Some technical notes on the latest line of thought. Artist-types and casual readers, please ignore.
A fan shape, or for that matter a simple accordion pleat or even a single mountain fold, can be embellished with “dimples”. These can be curving folds as in the Peacock’s Tail (last post), but they can more easily be straight ones: squares or diamonds with a valley-fold diagonal. They can be applied to one side of the paper—mountains only—or to two, mountains and valleys both.
If applied to one side only, the sheet will start to bend or curl toward that side. And if done to both sides, the curling of the surface can be made to balance out. With fans, the curvature starts to assume a shape resembling a clam-shell.
If a pattern is applied to both sides, there is an natural version with a nice mathematical aspect, with dimples on one side repeated every fourth corrugation line, and diagonally below it.
Dimples interrupt the straightness and destroy the rigidity of the lines in a corrugation. Indeed it sometimes seems the only thing keeping a corrugation from collapsing forward is the line itself—that frontal mountain fold: with very little of the walls behind it doing the work, let alone the valley folds in back. Once dimples of a certain (surprisingly small) size are put in, the structure will buckle with a little pressure.–I am sure this fact has been studied by engineers: since trusses too, and not just corrugations, are affected by the phenomenon. The point is one can put in such dimples if one wants a corrugated structure to give way in a sort of ‘controlled failure’.
What is true for all dimples but especially noticeable with curving (oval) ones, is that the first mountain fold (the one defining the dimple itself) creates a sort of ‘shadow’ valley fold area behind it, spreading out from its north and south poles like a magnetic field, but at one preferred location. This is the ‘optimal line’ for drawing in valley folds, but what defines it?
Whether as an accordion pleat or as a fan, dimples prevent a corrugation from closing flat. This is undoubtedly a loss of one nice feature of a corrugation, collapsibility. But there are certainly gains to be had too, aesthetically at least.
Corrugations allow curving bas-relief forms to stand out nicely, with ovals merely being one of the simplest forms to make (for some other examples see the insert in the last post). The fancier a bas-relief gets though, the more material it consumes and the more the corrugation or fan or mountain-fold will have to bend forward to accommodate it.
I don’t mean to say that curving forms can’t be also put directly on an initially flat sheet. Of course they can. But when there’s an initial mountain-fold or a series of them as with a corrugation, there is (a) more volume to begin with, so some 3-D effects are easier to generate, and (b) the flexibility of the corrugation can so-to-speak be drafted for the bas-relief form, so the overall effect on the surrounding area can be minimized or disguised.
And now a word about fans. A proper fan can’t be made from a square, if one uses the whole length of a side for the part that unfolds. Such a fan will only open to a maximum of under 57.3 degrees. The curve atop a fan-edge approaches that of a slice of a circle's edge; to make a full circle the length of the corrugated rectangle must be at least 2PI times that of the height. For a half-circle, at least PI times the height.
It’s interesting, though: a fan that opens only partway, can be made to open more by decreasing its radius. In origami, i.e. without cutting, there are only three ways to shorten a radius: by ‘trimming’ material from the cut edge (results in a cylinder forming at the edge), by moving up the point of origin, or by shortening the length in between--through dimples, pleats and so forth.
As the radius decreases relative to the circumference to below the 2PI threshold, the corrugations take up the slack and the cut edge will no longer be taut. But there are techniques that can make the cut edge not only not taut, but ‘ripply’, with wavy edges resembling ornamental lettuce or the edges of a torn plastic garbage bag—in short, resembling those forms that Eran Sharon [corrected link], in an article I am fond of citing, likes to study. Hopefully more on this in another post, when I've worked out the details.
Readers who have made it this far may enjoy the following new occasional feature of this Blog:
CURVE FOLD TEST QUESTION
Suppose you put in the following curve-folds (let’s call them ‘leaf’ folds) in a sheet, identically in the top and bottom but inverting the direction of the folds. Take a moment to look at what happens to the sheet around each ‘leaf’.
Now: are there any other folds—curved or straight—that can be put in, such that the top and bottom parts will remain completely symmetrical?
Feel free to write me privately with your answers, at: YoursTruly@YoursTruly.net, replacing "YoursTruly" with "saadya"
Tuesday, October 21, 2008
Around the time I launched this Blog, a question on my mind was: Is there any shape more beautiful in origami than the Paper Fan---that starts from the fan as its point of origin? In other words: Can the Fan be improved?
