There, an 'inside out dissection' of a polygon P is a defined as a dissection of P into another polygon P' such that (1) P' is congruent to P and the perimeter of P is internal to P' - ie the perimeter of P' is composed of the internal cuts made to P. It is also very nicely proved there that any triangle has an inside out dissection via 4 pieces, thus:
Let us consider doing an inside-out dissection of a polygon such that the total length of cuts is minimized. From the construction given on the above linked page, for any general triangle T, there is an inside out dissection with total cut length = half the perimeter of T itself; it is a 'perfect' dissection.
Likewise for rectangles or regular hexagons, we have inside out dissections where the total length of cuts is half the perimeter. Intuitively, it is obvious that half the perimeter of P is a tight lower bound on the total cut length.
Question 1: Which convex polygon P gives an upper bound on the length of cuts (compared to perimeter of P) for an inside out dissection? Basically, we are asking for polygons which need much more total cut length than half its perimeter for any inside out dissection.
Note: We observe that every convex polygon P can be dissected inside out with total length of cuts arbitrarily close to P (which is thus a loose upper bound on the total cut length). Proof: Form another polygon P' in the interior of P by insetting P by a small distance d. It is clear that the region between P and P' can be broken into parallelograms and triangles; dissect all these triangles inside out (as given in above picture) and reflect the parallelograms by 180 degrees and we have an inside out dissection of P. And as d goes very small, the total cut length arbitrarily gets close to P.
But we guess, the dissection just described must be suboptimal for any P and that the perimeter of P must be a very loose upper bound for total cut length. So, the question is which polygon approaches this upper bound closest for its "least cut length inside out dissection".
Question 2: For convex P, how will such an upper bound change if we allow P' to be non-congruent to P - insisting only that the perimeter of P should become totally interior to P'? It seems that allowing P' alone to be non-convex might not have an impact.
Question 3: Going to 3D, it is easy to see that a cube can be turned inside out by first cutting it into 8 cubes with three mutually perpendicular cutting planes - and the total area of the cuts is only half the surface area of the cube. It is not immediately obvious if for tetrahedra, a simple generalization of the triangle inside out dissection will go through.
Question 4: What happens to inside-out dissections in non-Euclidean geometry? Even the inside out triangle dissection doesn't seem to generalize obviously to hyperbolic geometry.
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Note: Regarding question 2 above, if P is allowed to be non convex and if P' is allowed to be non-congruent to P, the length of the cuts can be much less than half the perimeter of P. A simple example is shown below:
P is the non convex polygon to the left. It consists of the big yellow non concave region AND the 4 small white triangles at its fringes. We could do the following inside out dissection of it: as described in the above linked page, first dissect only each of the 4 white triangles inside out and keep them at the same place relative to the full polygon P. Then cut P along the blue 'equatorial' line and reassemble the 2 pieces to get the new polygon to the right (P'). Now, no portion of the perimeter of P is exposed. It can also be seen that keeping the 4 small white triangles fixed, if the number of 'teeth' in P are arbitrarily increased, the perimeter of P' is much less than that of P and the total length of cuts for our inside out dissection can be arbitrarily small compared to perimeter of P.
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