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)?

[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)?

[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)?

[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)?

[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)?

[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)?

[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?"

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.

[SQLGuru]

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

Layne

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!

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...

[Trogre]

... or the rather pretty xscreensaver hack.

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.

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:Mobius strip

[Vulva R. Thompson, P]

Relatively easy, just follow the guy with no face.

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.)

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

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).

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 /.

Mathematicians

[benhocking]

Well, them, and occasionally Star Trek writers.

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: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: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.

Intelligent General Reader write ups

[kgp]

Two easier to read commentaries in Nature and Science

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: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."

Re:Mobius strip

[Goaway]

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

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"?