The answer is – I suspect not. But there certainly exist highly fruitful progressions that have begun from this shape. I am referring to forms by, I believe, Kawasaki and Paul Jackson from an earlier generation, and in this one, the methodical “geometric” explorations of e.g. Ray Schamp and Goran Konjevod. All of these add a layer of complexity and visual interest and sometimes too a curving third dimension to the fan-shape’s basic two, but at a cost to the purity of that primal form, the sunburst. The cost is greater than the benefit, in my opinion. But what is to be done: we can’t remain virgins forever. It is the same problem with the Square, which invariably is more beautiful and pure than the tarantula or unicorn that is made from it.
Anyway, while seeking an answer to this question in my own language, I turned to the Peacock, which clearly does successfully ornament its fan tail (if success is to be measured in what works for Pea-hens.) This is my rendition of its pattern, using curved folds.
And lest we forget, here is an image of the original. Rather more sumptuous, I have to say.
Looking more closely at the Peacock itself, it is curious that in trying to woo the Peahen the male is using eye-spots (‘ocelli’), which generally in the animal world are ‘agonistic’ , i.e. warning signals.
Now this of course is not the only instance in nature where a ‘cold’ signal has been transformed into a ‘warm’ one. Indeed it happens all the time: a famous case from our own species is the smile—bared fangs being threatening almost everywhere else in the animal world. But usually, what is needed to “invert” the meaning of a signal is some change or added element in it. If it's the smile--you see this more clearly in the Mandrill, which also has one--rotating the direction of the teeth display, so that it's horizontal rather than vertical, is what changes its meaning from threat to appeasement. If it’s so-called “aposematic” coloration—warning colors, which are usually the two colors of yellow/orange/red against black, each region in large pools, spots or bands, divided clearly from the other color—that pattern needs to be replaced with colors other than the above, or with more than two of them, or with finer gradations between them rather than clear demarcations, sometimes carried through all the colors of the rainbow. And if it is eyes, these need to be softened with blush or lashes rather than outlined sharply. And then the signal will mean ‘Come Hither…’ instead of ‘Keep Off!’ (It is striking that the Peacock’s actual eyes, the one it sees through, are not softened like the ‘ocelli’ but rather made more fierce via cross-eye bands.)
Here with the Peacock, it seems to me, though there is a softening iridescence too in the tail's ocelli, the bird is counting on the further fact that these ‘eyes’ are also egg-shaped. Now eggs, for all their suggestion of mystery and fecundity and wholeness and expectation for us humans, are positively ravishing symbols in bird-language. The female when incubating has to stay fixated on & near to & worried about this exact shape for weeks or months on end, so she is primed to it. She has an ingrained weakness for just this oval-form, and a male who can display it in his body or in a pattern has a distinct advantage in sexual selection. (Or so I have suggested once before on this Blog. And how can the shining ovals of a displaying peacock NOT be read by a bird-brain as a shower or sunburst of fecundity?)
That, at any rate, was the theory. But while mulling these thoughts over I wondered what would happen if we used human symbols for the ocelli instead of peacock ones. Here is the first thing I came up with.
How about it, Ladies?
Tuesday, October 14, 2008
One of the reasons origami lends itself so well to the representation of animals is that animals are basically symmetric, and shapes made from a folded square—aligning edges, corners, flaps or other reference points—themselves tend to be naturally symmetrical.
In animals, the main symmetry is of course bilateral (reflective), but there are other symmetries as well. Hind legs and forelegs are ‘symmetrical’ in the sense of being similar to each other, so you have translational symmetry (copy and move) along with a sort of ‘allometric’ symmetry (plot features on a grid, stretch the grid). Digits, that is fingers and toes, are further branchings of limbs, and like branches elsewhere are a form of symmetry: ‘repeat the same thing, at another extremity, at a smaller scale’. In origami the similarity of the small-scale activity to the large-scale one is even more apparent.
