's pointing out that there's an unsupported statement that begs the questions. Whatever that "pointing out" might mean... If you read the two posts again, you'll see that one guy said something about making 3d prints of Moebius strip and the other guy asked in reply: "Are you sure?", i.e, he raised a question, which, according to the article you're referring to, could not have been "begging the question", because he asked something:

This is a common error of usage made by those who mistake the word "question" in the phrase to refer to a literal question. Of course, all this suppression of asking questions is militant:

While descriptivists and other such laissez-faire linguists are content to allow the misconception to fall into the vernacular, it cannot be denied that logic and philosophy stand to lose an important conceptual label should the meaning of BTQ [beg the question] become diluted to the point that we must constantly distinguish between the traditional usage and the erroneous "modern" usage. This is why we fight. Check out the twist (about which you might have wanted to know more, I dunno):

In the name of the purity of logic and philosophy, one is not allowed to ask when one is to say "beg the question". This means:

You're allowed to say: "This begs the question because X does not imply Y."
You're not allowed to say: "This begs the question: Doesn't X not imply Y?"

Because the second usage would be "raising the question", at the same level as:
You're allowed to say: "This raises the question: Is it really that X implies Y?"


The second twist is that the guy whom you're defending was kind of abusing the other guys argumentation via "begging"--English is sufficiently reach to allow for all kinds of other ways to ask, i.e:
"This 3d-printing thing begs the question, because you are not sure what you are talking about" (sound a bit harsh, no?); or "The 3d-printings thing raises the question: are you sure in what you're talking about?" (sound little bit less harsch but fake); he could have even raised the rhetorical question: "Are you sure in what you're talking about?" --in the sense: "the article has nothing to do with 3-d printing, because it is about distribution of energy of an 'optimally' bent Moebius strip, which does not imply that there aren't many other possible bendings...", etc... but he didn't, he begged something...

So we see the third twist: one can indeed use "begs the question" in some sense that might make some people upset and not without good reasons...

Re:What if I make an SLA (stereolithography)?

[pjt33]

"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?

Re:What if I make an SLA (stereolithography)?

[louiswins]

I'll settle this! Or, rather, Ryan North/T-Rex will settle this!

http://www.qwantz.com/archive/000693.html

Re:What if I make an SLA (stereolithography)?

[poopdeville]

The mathematical definition of a Mobius strip calls for a surface with zero thickness to it,

No it doesn't. That was my point. I even quoted a definition. The mathematical definition says nothing about thickness.

Re:What if I make an SLA (stereolithography)?

[SaXisT4LiF]

The definition you sited refers to a "square", a subset of a "plane", which has no depth to it.

Re:What if I make an SLA (stereolithography)?

[poopdeville]

You lack imagination.

Let I^3 be the unit cube in R^3. Let I^2 be the unit square in R^2. Define a topology on I^3: a set is open in I^3 iff it is the inverse of an open set in I^2 under the projection mapping onto I^2. This topology is homeomorphic to the topology on I^2. That is to say, I^3 is now topologically the same as the 'square' you mentioned has no 'depth'.

I^3 mod (the z-axis) is homeomorphic to I^2. There's a map that makes I^3 homeomorphic to a sheet of paper. There's a map that makes I^3 homeomorphic I^2. And there's the map that defines the Mobius strip. The obvious diagram commutes because of these homeomorphisms.

Re:What if I make an SLA (stereolithography)?

[poopdeville]

Note that the definition says "Topologically, the Möbius strip can be defined as the square..." This means that only the topological properties of the square are relevant. So if you find a different object with the same topological properties, you can make a Mobius strip out of it. As I showed in another post, you can construct a topology on the unit cube (has 'depth') that makes it have the same topological properties as the unit square (no 'depth').

I Can't find It.

[CastrTroy]

Looking at all the linked articles, I wasn't actually able to find the equation. Does anybody have the equation?

Re:I Can't find It.

[jimbug]

I believe it's along the lines of Paper+3rd Grader+Glitter Glue=Magic Circle Strip Thingy. Coffee Break!!

