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.

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.

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.

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.

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 by 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.

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.