Three-piece triangle-to-square dissection with nonpolygonal (curved) pieces

Establish whether allowing nonpolygonal pieces with curved boundaries enables a three-piece dissection between an equilateral triangle and a square under translations and rotations (and possibly reflections), without overlap; or prove that even with curved pieces, three pieces are insufficient.

Background

The main result rules out three-piece polygonal dissections between a square and an equilateral triangle (without flips). The authors explicitly raise the possibility that curved pieces might change the feasibility, suggesting the theorem may extend but leaving the question open.

Clarifying whether curvature can reduce the minimal piece count would test the robustness of the matching-diagram approach and the geometric invariants used in the proof, potentially requiring novel methods for nonpolygonal boundaries.