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.
My question:
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.
Best Answer
I think the answer is pretty much any shape. See Demaine, Demaine, & Lubiw "Folding and Cutting Paper", which describes a method that uses just one straight cut. (So I believe your condition on requiring a fold can be satisified by simply folding once more along any line perpendicular to the single cut to be made.)