# Mathematics of paper fold-cutting

## ... 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.

• The remaining piece in your question doesn't look like it has any folds left in it (unless I don't understand the noun fold). Sep 5 '14 at 7:20
• @barak: I think they mean the resulting shape must have a crease in it, that is, a fold passed through it previously.
– user856
Sep 5 '14 at 7:36
• It doesn't have to be a polygon, as you can generate shapes with holes. But surely it has to be a polygon or a difference of two or more polygons. Sep 5 '14 at 7:47
• @Rahul: What I actually meant is that the piece must have at least one fold left before unfolding.
– user139000
Sep 7 '14 at 15:44

## 1 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.)

• This is pretty incredible!
– user139000
Sep 7 '14 at 15:43