# Why can a piece of paper be folded along a straight line but not along a curved line?

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?

• 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. – Hartmut Braun Nov 26 '20 at 20:10
• "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. – sammy gerbil Nov 28 '20 at 0:11
• How does a classic fast food French Fry container get folded along a curve? Is it made from special stretchy cardboard? – arp Jan 13 at 20:39
• I don't understand this question very well. If you are talking only about elastic deformations and you have a strip of paper, then you have the problem of the elastica studied by J. and D. Bernoulli, Euler, Lagrange, et al. In the elastica, the inner part of the fold is compressed while the outer part is stretched but the central part (known is neutral line) remains isometric. If you have a wider piece of paper, then you have the problem of plate deformation. I believe the standard model for these deformations is the Föppl-Von Kármán model. – minmax Jan 28 at 13:37
• If the curvature is not very high you may make a lot of deformations. See for example this problem: "The shape of a Möbius strip" by E. L. STAROSTIN AND G. H. M. VAN DER HEIJDEN in Nature Materials 2007 doi:10.1038/nmat1929. This seems to be a counter example to your claim as stated above, at least as I understood it. – minmax Jan 28 at 13:43

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.)

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.

• 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? – Abu Saleh Musa Nov 26 '20 at 19:52
• to fold it along a curved line requires the paper to be stretched in some directions and compressed in others. – niels nielsen Nov 27 '20 at 1:24

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.

Paper can be more-or-less folded along a curve. You can do it with your fingers.

It could be argued that this is really just a lot of little short straight lines.

But it could be argued that any curve is just a lot of little short straight lines.

• That's a plastic deformation along the crease. In a plastic deformation, which is irreversible, paper can stretch, not by much though, otherwise it would tear. I believe the problem should be restricted to elastic deformations. – minmax Jan 28 at 13:17
• I probably disagree. You can make two straight creases that intersect. You can make a third crease that intersects the first two. You can make two more creases that intersect pairs of the first three. Etc. You can make lots of creases that approximate a curve. And a straight crease involves some stretching too. But I suspect you're making a philosophical point which is not affected by real paper. – J Thomas Jan 28 at 18:40
• If you allow for plastic deformations you might as well just crumple a piece of paper and make far more complex 3d curve creases. – minmax Jan 28 at 22:59