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.