A shallow dome pushes harder outward than a steep one of the same span: when the first dome of Hagia Sophia fell, it was rebuilt with a higher rise so that its thrust would be more nearly vertical.
rise raised by ≈ 6 m (about 20 Byzantine feet), 558–562
The thrust of an arch or dome grows as its rise diminishes: a flat arch pushes hardest, a tall pointed one least. The original dome of 537 was notably flat and its supports had already spread. Isidore the Younger, nephew of one of the original architects, gave the new dome a higher profile and strengthened the supports; it was completed in 562 and still stands, although parts of it fell and were rebuilt after earthquakes in 989 and 1346.
The episode is one of the first recorded cases of an architect revising a structure on the evidence of its failure: the empirical method by which masonry construction advanced until the nineteenth century.
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.
sphere radius R = S⁄√2; arches = sphere ∩ side planes (radius S⁄2); ring at h₀ + S⁄2 with radius S⁄2
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.
octagon side = S (√2 − 1) ≈ 0.414 S; squinch span = octagon side
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.
crown = h₀ + S⁄√2 (lower than a hemisphere on the ring, h₀ + S)
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.
drum height ≈ ⅓–1 × its radius (Byzantine); much taller in Russian churches
Pierce the base of the dome with a close ring of windows between buttressing ribs, so that the shell seems to float on light rather than to rest on the arches.
Hagia Sophia: 40 windows and 40 ribs; dome ≈ 31 m across, crown 55.6 m above the floor
To turn a domed square into a long church, open two of its great arches into semidomes of the same radius, and open the semidomes in turn into smaller exedrae, so that the space expands by stages from the central dome.
Hagia Sophia: central square 31 m; two semidomes of radius ≈ 15.5 m; four exedrae; interior ≈ 31 × 70 m
Cite this page Chris Paton, “Make a shallow dome steeper”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/dome-rebuilt-higher. Text, drawings and models licensed under CC BY-NC 4.0.