Two-piece self-rearrangements of convex polygons: decision theorems

A nontrivial two-piece self-rearrangement of a polygon cuts it into two polygons and reassembles them by isometries into the same polygon with a different positioned partition. For convex polygons with exact real-algebraic vertex coordinates and two arbitrary polygonal pieces, we prove a decision theorem in each motion range, proper motions and all isometries, without any hypothesis on the polygon. The positive polygons are those with a nonidentity rotational symmetry (with reflections allowed, any nonidentity symmetry), two explicit families of mirror-symmetric polygons, and the polygons passing one of finitely many tests. A terminating procedure decides membership, computes from the polygon a bound on the number of cut segments of some rearrangement, and constructs the pieces and motions. The proof sorts every rearrangement by the symmetry group of the polygon and the type of the relative motion, and proves a decision theorem for each branch. We also decide the versions with convex pieces and with a bounded number of cut segments, and give explicit answers for several families of quadrilaterals. Triangles are treated in a companion paper. MSC2020: 52B45 (primary); 52C20, 68W30. Files: the paper (PDF), its Supplement (PDF) with the proofs not given in the paper, their LaTeX sources, and a reproduction archive with the exact-arithmetic programs that check the computer-assisted results.