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 · Proportion · Geometry
The golden section
To divide a line in extreme and mean ratio, make the whole stand to the larger part as the larger part to the smaller; the ratio is φ = (1 + √5)/2 ≈ 1.618. Construct the golden rectangle by swinging the line from the midpoint of a square’s side to its far corner.
φ = (1 + √5)/2 ≈ 1.618; φ² = φ + 1; 1/φ = φ − 1 ≈ 0.618
Euclid defined the ratio as part of the construction of the regular pentagon and dodecahedron; it appears wherever pentagons do. Luca Pacioli’s De divina proportione (1509), illustrated by Leonardo da Vinci, praised its mathematical properties in almost mystical terms. A golden rectangle has the property that removing a square leaves another golden rectangle, and so on without end; quarter-circles drawn through the successive squares make the “golden spiral”, a close approximation to a logarithmic spiral.
The name “golden section” (goldener Schnitt) is not ancient but appears in German mathematics books in the 1830s, and most of its fame in architecture dates from the nineteenth and twentieth centuries.
Related rules
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
Proportion every step so that twice its height plus its depth equals the length of a natural pace on the level: two feet, or 24 pouces.
2 r + g = 24 pouces ≈ 65 cm (modern practice: 60–66 cm)
Hold that the beauty of the orders comes from fixed proportions that please by nature, as consonances do in music; where the ancients differ, the differences answer to circumstances such as distance and height.
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)
Cite this page Chris Paton, “The golden section”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/golden-section. Text, drawings and models licensed under CC BY-NC 4.0.