TITLE: Is the "Napkin conjecture" open? (origami)
QUESTION [52 upvotes]: 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.

REPLY [26 votes]: In addition to the answers above, here are some remarks from my paper in Russian; part of it used in the last lecture here.
(Sorry for self-advertisement.)
1. An other solution. 
It is based on idea of Yashenko [Math. Intelligencer 20 (1998)].
This way you can increase the perimeter just a bit,
but it is done by repeating one fold (which is very simple but not "simple" in the sense below). 

2. It is still not known if you can increase the perimeter by a sequence of natural folds;
i.e., folds like this:

I just learned that this problem also appears in Pak's book, Problem 40.16b;
it is marked by [$*$] which means that the problem is open.