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.

* * *

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)
  
  
  
  

David Ben Gurion

Sisters

Yoshizawa, Spain and Japan

[I had occasion, a year and a half ago, to hold in my hands for the first time in my life some original works by Akira Yoshizawa, in the home of a private collector of Japanese art in Haifa, Israel: a collector of discrimination and taste who had been given these works by Yoshizawa himself in Japan in the 1960s or 70s. Maybe one day I will write the full story of that visit. Meanwhile, here are some notes from my journal at the time. Still sketchy & inconclusive; possibly ill-considered: --but this IS a Blog.]


---------------------------

January 14, 2007.

The Yoshizawa things. Made an impression on me. But yesterday too I was singularly impressionable.

Resilience of the paper. Much stiffer than you can imagine but still paper-like. The paper itself does not seem excessively thick, perhaps 70-80 lbs, but it has the stiffness of thick Bristol. I must try this heavy wet-folding myself.

Looking at these works---one gets, transmitted from the touch on paper, an idea of the experience of the man: the winters, the poverty, the struggles of post-war & post-atom-bomb Japan, and this man’s isolated & defiant lifelong battle to make this nothing into something---all that comes across. Of course I am reading what I know and imagine, back into what I saw. Even so, there’s a world of experience, in that still-visible touch.

I’ve been thinking vaguely about these themes of late. Origami in the 19th century was already starting to bubble up, to break out of the confines of tradition it had languished in for centuries---both in Japan and in the West. But in the West, in Spain and Germany, there were more of these bubbles; and when the tradition turned creative in the hands of a few individuals, it did so first in Spain with Unamuno, and very shortly afterward with Solarzano (in Argentina) and others; while in Japan it was Yoshizawa by himself for a VERY long time. Unamuno and Solarzano were far less prolific than Yoshizawa was in his lifetime, were probably less talented; certainly they were less single-minded. They did not have to fight so hard to change origami into something else, that would command respect, in part because the tradition that considered this occupation childish or female or of no-account was not as strong in Spain as it was and is in Japan; and also because the idea of doing something ‘of-no-account’ held (and still holds) a certain charm to the Spanish mind, did not NEED to be defeated. Witness Gaudi; witness Miro.

And so in Spain, an origami developed that was more social from the outset: it was done not by one person but by several---by members of an intellectual class, who delighted in showing each other & teaching each other their new fold-sequences. (In this respect Spanish origami may have picked up a kind of social lightness and grace that it had possessed during Samurai times in Japan.) In Yoshizawa’s hands, conversely, the attempt was to change the production of origami into an ART; that meant, understanding paper qualities, pioneering paper stiffening techniques, and developing an evocative sculptural touch that only a master practitioner can have. In Spain, an origami develops that is more about straight lines and teachable folds, about elegance and delight of sequence; in Japan, an origami develops which is more about the end-result than the process, or where the process is something one is more secretive about.


[…]

Material and Immaterial

Last week I went to see the paper arts exhibit at the Eretz Israel Museum in Tel Aviv (“Material and Immaterial”—a nice title, will have to use it myself sometime) which was curated by Paul Jackson of the IOC. A solid, tasteful, well-arranged show of paper arts of various kinds, mostly by practitioners in Israel. Very little origami—three or four installations I think, depending how you count; but with one powerful installation of representational origami by French master Eric Joisel. Of his three faces I especially liked one that looked to be from leathery paper, presumably wet-folded. And the Joisel vases/bottles too were compelling, especially as another, permanent exhibit in the same Museum has historical glass bottles that form a nice counterpoint to his. (Glass has, over history, worked its way out of opacity and into translucency, as blowing techniques developed; and some of that same transition is kept in one of the Joisel bottles—with the layered paper at the narrow top being opaque, the thinner bulge below suggesting lightness & translucency. Interesting to compare glass and paper----)

It is not a fault of this show, and I mean absolutely no disrespect to the many & gifted other paper artists there: but like most exhibits even good ones of contemporary art today, there is little that sticks in the eye, heart or mind five minutes after seeing it.

The exception is the origami, and that one Joisel mask in particular.

