Huffmanesque

This is, I believe, the first successful reconstruction (with a minor extension) of the beautiful Concentric Circles form by the late David Huffman.

I wrote about this form once before, and was a bit brash and over-confident about it. It is NOT a simple cone with concentric folds about its center, as I then thought, but is slightly more sophisticated.

Mea culpa. In my own behalf I can say

1. It’s very hard to tell things from a photograph, especially a face-on one that flattens the object

2. None of the other people who have written about this, and who ought to know, got it exactly right either

3. If it weren’t for my mistake, I would never have come up with that neat wind up toy. Which it seems now I can even take credit for.




Invented March, 2008. I've written about this form elsewhere.



Here is the Huffman idea again. A pattern of serene geometry, an otherworldly form.

Saadya






added July 1, 2009: I just saw essentially the same explorations of cone-folding in Ron Resch's gallery, dated to 1969-1970. He and Huffman knew each other; but I have no information on who looked at these ideas first. Still, Resch's published explorations of cone-folding (no longer available on the internet) are much wider.

Dimple-Lock

[Revised Dec 1, 2008]

This is a way to join two sheets without glue, along a common mountain-fold, at a single point. It uses the ‘dimple’ method I’ve been discussing in recent posts. It's a pretty tight lock; for two free-moving sheets I don’t think I’ve seen one as tight.





1. Valley fold through both sheets, about 2 or 3 cm from the edge to be joined.

2. Remove one sheet, turn it over.

3. Reinsert.

4. Make a diamond dimple through both layers. (Put your hand between the layers from behind, and press in with a finger.) Sharpen the folds, both the diamond mountain fold and the valley at its center.

5. Close the dimple, sharpen the triangles that form next to it. (The far corners of the triangles are arbitrarily located so you may as well align one of them with the cut edge of the overlapping sheet and the other symmetrically).

Step 5 will also make both sheets conical. It is best if the cones of both sheets are oriented in opposite directions--otherwise you will have to invert one of them later.

6. Put a valley fold running through the apex of each of the triangles formed in step 5.

This is how it should look, front and back:





That's it! The lock is formed.

7. The next step is optional. The sheets are not flat; if you like, you can flatten them (from the tip of the cones) with a shallow triangle reaching to the edges of the paper. --For many purposes this step will not be necessary, and it does not add to the tightness of the lock.


By the way, I discovered this locking behavior while exploring what happens when you close a dimple on a single sheet (it locks up a section of the sheet), but it was obvious right away that the move could tie together two sheets too. And more than two as well; but the greater the number of sheets joined at a single point, the less sharp the folds of the exterior sheets and hence the less tight the lock will be.


Of course, because the bond is at a single point, it's more impressive to join two sheets at their corners, rather than at an edge as in the diagrams. The photo up top shows two squares joined at their corners with a Dimple Lock--its strength being tested via the famous 'Sandal Test'...



Saadya
Invented November 15, 2008.

Gifts

No, not new designs. But it was nice to have an excuse to make these cuties again. Would you believe—I hadn’t folded a bird-base in six months??

It feels good.

Dimples

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.

Dear Fans
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"

Yours
Truly

Peacock's Tail

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?


S.

Animal Symmetry and Representation

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:




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.


Saadya

Don Luis

Not much lately to show origami-wise, though a few things are cooking. Meanwhile here is a little study I made a few days ago of a Velazquez, oil and chalk on wood. Just keeping in shape. --S.

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.


[…]