The falsity of the following conjecture would be a nice counter-intuitive fact. 


Given a square sheet of perimeter $P$, when folding it along origami moves, you end up with some polygonal flat figure with perimeter $P'$.  

**Napkin conjecture**: You always have $P' \leq P$.


In other words, you cannot increase the perimeter using any finite sequence of origami folds.


**Question 1**: Intuition tells us it is true (how in hell can it increase?). Yet, I think I read somewhere that there was some weird folding (perhaps called "mountain urchin"?) which strictly increases the perimeter. Is this true?  

*Note 1*: I am not even sure that the initial sheet's squareness is required.


I cannot find any reference on the internet. Maybe the name has changed; I heard about this 20 years ago.


The second question is about generalizing the conjecture.


**Question 2**: With the idea of generalizing the conjecture to continuous folds or bends (using some average shadow as a perimeter), I stumble on how you can mathematically define bending a sheet. Alternatively, how do you say "a sheet is untearable" in mathematical terms?  

*Note 2*: It might also be a matter of physics about how much we idealize bending mathematically.



Is the "Napkin conjecture" open? (origami)
There is a general version of this question which is known as *"the rumpled dollar problem"*. It was posed by V.I. Arnold at his seminar in 1956. It appears as the very first problem in [*"Arnold's Problems"*](http://books.google.co.uk/books?id=LqvLnu8c3ToC&printsec=frontcover&dq=arnold%2527s+problems&source=bl&ots=l0ULwBNvsc&sig=mLNHb1upSiddEtfrHRjbjvJPfzE&hl=en&ei=6ZPpTIvqLcuaOozXvZEK&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBYQ6AEwAA#v=onepage&q&f=false): 



> 
> Is it possible to increase the perimeter of 
>  a rectangle by a sequence of foldings and unfoldings? 
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According to the same source (p. 182), 



> 
> Alexei Tarasov has shown that a rectangle admits a realizable folding with arbitrarily large perimeter. A realizable folding means that it could be realized in such a way as if the rectangle were made of infinitely thin but absolutely nontensile paper. Thus, a folding is a map $f:B\to\mathbb R^2$ which is isometric on every polygon of some subdivision of the rectangle $B$. Moreover, the folding $f$ is realizable as a piecewise isometric homotopy which, in turn, can be approximated by some isotopy of space (which corresponds to the impossibility of self-intersection of a paper sheet during the folding process). 
> 
> 
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Have a look at


* A. Tarasov, *Solution of Arnold’s “folded rouble” problem.* (in Russian) Chebyshevskii Sb. 5 (2004), 174–187.
* I. Yashenko, *[Make your dollar bigger now!!!](http://link.springer.com/article/10.1007/BF03025296)* Math. Intelligencer 20 (1998), no. 2, 38–40.




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A history of the problem is also briefly discussed in Tabachnikov's [review](http://www.math.psu.edu/tabachni/prints/Arnoldpr.pdf) of "Arnold's Problems":



> 
> It is interesting that the problem was solved by origami practitioners way before it was posed (at least, in 1797, in the Japanese origami book “Senbazuru Orikata”).
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