Re:I Can't find It.

[SQLGuru]

I found this: http://www.faculty.fairfield.edu/jmac/rs/halftw.ht m

Layne

Re:I Can't find It.

[Himring]

Surely that mobius guy has it. I think I played a game of wc3 dota with him once....

Re:I Can't find It.

[TheEmptySet]

The parent's link leads to is a parametrisation of the moebius strip (which turns out to require trig functions, and to be quite easy to come up with). I maintain, the actual Moebius strip problem (to find algebraic equations) is also not solved here.

Re:I Can't find It.

[Zencyde]

Where does one legally acquire a third grader? I'll be waiting for your answer at the local police station.

Re:I Can't find It.

[fixed.focus]

The actual paper is at:

http://www.nature.com/nmat/journal/vaop/ncurrent/p df/nmat1929.pdf

Re:I Can't find It.

[Nephilium]

You have to grow your own... it takes about ten years to do so... so it's a bit of a commitment...

It's also expensive... but you get to have fun at the beginning of the process...

Nephilium

Re:I Can't find It.

[Zencyde]

Can't you just purchase one? I don't think I'm willing to go through all of that for a mobius strip. What's it good for? Representing the concept of infinite? Showing that an object can theoretically exist with only one side? Bleh, useless I say!

Re:I Can't find It.

[fatphil]

Nope, that's a model for a ruled Moebius band which self-intersects in a straight line, as can be seen in the reference to American Mathematical Monthly, 91 (1984).

This one's different, it's not allowed to self-intersect I presume.

If only...

[InvisblePinkUnicorn]

Now if only they could build a little bridge out of matchsticks so those poor ants can get off that damn endless path.

Re:If only...

[CastrTroy]

In case anyone is confused, InvisiblePinkUnicorn is referring to this drawing by M.C. Escher

Re:If only...

[Anonymous Coward]

LOL +1 Funny please.

Re:If only...

[Trogre]

... or the rather pretty xscreensaver hack.

Re:If only...

[Anonymous Coward]

Ask, and ye shall receive...

Well... now I can sleep tonight

[xednieht]

"Here we use the invariant variational bicomplex formalism...." I can comprehend the first four words.

Who the hell talks like that?

Re:Well... now I can sleep tonight

[moderatorrater]

"Invariant" means unchanging. There's your fifth ;)

Not an algebraic equation

[TheEmptySet]

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.

Re:Not an algebraic equation

[coult]

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.

Re:Not an algebraic equation

[msuarezalvarez]

I would imagine an smooth semi-algebraic set over the reals would always be orientable...

Re:Not an algebraic equation

[TheEmptySet]

Not so, I think. At least in the projective case one should be able to make the real locus of complex algebraic varieties. So RP^2 for example. So my guess is that the problem comes when you try to algebraically get that RP^2 into R^3. Of course, once you've done that somewhat nicely, you allow yourself an algebraic inequality and cut it down to a Moebius strip. No?

Re:Not an algebraic equation

[TheEmptySet]

So, not really being an algebraic geometer myself, I just asked the nearest one (Pelham Wilson). He says that real algebraic varieties do not have to be orientable. But he also does not know how one would realise the Moebius strip as an embedded part of a real algebraic surface in 3D.

Re:Not an algebraic equation

[msuarezalvarez]

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 ;-)

Re:Not an algebraic equation

[TheEmptySet]

"You cannot embed RP^2 in R^3 at all, not even continuously."

You're right. So one needs to find a map (at least as nasty as the Boys surface) of RP^2 into R^3 such that the removal of the neighbourhood of a point in the domain of this map leaves an embedded Moebius strip. Smoothly this is possible, but I imagine it is hard or impossible algebraically.

Re:Not an algebraic equation

[msuarezalvarez]

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 get a proof, I'd start arguing that a semi algebraic subset of R^3 which is locally a smooth surface must be an open subset of a hypersurface and go on from there; I used the word `imagine' because my real algebraic geometry is quite poor.

