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Suppose that I have a rectangular piece of card or paper and it's lying flat on a table. Then, 2 of the opposite sides of the paper are pushed together slightly, completely smoothly and with the sides remaining perfectly parallel.

Q: Give a mathematical description (parametric, perhaps, for a given point on the paper) of the curve that the paper makes as the sides are pushed together by some fraction $n$ of the distance between the two sides?

There are many obvious questions about this: does the curve depend on the material (i.e. I said paper, because that's how I noticed it, but what about a similar thickness of steel etc.)?

What about when one takes a brochure and forms a triangular-prism with 3 of the pages and then imagine pushing the 3 long edges of the prism closer to the center of the triangle smoothly - what curves would the sides make? What about with a 4-sided or $n$-sided prism of paper?

[I'm preferably looking for an answer to the first question, but if you have the capability to do so, why not explore?]

Isky Mathews
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    Neat question! Maybe Euler-Bernoulli beam theory would be applicable here, but I'm not sure. – knzhou Dec 16 '18 at 15:46
  • @knzhou Thanks! I'll take a look - I'd never heard of this (with Euler and Bernoulli behind it, it must be pretty good!). I'm quite fond of questions that seem quite natural in that they ask about complicated things that we take for granted. – Isky Mathews Dec 16 '18 at 15:53
  • Just to confirm, you’re asking about the shapes shown here for sheets (as opposed to beams) and for large deflections? And you’re asking specifically about the pinned-pinned and fixed-fixed configurations? – Chemomechanics Dec 16 '18 at 17:48
  • @Chemomechanics Based on the notations of your link, I would think "Yes" for every question you asked. – Isky Mathews Dec 16 '18 at 17:52
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    Reason for voting to reopen: the other question addresses small deflections only. There is substantial literature on the large-deflection case; see here, for example. – Chemomechanics Dec 16 '18 at 20:54
  • @Chemomechanics It also doesn't really answer any of my more general questions, such as that of the triangular-brochure situation, at all. I vote also to reopen (especially given that you're probably going to give a whopping good answer). – Isky Mathews Dec 16 '18 at 21:34
  • To support this vote, I suggest editing your question to acknowledge the previous question but to ask more generally about the types of shapes that emerge for various edge constraints in both the small-deflection and large-deflection cases. – Chemomechanics Dec 16 '18 at 22:08

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