Two-piece self-rearrangements of triangles and of polygons in general position

A self-rearrangement cuts a polygon into polygons and moves them by isometries to tile it again; it is nontrivial if the set of positioned pieces changes. We classify the triangles that admit a nontrivial two-piece self-rearrangement, with arbitrary polygonal pieces and a cut of any complexity. With reflections allowed, they are exactly the isosceles triangles; every scalene triangle needs three pieces, also when no piece may stay in place. With proper motions, they are the equilateral triangles and the isosceles triangles whose apex angle lies in an explicit union of three sequences of rational multiples of π; for non-equilateral isosceles triangles, both pieces can be convex for exactly four apex angles. The proofs use invariants blind to the cut: the edge function, radial profiles and a periodized indicator. For convex polygons with at least four vertices we give six general-position conditions that exclude nontrivial two-piece rearrangements and hold for algebraically independent coordinates, and we construct, for every such number of vertices, convex polygons without symmetry that admit a nontrivial one with two convex pieces moved by rotations about one point. MSC2020: 52B45 (primary); 52C20, 37E05. Files: the paper (PDF), its LaTeX sources, and a reproduction archive with the exact-arithmetic programs that check the explicit constructions and the program that draws the figures.