3-rectangulations: an algorithm to generate box packings

3-rectangulations are spatial representatives of assemblies of boxes into a box. Algorithms to generate various classes of 3-rectangulations are developed. The method is extended to specify the generation of higher-dimensional <2-rectangulations, d > 4. 3-rectangulations I consider the following combinatorial problem: how many arrangements of boxes into a box are there subject to the proviso that they be 'spatially distinct'? Within an architectural context an assembly of boxes into a box may be regarded as 'a threedimensional building description', and may be seen as the natural extension of the 'floor-plan problem' which has been discussed elsewhere (see, for instance, March and Earl, 1977; Earl and March, 1979; Flemming, 1978; Krishnamurti and Roe, 1978). The solution presented here takes the form of a generative scheme or an algorithm which enumerates equivalent families of box packings defined in terms of line designs on a three-dimensional grid, namely the 3-rectangulations. The definitions and notations that appear herein are consistent with those for 2-rectangulations (Earl, 1978; Krishnamurti and Roe, 1978). The material presented in this paper is sufficiently self-contained. A 3-rectangulation is an arrangement of an (/, m, n) unit grid into nonoverlapping cuboids whose faces lie on grid planes. The faces are combined to form maximal planes (Earl, 1978). (A set of maximal planes is a set of plane segments the union of any two of which never forms a single plane segment.) Maximal planes may be defined by a set of boundary lines. Constructively a 3-rectangulation may be viewed as a finite set of maximal planes each of which is coincident with one of the grid planes. Of these, six share boundary lines (those of the bounding cuboid) whereas the rest have boundary lines coincident with other maximal planes but do not share boundary lines. Figure 1 illustrates this construction. 3-rectangulations may be classified into the following type categories. 1. A 3-rectangulation is proper if /, m, n > 2. [An improper 3-rectangulation may be considered as a solid 2-rectangulation of unit height.] Figure 1. The internal maximal planes of a 3-rectangulation.