Domes

Turn an arch about its vertical axis and it sweeps out a dome. The idea is simple, but the result is a structure unlike any other: curved in two directions at once, able to cover a great round space without a single interior support, and so self-sufficient that a masonry dome, once closed, needs no centring to hold it up. For two thousand years the dome was the largest roof that could be built, and it acquired meanings to match: the vault of heaven, the canopy over a ruler or a god, the crown of a city.

This chapter follows the dome from the concrete rotunda of the Pantheon to the brick domes of Byzantium, where builders found how to set a circle on a square; it ends with a glance forward to the domes of the Renaissance and of the Ottoman architects, which are treated in their own chapters.

How a dome stands

A dome can be thought of as a set of arches, its meridians, leaning against one another at the crown, held together by a set of horizontal rings, its parallels. Under its own weight a hemispherical dome is squeezed along every meridian. Near the crown the rings are squeezed too, and help. But below a certain level the meridians want to spread outward at the base, and the rings are stretched. For a thin hemisphere the change from compression to tension in the rings comes at about 52° from the crown.

℟The lower dome pulls apart

Expect a hemispherical dome to be in compression along its meridians everywhere but in tension round its horizontal rings below about 52° from the crown; restrain that lower zone with weight, thickness, rings or chains, or it will crack along the meridians.

thin hemisphere under its own weight: hoop force changes sign at ≈ 51.8° from the crown

Masonry and unreinforced concrete are strong in compression and weak in tension, so the lower zone of an ancient dome almost always cracks along its meridians. The cracks are not fatal. The dome simply becomes a ring of independent arches leaning on a compressed crown, and it stands as long as their feet are held. The history of the dome is largely a history of ways to hold them: thickening and weighting the haunches, burying the lower part of the dome in a drum, buttressing it with half-domes and piers, and, from the Renaissance on, binding it with chains of timber and iron.

Two further facts guide every dome-builder. A shallow dome pushes outward harder than a steep one of the same span, because its meridian arches are flatter; and weight near the crown adds to the outward push, while weight near the base steadies it. Domes are therefore made thick and heavy below and thin and light above, and an opening at the crown, an oculus, removes weight exactly where it does most harm.

Roman domes

The corbelled domes of the Bronze Age, such as the Treasury of Atreus at Mycenae, are not true domes: each course is a flat ring cantilevered slightly inward beyond the one below (see The corbelled dome: rings stepping inward). The true dome, whose stones or concrete work together in compression, was the creation of Roman builders using concrete. Vitruvius describes the domed sweating room of a bath, round, as high to the springing of its dome as it is wide, with a bronze disc hanging in the opening at the top to regulate the heat.

℟The sweating room: as high as it is wide

Make the sweating room of a bath circular, as high to the springing of its dome as it is wide, with an opening at the crown closed by a bronze disc that can be raised and lowered on chains to regulate the heat.

height to springing = width; oculus with a movable bronze shield

Bath halls were the Romans’ laboratory. The so-called Temple of Mercury at Baiae, a bath rotunda of the late first century BC, already spans more than 21 m; Nero’s Golden House in the 60s AD had an octagonal domed hall opening onto radiating rooms. By the early second century Roman builders were ready for a dome twice as wide.

The Pantheon

The Pantheon in Rome was rebuilt under Hadrian, around AD 118–128, on the site of an earlier temple founded by Agrippa, whose name it still bears in the inscription on its portico. Behind an octastyle porch of Corinthian columns with monolithic granite shafts, a rectangular intermediate block leads into a rotunda that is the purest geometrical interior of antiquity. The diameter is 43.3 m (150 Roman feet), and so is the height from the floor to the top of the dome. The cylindrical drum rises exactly to the centre of the implied sphere; the dome is its upper half.

℟A sphere within a cylinder

Make the height of the rotunda to the top of its dome equal to its diameter, so that the interior would exactly contain a sphere resting on the floor: the drum rises to the centre of the sphere and the dome is its upper half.

Pantheon: Ø = height = 43.3 m (150 Roman feet); drum = ½ Ø

The wall of the drum is about 6 m thick, but it is not solid. Eight great recesses (the entrance, the apse opposite, and three on each side screened by pairs of columns) are hollowed out of it, and brick relieving arches in the concrete carry the loads round them onto eight massive piers. Above the interior order runs an attic zone of blind windows, then the cornice from which the dome springs.

The dome itself is a masterpiece of weight management. It is some 6 m thick at the springing and only about 1.5 m at the oculus. Its concrete was graded as it rose: heavy stone aggregate at the bottom, tuff and broken brick in the middle, light volcanic pumice and scoria near the top. Its inner surface is hollowed by five rings of 28 coffers, each stepped back in several frames, which save more weight and make the vast curve legible. The 28 ribs do not line up with the eight bays of the drum below, so the dome seems to turn independently above the building, like the sky.

