Uniqueness of four-piece triangle-to-square dissections beyond Dudeney’s

Determine whether there exist four-piece polygonal dissections, other than Henry E. Dudeney’s 1902 construction, that transform an equilateral triangle into a square via translations and rotations without overlap; if such dissections exist, characterize or classify them.

Background

The main theorem shows four pieces are optimal for dissecting an equilateral triangle into a square (without flips), matching Dudeney’s classic solution. Optimality does not imply uniqueness, and multiple distinct four-piece dissections could exist.

Identifying whether Dudeney’s construction is unique—or classifying all four-piece solutions—would refine historical and technical understanding and potentially reveal new design principles for minimal dissections.