Historically, certain kinds of reciprocity between geometry and architecture have been used to bring about rational causes and practical means of formal innovation. Today, the digital medium is having unanticipated effects on this reciprocity. The results are profound innovations not only in the realm of form as such, but also in the process of translating abstract geometric concepts into building construction principles. Between the excesses of curved geometries and the economic constraints of building construction lies a seemingly incommensurable disparity. On the one hand, the digital medium affords the architect a means to model three dimensional forms that are entirely free of planarity. On the other hand, it aids the ongoing industrial production of materials according to the geometric extrusion of linear and flat components and surfaces. To operate in this breach, it is necessary to acquire a geometric vocabulary that operates between oblique and curved forms and the economically determined mechanical constraints of building assembly. This vocabulary, derived from projective and topological geometry, constitutes an auxiliary system of order, one which can ultimately serve to discretize curved surfaces into flat units, and thus translate complex surfaces into forms constructible at an architectural scale. Such an application of geometry produces a continually expanding repertoire of three-dimensional architectural form. Among the most far reaching effects is the recalibration of the long standing relationship between the part and the whole in architecture. Lectures tracing the lineage and technical bases of orthographic and perspective projection, projective geometry and topology will provide the foundations for the development of a vocabulary. Workshop exercises will apply these principles to particular problems of surface redefinition. The rendition of curvatures according to the logic of different patterns will demand critical assessment. The overall course objective is to provide the tools and critical instruments to imagine and represent with precision, dexterity, and virtuosity a continually expanding repertoire of three-dimensional architectural form.