But the objective was also to invent. The identification of the true regulatory principles at work in a given practice could lead to truly innovative combinations.5

Control and invention, these two objectives were also on the agenda of mathematician Gaspard Monge, the inventor of descriptive geometry. But tools can be mobilised to explore the yet unknown; they can serve invention. On the one hand, from Philibert De L’Orme to Gaspard Monge, this geometry of a projective nature was seen by its promoters as a means to exert greater control on architectural production. What has most of the time retained the imagination of designers are topological singularities, what mathematician Rene Thom has dubbed as ‘catastrophes’. First, he wanted to achieve a better control of the building production. Besides major architectural realisation like the castle of Anet or the Tuileries royal palace, his main legacy was the first comprehensive theoretical account of the geometric methods enabling designers to master the art of stereotomy. One of the best examples of this orientation lies in the way topology has been generally understood these days. Until the 18th century, most uses of mathematics and proportion had in practice to do with coordination and control rather than with the search for new solutions. From the Renaissance on, the geometry used for stone-cutting, also known as stereotomy, illustrates perfectly this ambivalence. Vitruvian proportion corresponded both to a way to ground architecture theoretically and to a method to produce buildings. In this domain also, a duality is at work. Whereas architects are usually interested in extreme cases that allow surprising effects to emerge, the mathematicians’ perspective is almost opposite. But this ambition was accompanied with the somewhat contradictory desire to promote individual invention. Let us turn now to mathematics as tools. For the architect, the aim was twofold. It has to do with conservation rather than sheer emergence. It is not fortuitous that De L’Orme was the first architect to be entrusted with major administrative responsibility by the crown. This interpretation of topology is at odds with what mathematicians consider as its principal objective, namely the study of invariance. De L’Orme was actually the first to understand some of the underlying projective principles at work in such a practice. control, giving precedence to standardisation upon invention. This explains the fascination exerted on architects by topological entities like the Mobius strip or the Klein bottle. From an architectural standpoint, the same mathematical principle can be simultaneously foundational and practical. Today, what might be often lacking is not so much

Figure 5. It might be necessary to reconcile, or at least articulate these two discrepant takes on the role of mathematics to fully restore their status. the capacity of mathematics to be on the side of invention, but rather its contribution to the framing and standardisation of design problems.