Dudeney Dissection of Polygons

Abstract

Given an equilateral triangle A and A Square B of the same area, Henry E. Dudeney introduced A partition of A into parts that tan be reassembled in some way, without turning over the surfaces, to form B. An examination of Dudeney’s method of partition motivates us to introduce the notion of Dudeney dissections of various polygons to other polygons.

Let A and B be polygons with the same area. A Dudeney dissection of A to B is a partition of A into parts which tan be reassembled to produce B as follows: Hinge the parts of A like a chain along the perimeter of A, then reassemble them to form B with the perimeter of A is in its interior, without turning the surfaces over. Using tilings of the plane, we produce Dudeney dissections of quadrilaterals to other quadrilaterals, quadrilaterals to parallelograms, triangles to parallelograms, parallelhexagons to trapezoids, parallelhexagons to triangles, and trapezoidal pentagons to trapezoids.