So one needs to find a map (at least as nasty as the Boys surface) of RP^2 into R^3 such that the removal of the neighbourhood of a point in the domain of this map leaves an embedded Moebius strip. Smoothly this is possible, [...]

I am not sure this works: the Boy surface has a curve of self intersection (three loops at mutual right angles) so it is not quite obvious that you can remove that and still have enough left of the topology to have a Möbius band...

Re:Not an algebraic equation

[TheEmptySet]

Thus, connected smooth subsets of real algebraic hypersurfaces are orientable.

That sounds convincing (that you need more than three dimensions).

I am not sure this works: the Boy surface has a curve of self intersection (three loops at mutual right angles) so it is not quite obvious that you can remove that and still have enough left of the topology to have a Möbius band...

Trace out a simple closed curve on the surface whose tangent bundle is non-orientable. This is possible since we may take a simple closed curve on the domain RP^2 and move it away from the three triple points, then whenever it touches two corresponding points in the double point locus we move it to the side a little near one of them. Now the restriction of Boys' map to a small enough neighbourhood of this curve is an embedding. And the complement of this neighbourhood in the domain RP^2 is a disk.

strange feeling

[mapkinase]

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.

Re:Mobius strip

[WwWonka]

pessimism and sarcastic remarks will get you nowhere in the scientific community.

Now leave me alone while I figure out how to get to the top of these stupid MC Escher stairs.

Interesting

[Anonymous Coward]

Interesting idea, but I'm having trouble seeing both sides of their argument...

Re:Interesting

[mcfuddlerucker]

Give it time, you'll get there eventually...

Um...

[StyxRiver]

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:Um...

[ivan256]

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.)

Re:Um...

[TheEmptySet]

If they really had found an ALGEBRAIC equation for the Moebius strip, as the post wrongfully claims, it would, as I understand it, be a significant advance in real algebraic geometry (study of spaces arising as the zero set of real polynomial equations). As it is we can only approach the Moebius strip in algebraic geometry as an object living in higher than 3 dimensions.

Re:Um...

[Nazlfrag]

A Möbius strip of half-width w with midcircle of radius R and at height z==0 can be represented parametrically by

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!

Re:Um...

[TheEmptySet]

What you've written is about the nicest way of embedding the Möbius strip in 3D, but it uses trig functions, not just polynomials (see my post again).

too bad though...

[SolusSD]

this kinda take s the *magic* out of my opengl mobias screen saver. :(

Re:Mobius strip

[Vulva R. Thompson, P]

Relatively easy, just follow the guy with no face.

Was the boundary value problem the riddle?

[bepolite]

The article mentioned solving the "boundary value problem" for the Möbius strip. I found this on wiki http://en.wikipedia.org/wiki/Boundary_condition. I'm not sure that this article presents anything new as it claims too.

Re:Was the boundary value problem the riddle?

[Verte]

It appears to be. The normal mathematical definition for the Mobius strip, IIRC, is a square with a pair of equivalence relations on the boundaries, but that's not a differential equation. I was actually under the impression that there was proof that the mobius strip could not be written in the way they speak of [I think they are talking about a Riemann manifold? Stress density, really now..] anyway, without access to the article, there's no way to know.

Obligatory link

[amstrad]

Obligatory link to Cliff Stoll's Klein Bottle site: http://www.kleinbottle.com/

Why did the Chicken Cross the Mobius Strip?....

[CaffeineJedi]

To get to the other

Re:Why did the Chicken Cross the Mobius Strip?....

[dargaud]

You mean: to get to the same side, no ?

The scientific principle of Möbius strippers

[jollyreaper]

Möbius strippers never show you their backsides.

algebraic equations for Mobius strips are not new

[coult]

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).

Correction

[TheEmptySet]

To a topologist these small differences do not matter, so any loop is a circle and any half-twisted flat loop is a Moebius strip. And the Moebius strip is specifically a (smooth) topological object.

