Dissecting Cuboids into Cuboids
Jeroen Spandaw · American Mathematical Monthly · 2004
1. INTRODUCTION. (Throughout this note we work in Euclidean space R.) Two convex polygons P and Q in the plane are said to be congruent by addition, written P Q, if we can dissect one of them into a finite number of polygonal jigsaw pieces that can be rearranged to form the second polygon. We denote a dissection of P by P = Pi + P2 + - - - + P,. By definition this means that the pieces P1, P2, ...., Pn are closed polygons whose union is P and that two distinct pieces have no interior points in common. Since the polygonal pieces can be cut into triangles, we can always assume that all pieces are triangles. Two polygons P and Q are said to be congruent by subtraction, denoted P - Q, if there exist polygons P' and Q' and polygonal dissections P' = P + P1 + - - - + Pn and Q' = Q + Q1 + - - - + Q, such that P', Pi,..., and P, are congruent to Q', Q1,..., and Q,, respectively. It is easy to see that congruence by addition and congruence by subtraction are equivalence relations. It is also clear that P Q implies P - Q, and P - Q implies that P and Q have the same area. Conversely, it is not difficult to show that P Q as soon as P and Q have the same area. Hence the two concepts of congruence by addition and subtraction are equivalent in the plane. A proof can be found in [3, Theorem 5.3.6]. One can readily generalize these notions to three dimensions. (In three dimensions one considers dissections into (irregular) tetrahedra.) A famous theorem of Dehn [2] asserts that a regular tetrahedron is not congruent by subtraction to a cube with the same volume, thus solving Hilbert's third problem. This result shows that it is impossible to devise a purely geometric theory of volume for three dimensional polytopes. We refer the reader to [3] and [5] for more information about Hilbert's problem and its history. 2. PROBLEM. We return to the plane and consider dissections of rectangles into rectangles instead of triangles. (In this note all rectangles are oriented so that their sides are parallel to the coordinate axes.) We consider the analogue of equivalence by subtraction that allows only rectangular pieces and denote this equivalence relation by =. For example, the rectangles in Figure 1 are equivalent.