The Shape of a Moebius Strip
Yes, mathematicians have known how to characterise a Moebius strip for aeons, but this recent paper by Starostin and van der Heijden [1] is concerned with characterising a developable Moebius strip: one that can be mapped isometrically to a plane strip. An isometric mapping is a mapping that preserves all distances. The most general mathematical characterisations of a Moebius strip, such as a subset of the projective space RP^2 or a normal vector that makes half a turn as it traverses a closed path, do not contain the restriction that the surface thus produced preserves all distances when mapped to a two-dimensional, flat strip of paper.
Starostin and van der Heijden characterise a developable Moebius strip by modelling it as an actual paper strip, complete with elastic properties, inextensibility, etc. This model approximates a developable Moebius strip because it requires much less energy to bend a paper strip than to stretch it, meaning that distances on the strip are, to a good approximation, preserved. Given the dimensions of the paper strip, the appropriate boundary conditions (imposed at s=0 and s=L/2, where L is the length of the strip and s is some measure which I'm not quite clear about of distance along the length of the strip), and the linking number (more on this later), the paper strip settles into a shape that minimises the deformation energy, which is due entirely to bending (since the paper is inextensible). Assuming the paper to obey Hooke's linear law for bending, the elastic energy can be calculated from the curvature and area of the strip.
Starostin and van der Heijden are not the first to tackle this problem: a certain M. Sadowsky had formulated the problem before, but did not attempt to solve the equations he had set up. Others after him had simplified the problem somewhat, but not to a point where it was feasible to solve the equations even numerically. Starostin and van der Heijden eased the computational problem using a geometric approach based on something called 'varational bicomplex formalism'. They were thus able to obtain numerical solutions for the curvature and torsion of a paper Moebius strip for several values of the width w (given a fixed length L). Their solutions confirmed a prior prediction that at and above a critical value of w/L the strip will collapse into a triple-covered equilateral triangle, like so:
The colours indicate local bending energy densities: violet for low densities ranging to red for high densities. One thing that I found interesting was that there were always three patches of low energy densities (three purple patches), but if you look at the profiles of the strips when cut apart and stretched out, you could see that there was always one largest purple patch and two smaller ones of about the same size:
Intuitively (without mathematical justification), I would have expected all three patches to be about the same size.
I wondered if perhaps the one large patch is a consequence of the linking number being 1/2 instead of, say, 3/2? They did solve the equations for a linking number of 3/2, but I could not tell from the figures provided the relative sizes of the three purple patches. I'm not entirely sure what linking number is, but intuitively it seems to have something to do with the number of times a closed cord is twisted in one direction. If the twists are in the other direction then they contribute negatively to the linking number: a strip with a linking number of -1/2, for example, is simply the mirror image of a strip with a linking number of 1/2. A Moebius strip must have a linking number that is an odd multiple of 1/2: 1/2, 3/2, -5/2, etc. The 1/2 factor simply means that one end of the cut-open paper strip must be joined to the 'opposite corners' of the other end, forming the half-twist that characterises the Moebius strip.
[1] Starostin E, van der Heijden G. The shape of a Mobius strip. Nature Materials, Vol. 6 (August 2007), pp. 563-567.
[Damnit, Blogger is absolutely hopeless with images, and doesn't support LaTeX either. I think I'm going to switch to Wordpress. I've just started something with the same name on Wordpress.com and imported all the posts on this blog there (they even successfully imported the images!). If you don't object, I can invite you two onto it.]
Starostin and van der Heijden characterise a developable Moebius strip by modelling it as an actual paper strip, complete with elastic properties, inextensibility, etc. This model approximates a developable Moebius strip because it requires much less energy to bend a paper strip than to stretch it, meaning that distances on the strip are, to a good approximation, preserved. Given the dimensions of the paper strip, the appropriate boundary conditions (imposed at s=0 and s=L/2, where L is the length of the strip and s is some measure which I'm not quite clear about of distance along the length of the strip), and the linking number (more on this later), the paper strip settles into a shape that minimises the deformation energy, which is due entirely to bending (since the paper is inextensible). Assuming the paper to obey Hooke's linear law for bending, the elastic energy can be calculated from the curvature and area of the strip.
Starostin and van der Heijden are not the first to tackle this problem: a certain M. Sadowsky had formulated the problem before, but did not attempt to solve the equations he had set up. Others after him had simplified the problem somewhat, but not to a point where it was feasible to solve the equations even numerically. Starostin and van der Heijden eased the computational problem using a geometric approach based on something called 'varational bicomplex formalism'. They were thus able to obtain numerical solutions for the curvature and torsion of a paper Moebius strip for several values of the width w (given a fixed length L). Their solutions confirmed a prior prediction that at and above a critical value of w/L the strip will collapse into a triple-covered equilateral triangle, like so:
The colours indicate local bending energy densities: violet for low densities ranging to red for high densities. One thing that I found interesting was that there were always three patches of low energy densities (three purple patches), but if you look at the profiles of the strips when cut apart and stretched out, you could see that there was always one largest purple patch and two smaller ones of about the same size:
Intuitively (without mathematical justification), I would have expected all three patches to be about the same size.
I wondered if perhaps the one large patch is a consequence of the linking number being 1/2 instead of, say, 3/2? They did solve the equations for a linking number of 3/2, but I could not tell from the figures provided the relative sizes of the three purple patches. I'm not entirely sure what linking number is, but intuitively it seems to have something to do with the number of times a closed cord is twisted in one direction. If the twists are in the other direction then they contribute negatively to the linking number: a strip with a linking number of -1/2, for example, is simply the mirror image of a strip with a linking number of 1/2. A Moebius strip must have a linking number that is an odd multiple of 1/2: 1/2, 3/2, -5/2, etc. The 1/2 factor simply means that one end of the cut-open paper strip must be joined to the 'opposite corners' of the other end, forming the half-twist that characterises the Moebius strip.
[1] Starostin E, van der Heijden G. The shape of a Mobius strip. Nature Materials, Vol. 6 (August 2007), pp. 563-567.
[Damnit, Blogger is absolutely hopeless with images, and doesn't support LaTeX either. I think I'm going to switch to Wordpress. I've just started something with the same name on Wordpress.com and imported all the posts on this blog there (they even successfully imported the images!). If you don't object, I can invite you two onto it.]
posted by Wowbagger at
12:53 PM
1 Comments:
seriously? ok. sure. i can use latex? send me the invite!! and adela too!
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