Existence of three-piece dissections between distinct regular polygons

Determine whether there exist pairs of distinct regular polygons—a regular n-gon and a regular m-gon with n ≠ m—that admit a common three-piece polygonal dissection under translations and rotations without overlap; if such pairs exist, identify and characterize them.

Background

While the paper resolves the triangle-to-square case for three pieces (without flips), the broader question remains for other pairs of regular polygons. Prior asymptotic results show linear growth in piece count for certain families, but small constant-piece cases may exist for some pairs.

Establishing whether any distinct regular polygons share a three-piece dissection would be a significant result in low-complexity dissections and could connect to symmetry and angle-compatibility constraints uncovered by the matching-diagram framework.