I ask myself whether I have a jaundiced view here owing to my fondness for paperfolds, or to my own specialty within it. But having looked into my heart, as they say, I think not; and I conclude that origami, alone among the paper arts, and almost alone among the contemporary plastic arts, has the ability, without shocking or social commentary, to linger in the brain, to cause delight and wonder. This is a quality, in other words, of the medium itself. Of course the particular works have to be good enough too. The choice of Joisel for this show in Israel was specially apt, for lots of reasons, but I imagine that works by other top designers could have had the same effect on this score, I mean would have shown origami to stand out from the rest of the paper arts, in a category apart.

But it was good to see these things together in one place, for it confirms a suspicion I’ve long had (and not me alone of course). Art-objects made from paper may be interesting, clever, colorful, ironic, contemporary, humorous, youthful, expressive, and all the rest. It really is amazing all the things you can make from this one immaterial material. And nevertheless and despite all that: the moment you take a scissors to the sheet of paper, all the magic runs out of it. It bleeds right out of the cut.


Saadya

Locks

At the AEP convention in Leon, Spain, earlier this month, I asked Miyuki Kawamura, the young Japanese rising star of modulars, the following question: How many types of lock are there in origami? Her answer: Hundreds. --Well, but in broad categories? --Maybe, twenty. 

Someone should publish a running list of lock types, with little sketches or photos. This would be an open database that anyone could later add to. It would not make that person money or much prestige but it would be incredibly useful for all of us in origami, for all sorts of reasons. 

I am not a modularist, but merely gaze on the field from a respectful distance. Yet even for us single-sheeters locks are important, and it is clear that modulars is where the subject is explored most thoroughly and directly. It has to be. 

I work on faces and much of the work tries to keep the sheet flat or nearly so, but sooner or later one wants to bend the sheet around and then the question is, how do you join the edges in back. 

Three-dimensional animal origami, which is all the rage nowadays, and rightly so, obviously also faces the same problem. Invariably there is a seam line, under or in back of the model. This is a consequence, almost mathematical, of the fact that the paper starts out with edges; and when you work flat, edges, though probably different ones, stay present every step of the way. You come to the end and still have them. If you started out with a tube you might have less of this problem, and with a sphere possibly not at all (nature’s clearest origami is indeed spherical--there is a blastula: it gastrulates), though even with these the problem of locking flaps exists. Also, such rounded forms are harder to work with: we actually need those edges of the flat sheet pretty badly. 

One has to admit, the sort of thinking that comes to the end of a process and asks ‘now what’, finding itself stuck with a problem it should have known all along it would encounter, is pretty defective. Though that’s the state a lot of us are still mired in. Komatsu in his owl, Diaz in his polyhedral/volume studies, and Joseph Wu in some of his 3D work have made efforts to carry us a little beyond this primitive condition. 

In any case locks are interesting, I want to say “satisfying”, all by themselves, quite apart from any pragmatic function they may serve in hiding ugly seam lines and as a replacement for glue. There’s a distinct pleasure when a flap fits into a slot and ties a form nicely together; when all the messy sliding about gets brought under single control; when all degrees of freedom suddenly disappear. --And it is origami’s job to study what is satisfying. 

So how about it, Miyuki? It would take you all of five minutes. (OK, five hours.) 

Saadya

Another bird---

Just seeing if I can't incorporate lessons from ‘technical origami’, here in the form of point-split feet, within the overall simpler, streamlined language that I still believe is more appropriate for birds in origami.

This model is under 30 steps long.

From a weird variation of the Preliminary Fold, which instead of dividing the center into 8 x 45 degrees, divides it into 12 x 30 degrees. More on this later.

S.

Concentric Circles

[Revised January 2010] There are lots of ways to get a sheet of paper to fold up nicely. Here is one way I would never have come up myself: that is, if I hadn't tried to copy somebody else's work. I failed (at first), but came up with a different interesting something which seems not to have been remarked on before.

The work in question was the piece of David Huffman's shown by the Institute for Figuring, who mistakenly label it a “Tower of Concentric Circles” -- when (a) it is not a tower and (b) Huffman has another model which is (!).  I imagined at first that this was simply a cone with concentric mountain-valley reverses, which is what I tried to make. You can see from the Huffman photograph that the cone-tip is twisting its orientation as it moves toward the center, which wouldn't happen with concentric-circle folds around a cone's apex, but I explained this to myself as either a trick of lighting or as some flexibility in the form that allows it to be bent that way post-folding (no such flexibility exists as it turns out).

My correct subsequent reconstruction of Huffman appears here. The rest of this article (except the final paragraph) proceeds as originally written.

