Complete Dissections: Converting Regions and Their Boundaries

Classical dissections convert any planar polygonal region onto any other polygonal region having the same area. If two convex polygonal regions are isoparametric, that is, have equal areas and equal perimeters, our main result states that there is always a dissection, called a complete dissection, that converts not only the regions but also their boundaries onto one another. The proof is constructive and provides a general method for complete dissection using frames of constant width. This leads to a new object of study: isoparametric polygonal frames, for which we show that a complete dissection of one convex polygonal frame onto any other always exists. We also show that every complete dissection can be done without flipping any of the pieces. Dedicated to the memory of Martin Gardner