Re:Correction

[jd]

Gaah. You are correct. (My excuse is that I'd failed to update my brain to the latest Linux kernel, causing an unexpected error.)

Re:Mobius strip

[Surt]

I know this is just flamebait, but you are aware that all of the modern disease cures are built on heavy amounts of basic math developed by previous generations of mathematicians, right?

Algebraic equation

[benhocking]

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.

Correction

[benhocking]

Energy is power times time, or force times distance.

Möbius trick

[ls671]

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.

My lack fo understanding denotes..

[i8myh8]

..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 /.

Re:My lack fo understanding denotes..

[shish]

please make links with a target of _blank so that we don't get taken away from our beloved /. Middle click.

Mathematicians

[benhocking]

Well, them, and occasionally Star Trek writers.

Re:The scientific principle of Möbius strippe

[Dunbal]

A lap dance could last forever, though...

I am not a topologist

[idontgno]

but the energy they speak of might be related to Willmore energy. 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.

Re:I am not a topologist

[florescent_beige]

If you were trying to make a joke so obscure nobody would ever even notice it was a joke, let alone get it, you have failed.

Step 2 is wrong.

[TheEmptySet]

You are right that you only get one object, not two as one might expect, but it is not a Moebius strip. Here's how to verify this fact (and teach the kids even more cool geometry while you're at it):

a) Trace a line around the surface of a Moebius strip, you will find yourself drawing on both sides before getting back to where you started, hence the strip only really has one side.

b) Now do the same with the object you made in step 2. You will find this behaves far more normally as it has two sides (i.e. you can't get to the other side without taking the pen off the paper).

Enjoy.

Re:Step 2 is wrong.

[ls671]

You are right, sorry it was a long time ago.

I now remember asking my aunt :

What will we get when we try to slice this Moebius strip in 2?

After the fact, she said : Oh ! still one Moebius !

Then, what will we get when we try to slice this Moebius in 3?

After the fact, she said : Oh ! 2 Moebius!

I just confused my aunt's words with realty ! I remember the drawing the line test also ;-)

Cheers,

In other news, a team from..

[LM741N]

Sweden just figured out the differential equations governing a noose.

Re:In other news, a team from..

[Sponge Bath]

Sweden just figured out the differential equations governing a noose.

A Nøøse once bit my sister ... No realli!

Re:In other news, a team from..

[Artifakt]

Does your sister by any chance know an Oslo dentist, who stared in "The Huge Nolars of Horst Nordfink"?

Re:In other news, a team from..

[Limburgher]

Nolars? Are those neeth? :)

Re:Sigh

[Ambitwistor]

Mod -1, NonSequitur.

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.

Re:Blantant Copyright Violation. Article Full Text

[omeomi]

The Möbius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180

Can I use metal?

;-)

Re:Mobius strip

[be-fan]

You'd understand the significance of this sort of work if you had a background in engineering. The utility of this work isn't just in understanding mobius strips. The methods used to understand such structures can be used to understand other types of structures.

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.

Brilliant

[wamatt]

a true slashdot classic!

Re:There's been a formula a mathworld for years

[The MESMERIC]

thank you!

wished they stopped doing that - post articles where you have to pay or sign-up to get to the real juicy part.

Re:Mobius strip

[Chris Burke]

What are you doing posting on slashdot instead of fighting AIDS or Famine? Hypocrite.

Re:Möbius trick

[gardyloo]

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 hard to figure out why the middle third turns into another strip, and the outer thirds produce a "regular" piece of paper, and why one resultant strip is longer than the other.

No one will ever solve my riddle!

[M0b1u5]

No one will ever solve my riddle!

MUHAHAHAHAHAHA

Re:Sigh

[PDAllen]

There is a difference between 'elastic energy exists' and 'here is the shape formed when a material is elastically deformed to satisfy certain boundary conditions, subject to minimum elastic energy criteria'.

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.

Never been done.... NOT!