Actually, symmetries in an animal can be even more subtle, and extend to body parts which look quite different from each other. One of the amazing discoveries from a quarter-century ago is the complex of “HOX” genes, a stretch of very similar genes which trigger cascades of embryonic growth, in animals as diverse as insects and mammals. The sequence of genes is lined up on the chromosome like beads on the string; each successive gene must have evolved initially as an additional copy of the one before it, which copy then underwent modification, causing its function to vary slightly (or greatly). Thus, antenna on an insect turn out to be modifications of feet: damage the gene and what grows on the head will in fact be feet. But an insect’s body parts, from labia to abdomen, and not just its appendages, also are controlled by repeats-with-variations of identical genes of the Hox complex, and that (on top of the basic segmentation) accounts for some of their self-similarity or symmetry. Amazingly, too, the effects on the body appear in the same spatial sequence as the genes in the chromosome—each next gene controlling the next segment of the body. Here for instance is how it looks in Fruit Flies:
[Image from this blog, which has a nice description of Hox and Homeobox genes. ]
Now, origami is often spoken of as being ‘biological’ in some way. Sometimes it is even referred to as yielding a ‘parallel embryology’. This analogy is actually quite deep, but it pulls in various directions, and it seems to me that at least part of it can be explored by thought about Hox genes and the repeating patterns of origami. Other aspects of the analogy, such as the crucial use in nature of ‘folding skins’ to create animal form, as in gastrulation; or some of the ways proteins like to fold themselves up, we will perhaps touch on in future posts (time, finances, war conditions, etc. all permitting, of course).
Getting back to symmetry: Generally speaking, a symmetrical origami design shows the animal in its most recognizable form. Clearly though, many animals assume postures—and some of them, even physiques—which are NOT symmetrical, a good deal of the time; and if one wants to represent these out of a folded square that can take special tricks and techniques . Robert Lang has a handsome Fiddler Crab with one arm much larger than the other, as it should be; and I seem to recall a Seated Lion someplace (by Giang Dinh?) with its body flung to one side, again a very typical posture for a lion. In these cases there is one characteristic asymmetry within an overall symmetrical plan, but the animal remains quite recognizable despite that.
Trouble starts, however, with those animals that lead most of their lives trying to avoid presenting a clear or symmetric outline. And here, Bernie Peyton’s expertise, both in his scientific career and as an origamian, becomes quite useful. Bernie is a wildlife biologist—in older parlance, a ‘naturalist'—who has spent 22 years studying Spectacled Bears in their native habitat in the Andes. Now, I don’t know about spectacled bears, but brown and black bears, along with quite a number of other furry animals, go out of their way to avoid showing an easy-to-read profile---most of the time. Most of the time, the head is lowered, the colors of the face and body parts blend in to the rest, so what you see is this lumbering mass that is not easy to judge the scale of from a distance or the emotions of even from closer in. Almost the only time the features become pronounced—with head raised, ears clear against the sky, arms outflung, the body too perhaps raised up on two feet—is when the bear needs to threaten somebody. It then becomes distinct, its size and intentions clear, and turns into just the sort of symmetrical, stick-figure shape that origami is so good at representing.
But that is not its typical posture; so if one wants to represent the animal as one is likely to encounter it in the wild—in a warm and not a confrontational context—that takes special efforts of design, observation and sensibility. In "Lying Bear", Bernie Peyton lets the animal be visually distinct by raising the ears just slightly over the line of the body, but the body itself is still massy and indistinct, limbs thrown akimbo in a casual asymmetric sprawl. So it is both amorphous and distinct, in precise balance.
One wonders: since this sort of representation is naturalistic for the animals but not entirely natural for origami, why choose the medium of origami to make it with? Why stretch paperfolding almost to its limits when, say, a wood-carving could have done the job more easily? Bernie’s answer no doubt will be that some of the special aesthetic qualities of origami animals --I mean their fragility, freshness, liveliness and transience--are important for him to convey, given the habitat destruction he has seen at first hand with such devastating effects on his animals. That is a noble reply but artistically, it's not entirely satisfactory. More work needs to be done, it seems to me, to make the sort of highly naturalistic, animated shapes Bernie hopes to capture, come to appear more natural to origami. —But he is already pretty far down this path.
Meanwhile, here is a different take on a Bear, in its less typical if more symmetrical form. On the warpath, in other words. Less naturalistic, if you like, but more natural for origami. The design is by Nicolas Terry; the fold is by Herman Mariano (who decided to make it a Brown Bear rather than a Black one). I had the privilege of showing both of these bears in the exhibit last year at the Tikotin Museum of Japanese Art.
A quick plug: Most readers know this already, but just in case you don't: both of these top-rank animal designers, Nicolas Terry and Bernie Peyton, along with the uniquely inventive and ebullient Vincent Floderer, will be present at the “Ultimate Origami Convention” in Lyon, France, between the 8th to the 11th of this November (2008). Get there if you can—you are in for a real treat.