Take a square of paper...

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... and fold it any number of times using consecutive straight folds...

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... then cut off any number of pieces using consecutive straight cuts...

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... and unfold the remaining piece*

* Condition: The remaining piece must still have at least one fold.

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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.

  • $\begingroup$ 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). $\endgroup$ Sep 5 '14 at 7:20
  • $\begingroup$ @barak: I think they mean the resulting shape must have a crease in it, that is, a fold passed through it previously. $\endgroup$
    – user856
    Sep 5 '14 at 7:36
  • $\begingroup$ 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. $\endgroup$ Sep 5 '14 at 7:47
  • $\begingroup$ @Rahul: What I actually meant is that the piece must have at least one fold left before unfolding. $\endgroup$
    – user139000
    Sep 7 '14 at 15:44

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.)

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

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