Nine-faced Jar

It was time to make another of these, so I did.

Folded from an uncut rectangle of foil-backed paper, two edges glued to form a cylinder.

9, because I wanted the viewer to see about 3 faces at once. (And because 8 is a boring number.)


Known inputs. A pot with faces I glimpsed at the Art Institute of Chicago in the 1990s, probably from bronze or ceramics; those little figures adorning portals to cathedrals, which from a distance look like pencil-marks or charcoal scraped on paper; David Huffman’s beautiful Tower Form. Thoughts about how the figures in a bas-relief are in process of freeing themselves from full immersion in their background (as in painting) to full independence in our world (as in sculpture)---and so thrive in this half-subservient state. Studies of face-making folds over the years, seeking always the minimum necessary lines . And technical investigations into curved folding.


About just the latter, a few words.

Apart from the pretty shape, I am showing off two kinds of curvature.

1. Curved hinges. Note the lines that descend from the top and taper inwards. Each line curves, but it bends to neither the left or the right: it is straight in the X and Y dimensions, curving only in the Z. And even so the planes to either side hinge along it. (Which fact is highlighted by contrast with the straight folds emanating from the bottom, which seem to behave normally.) This is not so common even in the world of curved-folding.

2. Bulges that emerge suddenly from flat areas, sometimes without the benefit of hidden folds, pleats etc.


A reaction I'm always happy to get to my curving origami is, ‘I don’t see how this is even possible’.

To which a response is: Sometimes, when you find that the one thing you have to do is impossible, it might still be that you can do two things, both of which are impossible, except together.


Saadya

posted also on Flickr

Ripple-Pop

Going through my files, I found this short video from a few months ago, that I thought you might enjoy.






By the way: In the month of August I will be traveling--by train through Central Europe, with ‘origami stops’ in Prague, Budapest and Zagreb; then on by plane to Toronto and by car to Algonquin Park (and by canoe to the interior). Then back to Israel.

If you or your group would like to host me or meet in the above cities or points in between, send me an email.

Saadya

Sphere from a Circle

Here is a new method for folding an uncut circle into a 'sphere'.

1. Fold concentric circles
2. Divide disk into equal-sized pie-slices (any even number will do)
3. Draw zigzags along the radii
4. Reverse all folds in alternating pie-slices
(or in a less elegant version, just crimp the radii)
5. Fold up
6. Secure with pins

In this technique the sphere is not hollow. The main difference from the more usual way of making spheroid shapes in origami is that instead of folding up 'walls', one folds polar cross-sections of the sphere. In fact this sphere has the odd mathematical property, that its center is the center of the circle, its "surface" the edge of the circle, and all points in between on the sphere are in the same pair-wise order (in distance from the radius) as in the circle.




Greater accuracy and surface coverage can be obtained with some effort by using many 'wedges' (=pie slices in the original circle; looked at side-on they are 'arms' or 'meridians'). But since the height of the sphere shrinks in direct proportional to the number of wedge-pairs, you'll get a perfectly delineated ball only at infinity, and then it will also be solid--and the size of a point.



____________
Added June 2010

Here are a few further points of interest about this form.

• This Sphere is related to the Concentric Winder. Imagine each of the layers of the Winder 'peeled': reverse-folded-along the same diameter line, at various angles, so that the Winder is 'rotated in space'. You will have something like this Sphere. (Some such idea was part of the conceptualization of this model.)
• Like the Winder, the Sphere also offers a negative illustration of the Albers Effect: Because of all the movement back and forth, the surface of each wedge can remain flat-on-average, without being forced to twist from its plane as it does with Albers disks or Yenn annuli.
• Less obviously, this Sphere is even more directly related to the folded Albers disk: for the latter too has circle-lines marking off approximate outlines of spheres. (Because of all the Albers twisting this is harder to see.)
• Perhaps the most surprising aspect, besides the directness with which the wedges rotate and condense into meridians, is that the initial paper circle has one curvature, the smaller sphere another, and yet the circle just magically adjusts its curve to oblige the needs of the sphere. Moreover it does so no matter how many wedges are put in, that is, regardless of the relative size of the sphere! Somehow everything always works out.
• I put my name on this Sphere, for its simplicity; that doesn't mean there are not more complicated things to be made by following its principles. This is new ground, there is considerable scope for variation--so go discover something for yourself!
  
