A geometric framework for studying optimal reactor design is developed. For a given feed and a prescribed kinetics (perhaps involving many reactions), focus is on the full set of product composition vectors that can be produced in principle by means of all possible steady-state designs that employ only reaction and mixing (including designs that transcend current imagination). This set, called the attainable region by F. J. M. Horn, carries the full range of outcomes available to the designer. Of special importance are its extreme points, for these determine the region completely, and reactor optima are often realized there. Although the attainable region is not generally discernible in advance, one can nevertheless prove that it has certain universal properties, which, in turn, provide information about qualitative designs that provide access to the extreme points. Despite the vast spectrum of designs the attainable region is intended to embrace, two theorems suggest that its extreme points will always be accessible by means of classical elementary reactor types taken in simple combination. These results suggest that any reactor product that is realizable can, in fact, be realized by parallel operation of those canonical reactor building blocks that give rise to the extreme points. This paper lays the groundwork for additional theory, in which special properties of reactors that access the extreme points will be studied in some detail.