Rules · Domes · Geometry · Structure
Draw the curve of a pointed dome with arcs whose radius is four-fifths of the span, their centres standing on the springing line one-fifth of the span in from the opposite side.
r = ⅘ S; centre at ⅕ S from the far springing; rise ≈ 0.74 S
The quinto acuto (“pointed fifth”) was a standard Tuscan way of setting out a pointed arch, and the building programme for the Florence dome adopted in 1420 prescribed it for the angles of the octagonal vault. A pointed profile pushes outward much less than a semicircle, and (decisively for a dome that had to be built without full wooden centering over a crossing some 45 m across) its lower courses lean inward less steeply, so that the brickwork could be laid ring by ring and remain stable at every stage.
In the model the curve is applied to the angle ribs, measured across the corners of the octagon; this is an editorial reconstruction of the principle, not a survey of the built curve, which varies slightly from face to face.
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)
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
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.
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.
Cite this page Chris Paton, “The pointed fifth”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/quinto-acuto. Text, drawings and models licensed under CC BY-NC 4.0.