* * *

Some tentative remarks on the Huffman piece recently appeared in Erik &Marty Demaine’s "history of curved origami sculpture". At least as of this writing, the Demaines neglect to mention a rather striking feature of such concentric circle patterns, the fact that you can do THIS:






Nice, no?

The math here is pretty straightforward. Think how if you have a disk with a radius cut, you can make a cone of it by tucking one cut edge under the other, and then sharpen the cone continuously by twirling the cut edge underneath. Now notice that the same reasoning applies also to all horizontal slices of the cone (=concentric circles on the disk), which can be made into mountain & valley folds. It’s a neat illustration of several kinds of symmetry, and a way of folding a sheet into a quite small shape that does not involve any straight folds. --How small? Mathematically you could wind the thing up forever, but physics as usual gets in the way: here in the form of the thickness of the paper, which causes the surfaces at some point to stick.

[Note, 2010: This is my "Concentric Winder." I have a few new thoughts about it, which I'll put in a separate Blog article.]

So far as I can imagine, this trick will work smoothly ONLY with concentric circles (though there is one other spiral form which almost works too, albeit with surfaces not exactly flush to each other). The mountain-valley pairs need not be equally spaced, but they do need to be circular and to have a common center. So, I will hazard the claim that this seems to be a means of compacting paper that is unique to sheets curved-folded by means of concentric circular folds.


Now, leaving Huffman aside, with this very same model you can also explore a different property of concentrics: the one that the Demaines are interested in, following work pursued early in the 20th century by Josef Albers with his design students in the Bauhaus, and later in the century by Thoki Yenn and Kunohiko Kasahara in the origami world.

Notice that when the form is wound up--with more than, say, one quarter of the circumference tucked under itself--the ridges of the mountain & valley folds add to the stability of the disk, which is flat on average. That is the familiar corrugation effect coming into play, the same method that gives the added stability to wavy plastic rooftops and corrugated cardboard.

When you unwind it though, a strange thing happens. The corrugations weaken---that is not itself surprising, as the mountains/valleys are growing shallower. But long before the disk becomes altogether flat, the surface will have LESS stability than a comparable disk without the mountain and valley folds. In fact, the disk starts to look for any excuse to break out of the plane: it refuses to stay flat! By the time the disk is unwound completely, it will naturally assume a contorted, saddle-like shape.

Why is that? Notice that the mountain and valley folds add some springiness to the paper, pulling the edges, and indeed every part of the interior, closer to the center. That means the circumference now has to occupy the same space as a smaller circle. It can’t do that while remaining in the plane, so it bulges out of it. The same reasoning applies to each of the smaller circles, so you get a nice uniform twisty shape that is saddle-like.

Why it's always 2 up, 2 down, and not some other number, remains a mystery to me.

Saadya Sternberg

Llopio’s Moment of Truth

This model with all its drama and flourish was designed by Neal Elias in the 1960s in the USA, and folded expertly a few years ago by Eyal Reuveni, here in Israel.

Full of suspense, presence and equipoise, it is the only ‘dated’ work I included in the Tikotin Museum exhibition, besides the Yoshizawas. But it not only holds its own against the four decades of advances in origami technique that came after it, it even remains distinct, like an island jutting from the sea. It’s kept its innocence too. Though it points in the direction of technical origami, this creation by the pioneer of box-pleating manages to avoid most of the clunkiness to which that technique is prone in lesser hands. It is not "showily technical", does just enough to get the job done, and so holds a quiet strength.

Elias was also the pioneer of the joined two-object model, and of course of the significant color change. I’ve noted elsewhere that having two objects of equal weight in a sculpture turns it from being a noun phrase (‘this is an A’) into a verb phrase (‘A is doing Y to B’). With two linked objects, besides an implied action or relation the viewer can shift his self-identification from one object to the other and thus invert the inflection of the verb. That may be part of the fascination. Here in origami the dramatic possibilities are still more pronounced: when everything is formed from a single sheet of paper, you perceive what the sculpture is meant to be, notice its different parts, and sense the fateful continuity between them, all in the same quick flash of recognition.

One does not need to have Spanish blood, or to have seen it shed in the bullring, to appreciate the drama of this moment. For this is what the viewers have come to see: the ‘Minotaur moment’, when matador and bull, before one of them cedes its life, suddenly become one. How perfectly this union of souls is expressed here, with matador and bull made of the same stuff and the same color: separated only by the sheet that joins them, with its metaphysical union and two-sidedness. (As in that beautiful English word which also means its opposite: to cleave, which is both to cut and to cling--both scissors and glue--and which term presumably derives from leaf, the primordial sheet.)

