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It seems that if something can be stretched then it can be folded along a curved line. Since paper can't be stretched I can't fold it along a curve. But it's just an observation not an answer to the question. Does anyone know the answer? Is it because of some kind of area conservation?

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    $\begingroup$ I‘m not sure about this thus I don‘t make it an answer. The conservation you are looking for is possibly the Gaussian curvature. When you bend paper without destroying its internal structure, the Gaussian curvature is always conserved. You can bend paper into a tube but not into a sphere because flat paper and the tube have the same Gaussian curvature but a different one than a sphere. $\endgroup$ – Hartmut Braun Nov 26 '20 at 20:10
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    $\begingroup$ "It doesn't stretch" looks like an answer to me. An observation can be an answer. Conservation of area is a property of materials which do not stretch. $\endgroup$ – sammy gerbil Nov 28 '20 at 0:11
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    $\begingroup$ How does a classic fast food French Fry container get folded along a curve? Is it made from special stretchy cardboard? $\endgroup$ – arp Jan 13 at 20:39
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Apparently, while you can't fold paper along a curve in 2 dimensions, you can do it in 3 dimensions.

This article from the journal Physics describes how a flat ring of paper, folded along its circular center, deforms into a saddle as the material buckles to absorb the stresses.

(Longer discussion at Physicsworld courtesy of an answer to a related question.)

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No, paper can't be stretched because its structure contains no slippage mechanisms which would allow it. To some extent, you can provide slippage by wetting the paper into mush, which unlocks the entanglements between the paper fibers and allows the sheet to deform.

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    $\begingroup$ I'm not asking why paper can't be stretched. But assuming paper can't be stretched, why can't it be folded along a curved line? $\endgroup$ – Abu Saleh Musa Nov 26 '20 at 19:52
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    $\begingroup$ to fold it along a curved line requires the paper to be stretched in some directions and compressed in others. $\endgroup$ – niels nielsen Nov 27 '20 at 1:24
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The short path between 2 points on the paper before folding is a straight line, all points of which belong to the paper.

Now suppose that we managed to completely fold it along a curve between these 2 points. All paths between them on the paper are now curves, with bigger lengths compared to the straight line, (which passes out of the paper).

So, it is not possible to fold it along curves keeping the same distance between all points. And it requires streching to change distances.

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