Francesco Borromini's San Carlo alle Quattro Fontane is the Baroque's answer to the static, symmetrical, circle-and-square geometry of the Renaissance: to build space not as a rigid mass but as a flexing, undulating, tensioned surface. Whatever the absolute calm of the Villa Rotonda is, San Carlo is its opposite — movement, rhythm, and continuity.

The church is tiny, yet its plan is extraordinarily complex. Borromini works not with circle and square but with ovals and triangles fused into one another; the inner wall never stays straight for a moment, undulating in concave and convex segments. On top of this undulating wall sits a dome with an oval base. The dome's inner surface is coffered with crosses, hexagons, and octagons that shrink as they rise; narrowing upward, this pattern makes the dome look taller and lighter than it is — lit by hidden windows, it seems to float. Structure here becomes a machine of mathematical illusion.

The façade (completed just before Borromini's death) carries the same logic to the street: instead of a flat surface, a sculptural stone curtain swaying in a concave-convex-concave rhythm. Borromini bends stone as if it were a flexible material; architecture begins to speak not in the static language of mass but in the dynamic language of continuous curves. San Carlo is the manifesto building in which Baroque architecture sets space in motion rather than freezing it, and the source for the curving architectural experiments of the next two centuries.