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 · Doors & windows · Ornament · Geometry
Bar tracery: windows drawn with compasses
Instead of cutting separate openings through a plate of wall, fill the whole window arch with glass and divide it by slender stone bars (mullions below, and circles, foils and sub-arches above) set out with compasses from the centres of the window’s own arches.
Bar tracery appears in the radiating chapels of Reims Cathedral, begun in 1211, and Villard drew one of those windows: two lancets under an oculus, all within a single arch. From Reims it spread quickly; by the 1240s the windows of Saint-Denis and the Sainte-Chapelle were subdivided again and again, two lights under each sub-arch, a larger rose above.
In the geometric window the oculus sits exactly between two sub-arches and the main arch. For an equilateral window of span s, its radius is s⁄4 and its centre lies about 0.56 s above the springing, a construction every apprentice could repeat.
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
Raise the central nave above the lean-to roofs of the aisles and pierce the upper walls with a row of windows, the clerestory, so that the middle of a wide hall is lit from high on both sides.
nave height ≈ 1½–2 × nave width; one window over each bay of the arcade
Cite this page Chris Paton, “Bar tracery: windows drawn with compasses”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/bar-tracery. Text, drawings and models licensed under CC BY-NC 4.0.