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 · Façades
Ad triangulum: the equilateral triangle
Alternatively, fit the section to an equilateral triangle standing on the width, so that the height is √3⁄2 of the width, and take the levels at equal divisions of that height.
H = W · √3⁄2 ≈ 0.866 W · Milan: 96 braccia wide → 83.1, rounded to 84 = 6 × 14
Stornaloco, a mathematician from Piacenza, was asked in 1391 to work out the triangulated section. The height of the equilateral triangle is irrational, so he proposed a practical rounding: six units of 14 braccia, 84 in all, which the builders could set out with a rod. The Milanese council later went lower still, to a squatter triangle, and the cathedral as built follows that compromise.
The difference between the two schemes is small on paper (the triangle is about thirteen per cent lower than the square), but it changes the proportion of every pier, window and vault, and at Milan it was argued over for years.
Related rules
A chain hanging freely takes the one shape in which it is in pure tension; turn that shape upside down and it is the shape of an arch in pure compression for the same loads.
free-hanging chain (catenary) ↔ ideal arch under its own weight
Draw a pointed arch from two centres on the springing line. With the centres at the springings, the arch is equilateral; with centres beyond them, a lancet; with centres between them, a drop arch. Late Gothic added arches of four centres and the reversed curve of the ogee.
r = s (equilateral) · r > s (lancet) · s⁄2 < r < s (drop)
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.
A round arch must rise half its span, so arches of different spans on one springing line have crowns at different heights. A pointed arch can be given any rise at least half its span, so arches of different widths can all be made to reach the same height.
round: f = s⁄2 (fixed) · pointed: r = (s²⁄4 + f²) ⁄ s for any f ≥ s⁄2
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
Cite this page Chris Paton, “Ad triangulum: the equilateral triangle”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/ad-triangulum. Text, drawings and models licensed under CC BY-NC 4.0.