[mark-t]

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 from a point on a circle of radius r, and rotating it around a vector tangent to the circle at that point. The rate at which you progress around the main circle is twice as fast as the rate at which you rotate around the point on that circle.

Varying theta from 0 to 2Pi, you got all the points in one complete strip, with opposite points along the edge differing by an offset of plus or minus Pi in theta. One could also vary from the point along one edge to the point on the opposite edge to obtain the set of points on the surface, parameterizing the surface in only two variables.

Re:Never been done.... NOT!

[JoshRoss]

Nice Pancake, I think that you missed something.

Re:Never been done.... NOT!

[JoshRoss]

Or maybe I missed something. I must say that my math strategy has a been more of a guess-and-check-and-guess type of proposition.

Re:Never been done.... NOT!

[mark-t]

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 we did in 2002... so I don't know how this could have been such a big riddle.

Re:Never been done.... NOT!

[aldo.gs]

Well, I guess that first of all you'd need to show that the parametrization you bring is really (or can be seen as) an algebraic equation. Please note that the use of sine and cosine functions disrupts the polynomial (although you could expand it as a series, but then you'd have a formal series instead of a polynomial).

Anyway, I can't read the article, but in the abstract they don't even mention the algebraic equation. It has more to do with tension and other material properties.

I beg your pardon

[baxissimo]

But if most everyone thinks it does, it might as well.

Re:I beg your pardon

[NeilTheStupidHead]

But if most everyone thinks it does, it might as well. After a quick survey of my immediate associates (about 15 people, so not a statistically large sample to be sure), the only one who thought 'begs the question' might be correct is the only one in the group that hasn't graduated high school (a 17yr old). As far as your actual attitude regarding language, I say to you: "You're right because clearly cabbage soppy wankel ebbeh gruntsponge." ^>^

Re:I beg your pardon

[moderatorrater]

Intelligent argument, other than the fact that it makes no sense. The point of language is to convey meaning and ideas, and as long as a phrase does that, why should we care about anything else?

Re:I beg your pardon

[gowen]

We already have a perfectly good phrase that means "raises the question". It's "Raises the question". If you co-opt "begs the question" to mean the same thing, your ignorance robs the rest of us of the only succinct way of saying "contains a hidden supposition which is contains the supposed conclusion."

WHY?

[yusing]

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, timeless as infinity?

Is there no human being, anywhere on the earth, tall, emaciated, daring, who will undertake to have his feet taped to his head, and report to us the nature of this invisible dimension? So many questions arise: when held to a mirror, will he see himself? Will he discover other feet-taped people there? Will it dark in the day, light in the night there?

(Continued next issue)

Re:WHY?

[!eopard]

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, timeless as infinity? Initially a mobius strip is two dimensional, but when you tape the two ends together you get those twists. This take the mobius strip into the third dimension - and THAT's where your other side went.

My Very Own Summary

[florescent_beige]

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 solution they are able to predict how different flat sheets deform into strips. Apparently, as the sheet becomes wider, the resulting strip becomes more triangular. The mechanical energy density of bending becomes more concentrated into creases.

Developable surfaces are of practical interest because it is easier to form sheet materials by bending them in one direction than in two directions. Bending in two directions requires stretching of the material and is harder to do. For example, to make a cylinder out of a sheet of tin, you could bend it with your hands. To form the same sheet into a sphere, you would have to bash it with a ball-peen hammer for several days.

The design software for composite structures, like the 787, includes programs that generate "flat patterns", which are the flat shapes the uncured composite fabric has to be cut into so it will form into the final shape of the part. Those are sort of developable surfaces, but not strictly so, because the fabric is a bit stretchy and the manufacturing process takes advantage of that fact.

cDonald's Theorem

[wakingrufus]

It turns out, the mobius strip is just a variation on cDonalds Theorem: n^2 + 9 + 9 Scientists who study MATHS (Mathematic Anti Telharsic Harfatum Septomin) have yet to find a name for the figure made by plotting its graph. If you can name it, the royal mathematics society would like to here from you!

