The Sections of a Parallelepiped and a Cube

The sections of a solid are doubly instructive from the point of view of the composition of correspondences. On the one hand, the role of transformations is reduced to a minimum. This is because, unlike the successive displacements or divisions, and so forth, examined in the preceding chapters, sectioning solids does not involve composing successive sections among themselves. In terms of transforming action, one section is similar to another. The only thing that differentiates them is the composition between the line one starts with and the path one travels according to variations of position and direction. To be sure, sections are transformations, but transformations that require a set of morphic comparisons for their compositions to be understood. On the other hand, because a section comes from cutting through a solid and thereby generating a plane form that is not necessarily equivalent to any of the external faces, the interplay of correspondences will be fundamental in defining these new surfaces. One can even maintain in this regard that the solid to be sectioned constitutes a tridimensional space in miniature that is comparable to external space in general, and that it includes its own internal visual figures that the sections will actualize. In that case, just as the psychogenetic evolution of geometric notions passes through three phases, ¹ intra-, inter-, and transfigural (algebrization), so the psychogenetic evolution of “subfigural” notions having to do with sections and their visible sides must must pass though phases of intra-, inter-, and transsubfigural correspondences. At least that is so insofar as the subject sticks to what is observable, devotes himself to inferences by taking account of the three dimensions, or goes on from there to a general structure whose deductive compositions are systematic. Thus, it is immediately clear what forms the usual three levels of compositions among correspondences will take. Either they will remain intramorphic, that is, prior to compositions, they will become intermorphic, that is, involve composition among correspondences of the same rank, or they will achieve the transmorphic, that is, achieve a higher degree of freedom.