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)?

[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)?

[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."

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:I Can't find It.

[SQLGuru]

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

Layne

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...

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?

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.

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:Mobius strip

[Goaway]

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