Ribbon physics

At March Meeting, there was a fascinating talk about the phase diagram of ribbons.  Yeah, just like the solid/liquid/gas phase diagram of water!  Their paper is called Helicoids, Wrinkles, and Loops in Twisted Ribbons (see on PRL if you have journal access).  Here’s the first figure:

Ribbon phase diagram

Left side: many possible responses of a stretched and twisted ribbon.  Right side: a phase diagram of the ribbon as a function of stretching force (T) and twisting angle (θ).

The researchers claim, “Our results can be used to develop functional structures”.  I don’t know what that means, but I’m already sold!  I am going to replicate this experiment!

The researchers use a rather fancy ribbon made up of polyethylene terephthalate.  I’m fresh out of that, so I cut out a strip of origami paper.  I gave it a little twist, and behold:

IMG_0879 (small)

You can already see a bit of the zig-zag crease pattern.  This is what the researchers call a creased helicoid.  Here’s another picture with the creases traced out:

IMG_0880 (small)

The zig-zags aren’t quite perfect, but I suppose that was to be expected given my budget (I couldn’t even afford the CT Scanner).  Origami paper isn’t an ideal ribbon, since it has some geometrical stiffness, as well as memory of past creases.

For my next experiment, I twisted the ribbon further and pulled a little harder.IMG_0884 (small)

This phase is called transverse buckling.  If you take a vertical cross-section down the middle, you find that the ribbon is bent.  In the zoomed image below, you can see that the white dot is not directly between the two black dots, but below them.  That’s what buckling is.

buckling 2

Please try this at home!  You can try a variety of materials, although paper was the only thing I found that could make a creased helicoid.  I bet you could do it with kids too!  It would show them how complex properties emerge from simpler ones.  That’s what condensed matter physics is all about.


  1. DonDueed says

    I don’t know, I think all that pulling and twisting might be a bit painful for the kids. And it might take awhile to get the creases out of them.

  2. Lee Rudolph says

    You can try a variety of materials, although paper was the only thing I found that could make a creased helicoid.

    Have you tried leather (e.g., an ordinary belt)?

    I’ve done some (purely theoretical) work on the kinematics of various structures that can be mathematically represented as linkages (so, e.g., not just actual linkages but also origami, flexagons, paper bags, etc.), with potential (and some actual…) applications (thanks to co-authors!) to motion planning for robotic hands, protein folding, etc., all via (somewhat novel) parametrizations of their configuration spaces. You’ve got me wondering if those methods might apply here! Thanks.

    P. S. In my knot-theoretical avatar, I prefer “unknot” to “trivial knot”, but I admit “trivial knot” is catchier and is probably preferable for your purposes.

  3. says

    @Lee Rudolph,
    I did try a (leather?) belt. If a creased helicoid is forming, it’s very subtle and I can barely see it. I’m not sure what material properties are most important to show longitudinal buckling and a creased helicoid, but I’m guessing incompressibility and flexibility.

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