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.