Intelligent General Reader write ups

[kgp]

Two easier to read commentaries in Nature and Science

Re:Möbius trick

[ls671]

Cool and corect, I already replied to the poster agreeing he is correct and explaning the reason of my mistake ;-)

Re:Mobius strip

[rhoder]

You're talking about friction, something Feynman studied with some fervor. Maybe the Journal of Friction and Wear [http://www.allertonpress.com/journals/faw.htm] would be a good place to start. Best of luck!

Mobius ring

[realsilly]

My dad gave my mom a Mobius strip wedding ring. Neither one of them had any idea what it was.

It's awesome....when she dies, it mine.... .... cuz I'm gonna take it. /look left /look right /vanish

Double-edged sword

[woolio]

I know this is just flamebait, but you are aware that all of the modern disease cures are built on heavy amounts of basic math developed by previous generations of mathematicians, right?

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.

Re:Double-edged sword

[somersault]

And once technology has advanced enough, people will no longer have to collect garbage. I doubt anyone aspires to becoming a garbage collector/janitor, it's most likely that they are only doing it because they don't have any recognised 'skills' for a 'better' job, or there are no free jobs available that utilise their skills. You can be thankful that the job is being done, but it's a bit strange to respect people for doing that job unless they actually chose to do it? Without doing R&D of any kind, we would be shivering naked in a forest somewhere..

Re:Double-edged sword

[bindo]

And once technology has advanced enough, people will no longer have to collect garbage. I doubt anyone aspires to becoming a garbage collector/janitor, it's most likely that they are only doing it because they don't have any recognised 'skills' for a 'better' job, or there are no free jobs available that utilise their skills. You can be thankful that the job is being done, but it's a bit strange to respect people for doing that job unless they actually chose to do it?

And you probably haven't worked alot in our life. (teenage ? 20 something?)
Besides, the world is full of people that "choose" not to work but to depend on somebody or worse.

You certainly NEED to respect people who do a job you wouldn't want to. To second guess their choice is childish. In fact, if you believe they had no choice but to do a job YOU would rather not this sounds alot like exploiting them .....

You are right on one count. Someday bots will do the dirty job.
When doing R&D you will have changed my life I will respect you.

Until then, in my personal hall of fame: Tesla, Marconi, Steve Jobs, janitors, garbage collectors.

Bindo ...

Re:Double-edged sword

[somersault]

Don't get me wrong, I think people doing those jobs should be treated with respect, but I think that of all humans unless they do something to show otherwise (such as having a lack of respect for others!). Yep, I'm 23, and have been rather lucky in which job I got - worked in the IT dept for my uncle's company over the summers that I was at Uni studying Comp Science, and now I'm the IT 'Manager' - basically because I'm the only IT staff here, but I'm in charge of all the servers, sorting any printer problems people have, writing apps for the departments to use (have done a timesheet system and equipment tracking app for the engineers and subsea divisions.. also wrote a task manager for the IT department - which used to be 2 guys - in my first summer here but they were too lazy to use it ;) ).

Anyway, I don't want to exploit people, and it sucks that people have to do the dirty jobs at all, though some people just don't have a choice. There may be some people who love the idea of emptying trash or just making the world a nicer place to live, but I doubt they make up the majority of binmen/trash collectors/janitors? :O Different people are suited to different careers though, and even though I would hate to work in marketing, I know that some people like that kind of thing. In fact if it comes purely down to jobs, I do respect the trash collectors more than those guys ;)

Re:Double-edged sword

[bindo]

Listen,

sorry if I'm patronising but you really get only half the point. I already knew you didn't want to exploit anybody. Fact is you have a choice when you rationalize all this:

1)it is morally RIGHT that precisely those people live with their hands deep in shit for your well being,

2)there is a free DEAL under which they do it for your well being. Never second guess why they took the deal, just respect the other party.

Anything else is exploiting them, and as you seem a decent person I guess you know option 1 is really not an option without some NASTY ideology/philosophy in support.
Even if they were free to get the deal only because they didn't have the freedom to get any other deal...

