Rules · Domes · Ornament · Structure
Divide the inner surface of the dome into sunk panels arranged in rings and meridians; let them diminish as they rise with the converging meridians, and stop them short of the crown so that a smooth zone surrounds the eye.
Pantheon: 5 rings × 28 coffers, each stepped in several recessions
Coffering (Latin lacunaria) lightens a vault by removing material where it does least work, and it articulates the vast surface so that its curvature can be read. In the Pantheon each coffer is stepped back in several frames, and the steps are cut more deeply on the lower side, so that the coffers look correctly foreshortened to a visitor standing on the floor.
The number 28 does not divide evenly into the eight bays of the drum below, so the ribs of the dome do not line up with the niches and columns: the dome turns independently of the wall, like a separate heaven over the earth. Twenty-eight, a “perfect” number to ancient mathematicians and the number of days in a lunar month, may have been chosen for its meaning.
In a bonded masonry wall the courses above an opening corbel over it of their own accord, so the lintel carries roughly the triangle of wall with sides rising at 60° from its ends; everything above that triangle arches onto the jambs.
load ≈ equilateral triangle on the span: W ≈ 0.43 · L² · t · ρ g
Build an arch of wedge-shaped stones (voussoirs) whose joints all point towards the centre of the curve: each stone can then fall only by forcing its neighbours apart, and so the ring holds itself up once the last stone, the keystone, is in place.
joints ⟂ intrados (radial to the centre)
An arch pushes outward on its supports as well as down. For the same load and span the horizontal push grows as the rise falls: halve the rise and you roughly double the thrust.
H ≈ w·s² ⁄ 8f (uniform load w per metre, span s, rise f)
Trace the path of the resultant force through the stones from crown to foundations. If a line can be found that stays everywhere within the masonry, the arch will stand; where it touches an edge, a hinge will open, and four hinges make it collapse.
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
An arch cannot support itself until it is complete. Build it on a timber centring shaped to the intrados, strike the centring only when the keystone is set and the mortar has taken, and reuse it from bay to bay, which is one reason to keep the spans of an arcade equal.
Cite this page Chris Paton, “Coffers in rings that shrink toward the crown”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/pantheon-coffers. Text, drawings and models licensed under CC BY-NC 4.0.