Rules · Vaults · Domes · Geometry
Cover a square bay with a hemisphere whose diameter is the bay’s diagonal, cut off by the four vertical planes of its sides; the cut edges are then the four semicircular arches of the bay.
vault radius = ½ × diagonal = s / √2; the side arches are semicircles of span s
The sail vault (vela) is a dome without a drum or pendentives: a single spherical surface resting on the four corners of the bay. Because a sphere cut by a vertical plane gives a circle, each side of the bay receives a true semicircular arch, which is why the vault fits Brunelleschi’s arcades so neatly. Seen from below, each bay of the loggia is covered by a shallow cupola; seen from above (in the figure with the upper storey removed) the vaults form a row of small domes.
The same geometry, raised on pendentives and crowned by a separate hemisphere, gives the full “pendentive dome” of Brunelleschi’s Old Sacristy and Pazzi Chapel and of countless later churches.
To find how thick the pier of an arch must be, divide the intrados into three equal parts; draw a line from the springing through the first division, and produce it beyond the springing by its own length. A vertical through the end of that line marks the outer face of the pier.
pier ≈ ¼ span (semicircle) · ≈ 0.23 span (equilateral) · less for a lancet
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
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
Lay the ideal garden out on a circular plan in concentric rings, woods outside, then meadows and orchards, then gardens of knots and clipped evergreens divided by walks, with water and the fountain of the goddess at the centre.
Keep the lowest third of the shaft cylindrical; above it, reduce the radius along a curve found by dividing a quarter-circle and the upper two-thirds of the shaft into the same number of equal parts.
r(k) = R · cos(α·k), cos α = r_top ⁄ R, k = 0…1 over the upper ⅔
Cite this page Chris Paton, “The sail vault over a square bay”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/brunelleschi-sail-vault. Text, drawings and models licensed under CC BY-NC 4.0.