TITLE: Fold-and-cut problem in three dimensions
QUESTION [11 upvotes]: The fold-and-cut theory states that "Any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut. Such shapes include polygons"
My question

Question 1 What is the time complexity for finding such a crease pattern?
Question 2 How do we express; $\chi(P)$ ie the minimum number of  folds required to make a crease pattern from which the polygon $P$ can be cut out?
  Question 3 Does the fold-and-cut-theorem have a three dimensional analog?

Would also request for a reference to the proof.

REPLY [5 votes]: In response to @redhound.
Now updated with step-by-step instructions online:

         


         

The turtle nearly fully folded.
The black lines all align.


         


         

The hole remaining after 1-cut.