recutting a polygon to obtain its flip

Question

I am curious if it is possible to cut the triangle with vertices (0,0), (0,1), (1,0) into many polygonal pieces and only translate each piece appropriately to construct the triangle with vertices (1,1), (0,1), (1,0). I don't see a simple reason why this is not possible.

Answer 1

The Bolyai-Gerwien Equidecomposability Theorem states that, if $P$ and $Q$ are polygons of the same area, then $P$ and $Q$ are equidecomposable: $P$ can be cut into pieces that re-arrange to make $Q$ (and vice-versa). Proofs tend to follow a route that actually proves something stronger: $P$ can be cut into pieces that can re-arrange via translations and $180^\circ$ rotations to form $Q$ (and vice-versa). A caveat is then made that we generally can't do better than this: Translations alone are not enough.

You can read this type of discussion in these passages from Miklós Laczkovich's "Conjecture and Proof" . The author proves specifically that a triangle is not translation-equidecomposable with any rectangle; the argument applies to your flipped-triangle case as well. I'll try to summarize.

(It'll take me a while to type that up. I'll post my as-yet-incomplete answer now, so that you can consult the Laczkovich paper in the meantime.)