The pointed arch, by contrast, is the product of Platonic geometry. It results from the intersection of two arcs drawn on the same straight line—for French builders like the ones who built Sens and Saint Denis, essentially two quick swipes of a compass. Any builder worth his pay could then use a set of calipers to reproduce a series of those same arcs within the regular rectangle of the church’s outside wall, or crisscross a pair of pointed vaults inside a series of perfect squares in the interior, enabling him to prop up the roof with far less stress than the old barrel vault of the Romans.
