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

There are lots of ways to get a sheet of paper to fold up nicely. Here is one way that I would never have come up with myself.

It is based on a simple fold-pattern by David Huffman: a disk with concentric mountain-valley circles, a single radius cut, and a tiny donut-hole in the middle. Based on? No, it is the same fold-pattern, the one from among the Huffman photos shown by the Institute for Figuring. (Unfortunately it was mistakenly labeled a “Tower of Concentric Circles”— although (a) it is not a tower and (b) Huffman has another model that is.)


Some tentative remarks on this form recently appeared in Erik & Marty Demaine’s useful if rather partial "history of curved origami sculpture". Yet at least as of this writing (mid-March 2008), the Demaines neglect to mention the most striking feature of such concentric circle patterns--and the one Huffman, surely, was exploring: 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 go on forever winding the thing up, 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.

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 which as they say was pursued over the last century by Josef Albers and his students in the Bauhaus, and 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 is used to give wavy plastic rooftops and corrugated cardboard their added stability.

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 does. 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 are adding some springiness to the paper, pulling the edges, and indeed every part of the interior, closer to the center. That means that 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 likely to be saddle-like.

If you hold a sheet of paper taught in your hands, and then move your hands closer together, the sheet will also bulge from the plane, for the same reason. It has nowhere to go but up or down.

The Demaines elsewhere write that “We know almost nothing about curved creases,” and in this context state that "forms that we are just beginning to understand" can be made by various new permutations of the concentric-circle technique. Presumably these expressions of ignorance or humility aren't being made only on the authors' own behalf, but for all of MIT, perhaps for all of Computational Origami so far as they are aware, or for 'art and science as a whole'. But the math, at least, that governs transitions in surfaces that undergo differential expansion (which is what is forcing the curvature) has been known for some time, and the physics too has been investigated at some depth--among others by Eran Sharon at the Hebrew University of Jerusalem (for a general discussion see American Scientist, 2004). More to the point for my purposes, this sort of curling or bending behavior is not unique to concentric circles: ANY folds that squeeze the interior of a sheet faster than the exterior, are going to cause the remoter parts to warp and bulge from the plane. The curling phenomenon will happen with concentric circles that don’t share a common center, with ellipses, with spirals, with suitable non-parallel curves, curves that do and that don’t intersect, indeed with suitable straight folds too. The outcome is, I grant especially elegant and ‘pure’ with concentric circles, but unlike the phenomenon Huffman was exploring this is not a property unique to concentrics or even to curved folding as such.

The results--as nicely exhibited by the Demaines in their work in the show presently at MoMA--are interesting and visually striking nevertheless. So it remains to be seen how this sort of technique can be adapted for the purposes of an origami that is not only mathematically inspired, but also expressive.

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 show, 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 that juts 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 shifts the emphasis in it from a noun phrase (‘this is an A’) to 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—as can happen in a lingering dream—and thus alter 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 instant. 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 is this union of souls expressed here, with matador and bull made of the same stuff and the same color: separated now only by the sheet that joins them, with its metaphysical thinness of the color-change. (As in that beautiful English word which also means its opposite: to cleave, which is both to cut from and to cling to--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 South Africa. Some place locked within fields and mountains, where the news of the world does not fast filter in. For when one is disheartened 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 on a day you ran off from school; the dexterous grace of the knife-thrower; the joys, joined, of intelligence, simplicity and magic.


A moment of truth.

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.

Curved Folding, revisited

Work in progress. (i.e., I haven’t ruined it yet.)

The aim and method is the same: the minimization of lines; the exploration of the inherent softness there is in smooth paper surfaces joined by curves and open folds. I think in the end paper has more possibility for expressing tenderness, or at least receptivity, than even skin on flesh, nature’s original. But for that one needs to get away from the angularity of most origami.

It’s been about a year since I’ve done anything new in this direction. I guess we all have stages where that keening high origami brings, fades of itself or is forcibly set aside. I have Joseph Wu to thank with his invitation to exhibit at Vancouver's Pendulum Gallery (in tandem with PCOC 2007) for getting me jump-started: What I sent him may have been older material, or things derived from older ideas, but the juices now seem to be flowing. I doubt that anything less than the knowledge that people like Joseph and Michael LaFosse and possibly Roman Diaz will be there (among other luminaries) to see those things and harrumph at them, could have got me going---

Cheers!

Saadya