On Mon, Sep 7, 2015 at 11:52 AM, Leong Cheng Chit <[email protected]> wrote:
> Here is my attempt at a definition: > > When you bend a sheet of paper, the fibres on the mountain side are > stretched and the fibres on the valley side are compressed. This causes > tension in the paper which tends to revert itself back to its original flat > form. Wetting the paper reduces the tension and the paper when dried will > keep to its bended form. > > How then can you fold the paper such that the bended or curved form is > maintained without wetting the paper? You can fold the paper such that you > have two curved surfaces intersecting along a curved line. This is done by > folding the paper with an intrinsic straight and an intrinsic curved crease. > There is an overlap of paper between the curved and straight crease and the > tension on the two curved surfaces in shape without wetting and drying. You > can have more than two curved rigid surfaced by such tension folding. > Below > is a form consisting of three curved surfaces by dry tension folding: > > https://www.flickr.com/photos/chengchit/920472460/in/dateposted-public/ Dear Cheng, The business of paper fibers stretching and compressing seemed wrong to me, so I immediately went and took a sheet of thick aluminum foil, and tried the fold you've described--one straight and one curved crease segment, like a shallow half-moon, all in the interior of the sheet. (Of course it's a fold I've made plenty of times before). Now, foil has no fibers. But the shapes formed for all that, just like in paper; moreover, it is stable! So it is wrong to say that the sheet is "in tension", but quite right to say that a great deal of tension, that is, pulling, orthogonal to the crease pair, is needed to undo this form. The curved surfaces are locked in place. Why is that? I am unsure. Perhaps our mechanics experts can weigh in on this? I will forward this thread to Itai Cohen at Cornell who does exactly studies of this kind. I'll give a clue, that if you make a fold of this kind using a shallow isosceles triangle instead of a straight and a curve, a weaker lock is formed; that lock becomes stronger if the cone-wells are bent across a mountain fold. And another clue is that if you make this fold with the shallow moon stretching all the way to the sides of the sheet (and not floating in the interior) the lock is weaker. So apparently the formation of cone shapes at the apexes of the moon is involved in the locking. Grab some paper and see! Thanks to Jorge Jamarillo for launching what's turning out to be an intriguing thread, though as said, the term "dry tension" really has to be trashed. Saadya
