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
Rules · Vaults · Geometry · Ornament
The fan vault: identical cones of tracery
Make each springer the foot of an inverted half-cone (a conoid) whose ribs all have the same curve and are cut across by horizontal rings; where the conoids stop, fill the centre with flat panels.
The fan vault is the last invention of English Gothic and has no parallel abroad. It appeared in the Gloucester cloister in the 1350s and culminated in John Wastell’s vault over King’s College Chapel, Cambridge, and in the pendant vault of Henry VII’s Chapel at Westminster. Because every rib of a fan is the same, the whole can be carved from a few repeated blocks: the “ribs” are tracery cut on the surface rather than arches built separately.
Related rules
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
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, “The fan vault: identical cones of tracery”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/fan-vault. Text, drawings and models licensed under CC BY-NC 4.0.