Contemporary geometry for the built design?

I explore the terrain that lies between architecture and geometry, from the perspective of a structural scientist with no professional architectural expertise. The divide between these disciplines perhaps stems from an ancient dichotomy between the art versus engineering schools of architecture, fertilised by the current dogma that art and science can never meet. Architects stand to gain much from study of the spectacular advances in geometry in recent decades, such as the growing understanding of cellular patterns in space, tiles, nets and curved surfaces. Some examples of those advances are discussed in detail. I conclude that both architecture and geometry would benefit from a renewed mutual interest.