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 · Geometry · Proportion · Ornament
A pinnacle from one square
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
The procedure needs only compasses and a straight-edge and yields every dimension from the first without measuring. Scale the first square and the whole pinnacle scales with it, which is how a lodge could carry one design from a buttress to a tabernacle to a pulpit.
Roriczer takes all the heights of the elevation as multiples of the base square. The model uses illustrative multiples (five for the body, six for the spire), which the sliders change; whatever multiples a master chose, the whole pinnacle still followed from the one square.
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, “A pinnacle from one square”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/roriczer-pinnacle. Text, drawings and models licensed under CC BY-NC 4.0.