How did Francesco Borromini use concave and convex surfaces in his buildings?

Francesco Borromini, a central figure of the Baroque style, utilized concave and convex surfaces to transform traditional architectural forms into fluid, active, and dynamic masses.1

Rather than relying on the static, anthropometric proportions favored by his contemporaries, Borromini based his designs on complex geometric configurations. He employed these undulating surfaces to create a sense of movement and to challenge the traditional, restrictive nature of walls. By treating space and light as primary architectonic components, he allowed the surfaces of his buildings to expand and contract, a technique that drew inspiration from the dynamic potential he observed in the works of Michelangelo.2314

A primary example of this approach is the church of San Carlo alle Quattro Fontane, where Borromini integrated concave and convex masses to create a "curious, undulating lower zone" that electrified the architectural landscape of Rome. His innovative use of these curves helped define a new architectural vocabulary that influenced designers throughout Europe for the following century.314