Monday, September 01, 2008

Concentric Winder

[Revised August 2010]

I promised some comments on this basic shape---which is more a geometric demonstration than an origami model, but one that still needed inventing, discovering or just someone to "claim it" as their own.
By the latter I mean, that I'm pretty sure that all the people who worked on cone-folds over the past half-century---most notably Ron Resch and David Huffman---would have come across this idea themselves, in the course of fiddling to find a cone-fold state that is aesthetically most pleasing to them. But if so, none of these pioneers paused and said "this is sufficiently interesting to put my name to it." For it is a "stupidly simple" idea, maybe too stupid for David Huffman, an engineer who liked things to be simple but to at least seem complex. Yet as you'll see, here and especially later, there are a lot of consequences to this one form, which you don't realize UNLESS you embrace it, stupidity and all.  --Take it in and feed it, like you would a stray cat.

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First off: How does it work? Well, if you had just a paper disk and made a cut along a radius, but no mountain-valley folds, you could twirl one layer up indefinitely underneath the other to form a cone—and the more you twirled it the more acute the cone would get. Now you've added mountain and valley folds, but so what? The cone reverses its direction along the concentric folds: that's the same as turning a cone upside-down in the air. Nothing in the above logic has changed. But it does look a lot stranger.

Here are a few other thoughts which this somewhat hypnotic model prompts.
  • Align the slits, count the layers—and you’ll know just how much you’ve shrunk. If the slit on top and the slit on the bottom line up, you’ll have made at least one complete turn. The width of the Winder, compared to the initial disk, shrinks exactly in proportion to the number of turns: 3 layers or turns yields a Winder that is one third the size of the original disk, and so on. (The outer edge will consist of three full circles on top of each other, so it is dividing the original outer edge into 3. Since C= 2pi r, if C is divided by 3 so is r.)
  • Negative illustration of the Albers Effect. Some of you may have seen the twisty, contorted shape you get when you fold a paper disk that has mountain and valley folds, but no radius cut. That is a phenomenon explored and probably invented by Josef Albers, in the 1920s. The contortion happens because the radiuses of all the circles are shrinking (toward the center, along the folds), but the circumferences are not (there are no folds interrupting them), which is not allowed by the above law, C = 2 pi r. This "Albers Effect" contortion is avoided here--the shape stays flat on average--because the excess circumference is allowed to slide over the layer below. This point was stressed recently at the Italian origami convention, by Herman Van Goubergen.
  • Positive illustration of the Albers Effect. If you unwind a tightly-wound Winder, eventually it will start to approach the condition of a disk without a radius cut. That is, at some point it will stop being satisfied to be flat-on-average, and the surface will begin to wobble. Unwind it further, and the contortions of the Albers Effect will begin to form. (Question for extra-credit: why does this happen when it happens, and not before?)
  • Illustration of Curve-fold Law II. One of the laws of curved folding which is most significant for origami is that the surfaces to either side of the crease—which are also curving—can never through continuous movement be brought flush to each other. With these curves which are perfect circles you get a limiting case illustration. Here the walls to either side of the crease not only never become flush, they never even touch—yet in principle they could be brought closer ad infinitum.
    This too is an answer to a question that many of you (OK, some of you; OK, just me) may have asked, namely is there anything special about circular curve folds, compared to folds that are curving but not circular. Clearly this sort of infinite winding maneuver is possible only for sets of concentric circles; it would not work for any other set of curve-folds, and not for sets of circles that are not concentric.
  • Vary the spacing. Notice that while the concentric circles drawn here are equidistant, they didn't have to be. Instead of a flat-on-average surface you could make, for instance, a dish shape that grows shallower as you unwind it—or any number of other shapes. Try something new!
  • Complaints about the Cut. I showed this form in Chicago to Bradford Hansen-Smith, who is easily the world's most accomplished, and certainly its most obsessive, explorer of folds that begin from circles. His first reaction was: I tell my students, NEVER cut the circle.
    My answer (now: in person I was more polite): Sheesh! Do you think that as an origami person I LIKE making cuts of any kind? If you insist on avoiding a cut, you can fold the circle in half after scoring the concentric circles, then do the same maneuver along the folded edge. It will fold up to a tiny shape in the same way. But it won't reopen to the full size of the initial circle, only back to the semi-circle, which is a loss, it seems to me. Besides, this form is saying something important about the relations of circles, cones, the dynamics of curve-folding, AND a radius---which is properly conceived as a cut.
  • Similarity to other forms. This Winder is related to other forms that exist in origami today, in particular, a certain nice extension practiced by the Demaines (and invented by them?) to the idea behind “Thoki’s Hat”, by Thoki Yenn. In that variation, instead of starting from a single disk, you start out from a flat “disk” with an “extra-long circumference.” Take several paper disks, stack them on top of each other, and cut them all through at one radius; then join the radius of the top disk to the radius of the disk underneath, and so on going around like circular parking garage. Finally corrugate the whole stack with mountain-valley concentric circles: this gives you a single, ‘extra long’ disk to work with—or an extra-long annulus, if there’s a hole in the middle.  --In this context it may be worth pointing out that the Concentric Winder creates a similar sort of a ‘stack’ without having to glue disks together. It also follows that some of the forms produced by one method, should be imitable in the other.
    The point to bear in mind is that when you wind up this form, you get a stack of disks that happened to be joined, each to the one below it, along a line. The existence of these mostly separate layers also creates possibilities for manipulating them separately---the significance of which, we'll see with the Sphere-from-a-Circle.

    [Added March 10, 2013.]  In response to a query on the O-list about folds that have great compressive strength, I would venture that with a few winds (say, 6 or more) this form seems to me to be the strongest one possible that can be achieved via folding, especially given that one can wind it up further indefinitely, as needed.  Winding it narrows the angles of the corrugations, bringing them closer to the vertical, and also adds to the number of layers of the paper. Meanwhile the fact that the corrugations are circular means that there isn't some way the walls can move in any direction in response to pressure from above or even vertical pressure + perturbations from the side.  (Compare this to what may be the next-nearest competitor--the  fold named for its chief promoter, Professor Miura.)

    Related Posts:
    Concentric Circles (March 2008)
    Huffmanesque (January, 2009)
    Organic-Circle Fold (March, 2009)
    Sphere-from-a-Circle  (May, 2009)