So what? If You are free to choose it's mostly your family's hard work. Not your's. And its not these peoples' fault if they weren't educated. Even the most low IQ person you now can do more than cleaning the floor. Alot more. Besides most of them ARE educated at least enough to do a different job. Though choices happen, people fail. These guys are those fighting back.
That is why you must respect them.

Talking about marketing, I hear your pain :) but the same goes on for them, I guess you have to pay the rent and if you miss technical skills...
Still if Edison wan't in my list there was a reason :)

Bindo

Re:Double-edged sword

[somersault]

Yeah, I don't think that just because someone doesn't perform well in educational tests that it means that they should have a crappy job, because as you say they can do a lot more.. for example someone with ADHD may not be able to concentrate to learn things in class, but could still perform very well with problem solving, others may be crap at math/english but good at art, some people may be good at sports (of course to be fair, there will be some less fortunate people who aren't particuarly great at any one thing - but as you say, any human can learn/do more). I guess what I'm saying is kind of obvious too, thanks for clarifying anyway.

I think there is a skill to marketting too, and some people genuinely actually want to work there ;) If it wasn't for good marketting, Windows wouldn't be in such a dominant position though, VHS wouldn't have won, etc. That's pretty much the reason I have such negative opinions of that whole area - good marketing can unfortunately drive a bad/mediocre product to succeeding over a better product.

Re:Mobius strip

[Nazlfrag]

You should turn back, it's taken me twice as long of a hike to get back to the bottom, and I'm still going.

Re:Mobius strip

[Goaway]

Slashdot: News for anti-intellectuals, stuff that confuses us.

Mobius strip

[ronaldg]

What if it were molded to shape? There would be no bend stress.

Re:Mobius strip

[Toad-san]

Very good point. Because I don't see why those stress areas are "required" at all! Given a thin enough (and long enough) strip, I submit that a Mobius strip could be formed that had any stresses evenly distributed along its entire length! There's no need for "kinks" anywhere!

No stress, no math. bidda bing ...

Misleading

[joeyblades]

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 and Quantum mechanics). The actual discovery from Starostin and Heijden relates materials science to deformations of the Mobius shape. Interestingly, even this seems to be quite similar to Penrose's work tiling Mobius shapes. Actually, it also looks a lot like the work of Andrzej Sitarz in 2001... I'm starting to wonder where is the inovation of this "discovery"?

Re:Misleading

[singingjim1]

Dude...you're smart. Maybe TOO smart. I got my eye on you joeyblades.

Re:Misleading

[joeyblades]

'cept for the part where I misspelled "innovation"...

Re:what assumption?

[Hatta]

Excellent link. I'm just wondering about a point of clarification. If you were to say "I can't tell you because it's secret", would I be right to respond "That begs the question, why can't you tell me?"

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.

That, too, of course

[benhocking]

But the intention was to warn you that I am not a topologist.

New and only Moebius Strip sci-fi novel!!!

[Moebius Tripper]

I just recently read D. Richard Lewis' sci-fi novel called: "TIME TRIP ON A MOEBIUS STRIP," and it ties in, or should I say, "twists in," pretty well with this new theory these two scientists came up with explaining the riddle of the Moebius strip... The plot in this sci-fi novel is quite interesting...The author has this marine biologist discover a giant nautilus shell on the beach and then with the help of the great grandson of Professor Moebius, constructs a giant metal Moebius strip in the shell...The marine biologist then rides a vehicle upon the strip and enters another dimension where he then meets 16 famous missing people of history who have arrived in this time-less domain via a cloud...There is an angel in the story as well... The author has also discovered many amazing connections that these lost famous people have with eachother...I was in awe by them...Carl Jung would have probably tried to tie these connections in with his theory of "archetypes," which is what the author does through one of the lead characters who is a woman psychiatrist. The novel was a great feat of historical research and quite original...A++++

I stand by the usage

[PCM2]

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.

Tape reels

[shish]

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?