If false 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.
Q1: Intuition tells us it is true ( how on hell can it increase?). Yet I think I read somewhere that there was some weird folding (called "mountain urchin"??) that strictly increases the perimeter. Is this true?
Note1 : I am not even sure that the initial sheet's squareness is required.
I cannot find any reference on the net, maybe the name has changed, I heard about this 20 years ago.
The second question is about generalizing the conjecture.
Q2: 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 maths?
Note2: It might also be a matter of physics about how much we idealize bending mathematically.