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 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 increase the perimeter. Is this true?  
*Note1* : I am not even sure that the squareness of initial sheet is required. 
 
I cannot find any reference on the net, may be 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 mathematically can you define bending a sheet, otherwise said : 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.