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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."
I found this: http://www.faculty.fairfield.edu/jmac/rs/halftw.ht m
Layne
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.
In case anyone is confused, InvisiblePinkUnicorn is referring to this drawing by M.C. Escher
Anthropic principle: We see the universe the way it is because if it were different we would not be here to see it.
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.
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.
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.
Energy is power times time, or force times distance.
Ben Hocking
Need a professional organizer?
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).
After all, I am strangely colored.
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.
Welcome to the Panopticon. Used to be a prison, now it's your home.
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.
After all, I am strangely colored.
... or the rather pretty xscreensaver hack.
"Nine times out of ten, starting a fire is not the best way to solve the problem." - my wife
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.
File under 'M' for 'Manic ranting'
Two easier to read commentaries in Nature and Science
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"?