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