Finiteness and enumeration of rectangles obtainable from a triangle via three-piece dissections

Ascertain whether the set of rectangles that can be formed by dissecting a triangle into exactly three polygonal pieces and reassembling them via translations and rotations without overlap is finite; if the set is finite, develop a complete enumeration or characterization of those rectangles.

Background

Beyond the specific triangle-to-square case, the paper suggests studying which rectangles can be obtained from an arbitrary triangle using only three pieces. The result for the equilateral triangle and square shows that three pieces are insufficient in the no-flip model, prompting inquiry into other targets and whether only finitely many rectangles are attainable.

Resolving finiteness and providing an enumeration would advance structural understanding of low-piece dissections and could inform broader classification efforts across polygon pairs.