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.

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Some tentative remarks on the Huffman piece 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 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.

If you hold a sheet of paper taut in your hands, and then move your hands closer together, the sheet will also bulge from the plane, for about 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 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 truth is that the subject of the differential expansion of a sheet, which is what is causing the buckling or rippling, is a reasonably well-studied phenomenon in contexts other than origami (for a layman's  discussion see American Scientist, 2004). More to the point for my purposes, this sort of rippling behavior is not unique to concentric circles: ANY set of folds that squeezes the interior of a sheet faster than the exterior, is going to cause the remoter parts to warp and bulge from the plane. Saddle-shapes will form with concentric circles that don’t share a common center, with ellipses, spirals, suitable non-parallel curves, curves that do and that don’t intersect, indeed with suitable straight folds too. The outcome is especially elegant and ‘pure’ with concentric circles, but this is not a property unique to concentrics or even to curved folding as such.

The results of the Demaines's explorations of concentric folds, now in an exhibit at MoMA, are interesting and visually striking. I am pleased for them but chagrined by the title they chose for their installation: "Computational Origami". So far as I can tell their objects were made by a process no more and no less "computational" than the very similar works taught 40 years ago by the late Thoki Yenn (followed by Kasahara, but preceded by Josef Albers), which the Demaines now have extended--nicely, but not in any way 'computationally'. Given the rarely seen look of such origami, the public can't fail to conclude, quite mistakenly, that "this is what origami looks like when computers are involved."  And given that Erik is a professor of computational origami at MIT as well as manager of the archives of Thoki Yenn, that title gives the unfortunate impression that the field is being grabbed for himself at his teacher's expense. Sigh....  The Thoki Yenn twisting-origami style is clearly eye-catching and I can only hope the Demaine's work will prompt more origami artists to explore it. For myself, I am curious to know whether this technique can be adapted for the purposes of an origami that is not only abstract/geometric, but also figurative and expressive.


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.