  
  Finally I have to admit---I got lucky here (though it wasn't just luck, as I was actively looking for a Sphere in the month before I found it). --If anyone should have discovered this principle before, it should have been Josef Albers 80 years ago. He after all was asking the right kind of questions, viz., what can you do to a sheet or a pattern that will cause the surface to change its expected behavior. This led to his discoveries of both the Concentric Circles fold and of the Hypar. In retrospect I see now that this Sphere is a cross of those two Albers shapes, though that's not at all how I came to it. But Josef Albers did not take this step: his loss, and my gain.

Cheers!

Saadya


Related Posts:
Concentric Circles (March 2008)
Huffmanesque (January, 2009)
Organic-Circle Fold (March, 2009)

Organic Circle-fold

Faced with a Circle, one draws a blank. Where to start? The shape is too symmetric: a blind eye. How, where, will you find different parts in this Unity? And supposing you do: what creases can you put in, that will possibly preserve anything of the circle's original circularity?


Some of these problems, as it happens, are faced by living forms as well. By the fertilized ovum, a single pregnant cell, or by its descendant the blastula, the nearly homogenous sphere of 1000 or so identical cells. These too must divide or fold, to create form. The sphere in nature solves its problem, breaks its symmetry—by gravity perhaps, telling it which side is up and which down, or by some accidental imperfection on its perimeter. In the blastula a fold line arises (the ridge of it assuming special significance later on) and the whole magic of gastrulation begins: nature's origami on steroids. And then differentiation, growth of the different regions can proceed, each part developing according to its own strict rules.

I tried something similar with this circle, making the most obvious division and then applying the most obvious distinct pattern-folds to each part.

The resultant form is ‘organic’ --- reminiscent of how organs emerge in plants and animals, with intimations of branching and budding and leafing and embryo-curling. ‘Organic’ too in the sense more usual in art, of a thing that is like a living form but is not specifically representational.

Curve-folds, and corrugations generally, have this quality that is present in some but not all origami: in principle you can see all the folds at once. There is nothing tucked away, nothing up your sleeves--there are no sleeves. The magic is all out in the open. So the continuity that is key to origami's aesthetic, the sense that the thing is an integral whole despite all its details, is even more apparent than usual. That too can help make an origami shape seem more "organic".


So, this is one thing that can be done with a Circle. What else can?



Related Posts:
Concentric Circles (March 2008)
Huffmanesque (January, 2009)
Sphere-from-a-Circle  (May, 2009)

.

Curve-fold Fountains

(or: "Feather Caps")










Made from an uncut... semicircle.

As usual I am exploring the interactions (geometric, aesthetic) between curved folds and straight ones.



Saadya


-------------------
Added March 7:



"Angels Instead"

Made from a circle with a radius cut.

toward a Buddha paperfold

Borrows an idea from Giang Dinh.
Made, as it happens, from a square.

S.


--------

Added Feb 28:
And since the technique exists anyway, one may as well apply it.

Fanwheel Flasher

This is in the ‘so simple it seems to have been overlooked’ category. Or maybe it was just too obvious to mention. In any case I haven’t seen it before.






It is just an ordinary fan from a rectangle, with one edge glued to the other to make a full circle. Only, instead of the corrugations meeting the long edge at 90 degrees, they meet it at slightly less. That small change is enough to allow the whole thing to collapse, as a regular circular fan cannot.

This is another instance of the guideline--I hesitate to call it a "law"--that in collapsible origami, skewed angles work better than perpendiculars. Perpendicularity is all about equilibrium, among multiple options and stresses; skewedness is all about disequilibrium--and decisiveness.

If you make this, don’t forget that for a fan to form a circle, the long edge of the rectangle must be at least 2pi times the short one. A ratio of 6.5 to 1 is a reasonable approximation.

How it came about: I’d been studying fan disk shapes made with curves instead of straight lines, some of which are like those which Nishimura and Christine Edison have also been making. These disks collapse to some extent---that is one of the first things one notices about them. Always on the lookout for “special properties of curved folds”, for a while I thought the collapsibility was a function of the curviness. Alas: closer inspection reveals that it is because curves are, by definition, not everywhere perpendicular to the long edges. Simple curves either hit the bottom edge at an angle, the top edge at an angle, or both. It is that angle, and not the curviness as such, that is doing the work of allowing the collapse. And an angled straight fold does a better & purer job of it than a curved one.

It is hard for me to believe that the familiar paper fan, one of the oldest of forms in origami, can give up ANY new tricks at this point. But you never know. If anyone has seen this before, please drop me a note. --Until I hear otherwise I’ll claim this one as my own.

invented December 3, 2008

Saadya



P.S. Ray Schamp just pointed me to the beautiful and comprehensive images in Nojima Taketoshi 's M.A. thesis on collapsible forms. This shape does not appear, but others using a very similar principle (from a flat disk instead of a rectangle) do.

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