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)
Rules · Domes · Geometry · Proportion
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 Pantheon’s interior is the plainest possible statement of geometry: a cylinder whose height is its radius, capped by a hemisphere of the same radius. The sphere it implies touches the floor at the centre and the rim of the oculus at the top. No other ancient building makes a single solid so legible from inside, and none had so large a clear span. It was not exceeded until the domes of the Renaissance, and it is still the largest unreinforced concrete dome in the world.
The portico in front, an octastyle Corinthian porch of monolithic granite shafts, is joined to the rotunda by a rectangular intermediate block that carries a second, higher pediment. The junction is awkward, and has prompted the theory that the portico’s columns were planned taller than those that could be obtained. The famous inscription naming Agrippa was copied from the earlier temple on the site, built around 25 BC.
Related rules
Classify arches by the geometry of the intrados: the number and position of the centres (one for round and segmental, two for pointed, three or four for basket and Tudor arches), or a single continuous curve (ellipse, parabola, catenary), and distinguish true arches, with voussoirs, from corbelled “arches” of oversailing courses.
Divide the inside of a dome into coffers that shrink in step with the meridians as they rise, so that their perspective deepens the dome and draws the eye to the light of the lantern.
Make the corner columns thicker than the rest by a fiftieth of their diameter, because, seen against the open sky, they look thinner than they are.
D corner = D × (1 + 1⁄50); Parthenon: 1.95 m against 1.905 m
A Doric frieze cannot obey all three of its rules at a corner: triglyphs over the column axes, a triglyph turning the corner, and metopes as broad as they are high. One of them must give way.
displacement of the corner triglyph = e − t⁄2 (e = half-depth of the architrave, t = triglyph width)
Divide the height from floor to ceiling into three and a half parts and give two and a half to the opening; make a Doric door five and a half twelfths as wide as it is high (an Ionic one two-fifths), and narrow it slightly at the top.
h = 2½⁄3½ H; Doric w = 5½⁄12 h, Ionic w = h ÷ 2½; top narrowed by ⅓, ¼ or ⅛ of the jamb as h ≤ 16, ≤ 25, ≤ 30 ft
Cite this page Chris Paton, “A sphere within a cylinder”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/pantheon-sphere. Text, drawings and models licensed under CC BY-NC 4.0.