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)?

[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?"

Obligatory link

[amstrad]

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

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.

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: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.