The following is based on Inside-out polygonal dissections
Definition: We say that a polygon P has an inside-out dissection into another polygon P' if P′ is congruent to P, and the perimeter of P becomes interior to P′, and so the perimeter of P′ is composed of internal cuts of the dissection of P.
Question: What if we further insist that no point on the boundary of P should be on the boundary of P'? In the example inside-out dissections shown in above linked discussion, there are isolated points of the boundary of the input polygon P that are also on the boundary of P'.
Note 1: The Existence of such 'totally inside out' dissections is not hard to show for any triangle and hence for any polygon. But the least number of intermediate pieces even for a "triangle to congruent triangle" dissection seems considerably more than 4 (the number conjectured on above-linked page) - especially if we insist that all intermediate pieces need to be simple and convex polygons (as is indeed the case in the dissections shown in the above-linked page). A simple-minded method can turn a triangle totally inside out via 7 intermediate pieces out of which 6 are convex and 1 piece will have 3 holes - and this piece will have to be cut into several convex pieces.
Note 2: The inside-out and totally inside-out dissections appear to have interesting analogs in 3D - the simplest example is that a cube can be turned inside out (but not totally inside out) via 8 smaller cubes.
Further question: If we add this totally inside-out condition to general dissections where P and P' are only equal area planar regions (not congruent), will it impact classical results such as the Wallace-Bolyai-Gerwein theorem, Tarski circle-squaring etc..?
