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 · Cities · Plans · Geometry
Devise, order and array the new town
Lay out a new town with the cord on open ground: straight streets crossing at right angles, running through from gate to gate, blocks of equal size between them, and a square in the middle; then, when the town can afford it, enclose the rectangle with a wall.
The bastides were founded on new sites, usually by a contract between a prince and a local lord, and needed to be attractive to settlers quickly. A grid was the fastest and fairest way to divide land: the plots could be equal and the streets straight, and the whole town could be surveyed with cords and stakes in days. Monpazier (1284), Villeréal, Monflanquin, Mirande and Aigues-Mortes show variants of the pattern.
That planning was understood as a craft is shown by Edward I, who in 1296 summoned from the English towns men who knew best how to devise, order and array a new town, to plan one at Berwick. Walls often came later, when the inhabitants obtained leave and means to build them.
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
Draw a square the width of the pinnacle; inscribe a second square on the mid-points of its sides, and a third on the mid-points of the second. Turned back into line, the squares give the widths of the body, its panels and the spire; every height is a multiple of the first side.
widths: s, s⁄√2, s⁄2, s⁄(2√2) … · heights: n × s
Cite this page Chris Paton, “Devise, order and array the new town”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/bastide-grid. Text, drawings and models licensed under CC BY-NC 4.0.