Pendentive and squinch
Two ways to set a round dome on a square bay: the pendentive, a spherical triangle cut from a sphere through the corners, and the squinch, an arch across each corner.
The rules it illustrates
The pendentive: a sphere cut by a square. To set a round dome on a square bay, imagine a sphere passing through the four corners of the square; cut it by the four vertical planes of the sides and by the horizontal plane of their crowns. The four spherical triangles left in the corners are the pendentives; the circle at their top carries the dome.
The squinch: from square to octagon. Throw an arch diagonally across each upper corner of the square, backed by a small conch or cone; the four squinches and the four walls between them make an octagon, on which a round dome can sit.
The sail vault: one sphere, uncut. If the sphere that forms the pendentives is simply continued above the ring, the bay is covered by a single shallow dome, the sail vault or pendentive dome, which springs directly from the four corners.
Raise the dome on a drum. Lift the dome on a cylindrical or polygonal drum pierced with windows between the ring of the pendentives and the springing of the dome, to light the centre of the church and to make the dome visible from afar.
Where it appears in the book
In the styles Byzantine.
In the chapters Domes, Byzantium and the Early Church.