It pleases me to think that this old work on a Spanish theme might give courage today to folders in far-off places: to people in sunny or southern climes, in Spain itself perhaps, or in South America, or even southern Africa. Some place locked within fields and mountains, where news of the world does not fast filter in. For if one is dismayed by how much is made of technical origami in the various media today, it is good to remember what paperfolding once was, and still is, about: Firmness of purpose in the cleanness of line; purity of soul in the expanse of clean surface; whimsy and lightness, as in the dexterous grace of the knife-thrower: the joys, conjoined, of intelligence, simplicity and magic.

at the Tikotin Museum of Japanese Art.

Three Tenors

A homage of sorts to Carlos Corda.

S.

"Japanese Art"

I’ve been spending so many hours lately in the company of some great Japanese and Asian Art--for example, the okubi-e prints of Kitagawa Utamaro. It would be strange if nothing whatsoever of that rubbed off on me.... Here are a few things where the influence has, hopefully, been favorable. --I post them with some hesitation, for clearly there is still a long way to go.

These are all variations on a theme, wet-folded from paper of various kinds. Actually, I’ve been wanting for some time to branch away from foil-paper folding, my main medium for face-work. Foil-paper is a beautiful & rich medium unto itself, which is either a department of or a field adjacent to origami; but there are certain paperfolding values that foil manipulation just can’t capture. And in any case it was high time I learned how to wet-fold.

More is coming.

I’ve held off doing figures that are viewable 360 x 360, not because that is not a desirable result in itself, but because I wanted first to be sure I have a technique which preserved clean surfaces in the face, so that it would not invariably be grimacing, grotesque or “fantasy-oriented” as it is with some of the others who work onhuman figures. Now that I have such a method I can proceed to the rear and work down to the rest of the body, if need be. But I am in no hurry to get there. As I see it, the technical problem to be solved is not how to go all the way round or get all the way to the toenails. It is how to assure that each step does not “injure the paper”, to adapt a quaint concept from Yoshizawa.

The last few works are less “Japanesey” but still perhaps “Asian”: I may have had in the back of my mind (I certainly wasn't directly copying anything) some Tang Dynasty sculptures with the clump on the top of the head and that beautiful air of disarming tranquility, that you can make with a dollop of clay. That is possible with origami too. And since it is possible---it is necessary.


Saadya

Smell of a Bird Base

Been having trouble lately with my Birds. Partly this is nature’s fault: Many birds have at least three colors, rather than two: a topfeather color, an underbelly off-white, and red or brown for foot & beak. That’s not even counting the black of the eye. With origami you’ve got two basic colors to work with plus a pseudo-grey from shadow-pockets—not quite enough for this particular job.

One could, I suppose, bite the bullet, and add another sheet with one or two new colors. Joseph Wu has gone in this direction with his elegantly-marked Frigate Bird. And I seem to remember Nicolas Terry doing something similar to get a multicolored frog.

An option for those of us stuck, religiously, with single-sheet color-faithful origami is to confine ourselves to those birds that do follow a 2-color scheme. Pigeons, for instance, which run the gamut of gray-scales and blended patterns, sometimes come in an all-gray or all white body plus reddish feet. Likewise there are quite a few avian species in which the females, who tend to be interested in crypsis rather than showiness, keep their color-numbers to an origami-manageable minimum. (There are, of course, always reasons why an animal has the color numbers it has.)

Another problem is idiosyncratic---or maybe just personally idiotic. A long time ago (20 years… sheesh!) I convinced myself that a standing bird is fundamentally a four-pointed creature, from which it follows that it should be designable from a bird base (5 points: hide one). If you want to use the fancy techniques that have since come on line, to add complexity or detail---open a beak, pry apart toenails, start the wings a-flutterin---this ought to be doable optionally at the last minute, modifying extremities to taste, rather than by designing complexity in from the outset or inventing new bases as I believe Roman Diaz once argued is necessary. All the above poses a certain challenge, since to date I have not been able to pull this off entirely successfully.... Roman in a recent letter even claimed that “The smell of a bird base is difficult to disguise on a model.” But what to do? I happen to like the smell of a bird base.

So here's where we are, on a cold November night.