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It is well known that, for any two rectangles of the same area, the first can be cut into a finite number of polygonal pieces and reassembled into the other (for example, by Montucla's dissection).

Questions: Can we do the same if only rectilinear polygons can be used as intermediate pieces? Or if only rectangles can be used as intermediate pieces?

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There is an affirmative answer for rectangles (and therefore also for rectilinear polygons) in the paper "Rectilinear Glass-Cut Dissections of Rectangles to Squares", by Czyzowicz, Kranakis and Urrutia.

Note the different uses if the word “rectilinear”: Wikipedia defines a rectilinear polygon as one with rectilinear sides. The paper above uses rectilinear to mean that the rectangles can be produced using only rectilinear cuts, without needing to make any right-angled turns in the middle of a cut.

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    $\begingroup$ It seems to me that the answer in the paper is that a decomposition with rectangle pieces exists if and only if the ratio of the lengths is the square of a rational ($w/h=r^2$). I don't see where rectilinear polygons are mentioned. $\endgroup$
    – Pierre PC
    Commented Sep 7, 2020 at 23:14
  • $\begingroup$ Thanks, I corrected this. $\endgroup$
    – user44143
    Commented Sep 8, 2020 at 0:20
  • $\begingroup$ Thanks very much! The paper linked above has a reference to Stillwell's 'Numbers and Geometry'. It has been established that say a sqrt(2) X 1/ sqrt(2) rectangle cannot be converted to a unit square without oblique cutting and pasting. And now, it is also clear that if two rectangles can be dissected into each other via rectilinear intermediate pieces, then they can be dissected into each other via rectangular pieces. So, in general, one can dissect rectangles to rectangles only by permitting oblique cuts - as for example Montucla does. $\endgroup$
    – Nandakumar R
    Commented Sep 8, 2020 at 8:31

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