Take a square of paper...
... and fold it any number of times using consecutive straight folds...
... then cut off any number of pieces using consecutive straight cuts...
... and unfold the remaining piece*
* Condition: The remaining piece must still have at least one fold.
Which shapes can be made this way?
Some initial ideas:
- The resulting shape is always a polygon, but
- it can be concave or convex and
- does not need to have any symmetries.
- The condition (*) is necessary because otherwise it is trivial to make any polygon by simply cutting it out and discarding everything else, at least if incomplete straight cuts are allowed.
The procedure leaves a strong feeling that it somehow limits the class of polygons that can be created but alas, I have not been able to find any polygon for which I can prove that it is not an element of that class. Neither have I shown that every polygon can be created that way.
Every regular polygon ($n$-gon) can be created by collapsing the square into $n$ radial sections around its center and then making a single symmetric cut.