Outside, the lower third of the dome is hidden inside the drum, which rises well above the interior springing, and a series of step rings loads the zone where the dome would otherwise spread. The oculus, 8.2 m across, is the only window. A disc of sunlight moves across the coffers and down the walls through the day, and rain falls onto a floor gently sloped to drains. No larger unreinforced concrete dome has ever been built.

A circle on a square

A rotunda gives a dome the round wall it needs. Most buildings, however, are made of square or rectangular bays, and a dome over a square must somehow bring the corners of the square round to a circle. Two solutions dominate the history of architecture.

The pendentive is the more elegant. Imagine a sphere passing through the four corners of the square bay. The four vertical planes of the sides slice it in four semicircles, which are the four arches of the bay. The horizontal plane through the crowns of the arches slices it in the circle inscribed in the square. What remains of the sphere between the arches and below that circle is four curved triangles, one in each corner: the pendentives. They are part of one continuous surface, so they carry the ring and whatever stands on it smoothly down to the four corner piers.

℟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.

sphere radius R = S⁄√2; arches = sphere ∩ side planes (radius S⁄2); ring at h₀ + S⁄2 with radius S⁄2

Once the ring exists, the builder has three choices. He may continue the same sphere upward, covering the bay with a single shallow sail vault; he may set on the ring a separate, taller hemisphere, the dome on pendentives; or he may raise a windowed cylinder, a drum, between the ring and the dome, to light the centre of the building and lift the dome into view.

The squinch is older and more straightforward. An arch, a niche or a small cone is built diagonally across each upper corner of the square. The four squinches and the four walls between them form an octagon, and an octagon is near enough to a circle to carry a dome, or it can be turned into a sixteen-sided figure by a second tier of smaller squinches. The squinch was the Persian solution, used in Sasanian palaces, and it passed into the churches of Armenia and Georgia, into some Byzantine buildings of the eleventh century, and into Islamic architecture, where it was elaborated into the honeycomb vaulting called muqarnas.

Hagia Sophia

The pendentive reached its monumental form in the church of Holy Wisdom, Hagia Sophia, built for the emperor Justinian in Constantinople between 532 and 537 by Anthemius of Tralles and Isidore of Miletus, a mathematician and a geometer rather than master masons. Their task was to combine the long nave that the liturgy required with a dome that would crown the capital. The solution was a central square about 31 m across, defined by four great piers and four great arches, with a dome on pendentives above. To east and west two of the arches open into semidomes of the same radius, and each semidome opens in turn into smaller exedrae, so that a space some 70 m long is covered by a cascade of curved shells with no columns in its centre.

℟Semidomes lengthen the dome

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

At the base of the dome forty windows are set between forty short piers, each of which continues one of the forty ribs of the dome down to the cornice and is backed outside by a buttress. Justinian’s historian Procopius described the effect by saying that the dome seemed to hang from heaven rather than to rest on masonry. The windows occupy exactly the zone in which a hemispherical dome tends to split along its meridians; the builders turned the weakness into the building’s most famous effect.

℟A ring of windows at the foot of the dome

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

On the north and south, where no semidome takes the thrust, the great arches are filled only with thin walls pierced by windows, and the real support is outside: four enormous buttress piers, joined to the main piers by arches over the galleries, rise above the roofs. Even so the piers leaned outward as the building rose, and the first dome, which was notably shallow, fell after earthquakes in 558. Isidore the Younger rebuilt it with a higher profile, about 6 m higher, and it was completed in 562. It still stands, though parts of it collapsed and were rebuilt after earthquakes in 989 and 1346.

After Hagia Sophia

No Byzantine church after Justinian tried to rival Hagia Sophia’s span. The middle Byzantine church domed a small central bay over four columns and raised the dome on an ever taller drum; five domes crowned the richest churches. In Russia the drum grew taller still and the dome swelled into the bulbous onion dome, an outer shell of timber and metal over a lower vault: a dome made for the skyline rather than for the interior.

The great domes of later centuries returned to the Pantheon and to Hagia Sophia. Brunelleschi’s dome over the octagonal crossing of Florence Cathedral (1420–1436), about as wide as the Pantheon, was built without centring as two linked shells bound by chains. Michelangelo’s dome of St Peter’s in Rome, completed by Giacomo della Porta in 1590, stands on pendentives and a colonnaded drum. And in Istanbul the Ottoman architect Sinan took Hagia Sophia as his starting point, surrounding central domes with ranks of semidomes at the Süleymaniye (1550–1557) and setting a dome of about the same span on an octagon of piers at the Selimiye in Edirne (1568–1575). Those buildings belong to later chapters; the geometry that made them possible was worked out here.