Flying Buttresses

Domes and arches exert a horizontal component of force on their supports. So to build big ones, you have to counter that component. Big cathedrals have either very thick walls or big external piers pressing their sides toward the middle; these are called buttresses. A flying buttress is a buttress with a large section missing to reduce weight without compromising strength too much.

The introduction of the flying buttress enabled builders to both make the walls of cathedrals much higher and to install large stained glass windows in the walls.

Compare Il Duomo in Florence, Italy (massive walls, feels like a fort) with Notre Dame in Paris (feels almost like you are outside, but with heavenly colored lights playing on you).