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Ref:

  1. Inside-out polygonal dissections
  2. Further queries on inside-out polygonal dissections

Question: Consider any two polygons P1 and P2 with equal area and equal perimeter. Is it always possible to dissect P1 to P2 in such a way that the every point on the boundary of P2 was also a boundary point on P1? If the answer is "yes" (likely), can some bound be derived on the number of intermediate pieces if the number of sides of P1 and P2 are n1 and n2 respectively? Will convexity have any bearing on the result?

Note: an inside out dissection of P1 to P2 seems a less constrained problem. Not sure about a totally inside out dissection.

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    $\begingroup$ The answer should be yes: partition the boundary of the first polygon into segments which can be used to cover the boundary of the second one. Now for each of these segments, cut out a thin triangle with it as base, so that these triangles fit into the target polygon without overlap. You've now covered the entire boundary with boundary pieces, the remaining polygon has the same area as the remaining hole in the target, so the rest can be dissected to cover the remaining hole. I'll leave this as comment for now, since I don't know how to say anything about the required number of pieces. $\endgroup$
    – Achim Krause
    Commented Sep 25, 2022 at 6:58
  • $\begingroup$ Thanks. At least the basic existence question now has a simple answer. $\endgroup$
    – Nandakumar R
    Commented Sep 27, 2022 at 9:44

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