Rules · Geometry · Roofs · Structure
A saddle surface in which two opposite corners are raised and two lowered is a hyperbolic paraboloid; it contains two families of straight lines, so it can be formed with straight boards and yet has double curvature.
y = k·x·z (or any bilinear patch through four points not in one plane)
Take a square frame, lift two diagonally opposite corners, and connect opposite sides with straight strings: the strings describe a hypar. The surface curves upward in one diagonal direction and downward in the other (it is anticlastic), which makes a thin shell very stiff; and because it is ruled by straight lines, the formwork for a concrete hypar can be built of straight timbers.
Candela built most of his several hundred shells from combinations of hypars: umbrellas of four on one column, groined vaults of four meeting on diagonal valleys, and the eight lobes of the Los Manantiales restaurant.
A thin curved shell carries load by the membrane forces in its surface, not by bending; so shape it to its loads and supports, and it can span great distances with a thickness of a few centimetres.
span ÷ thickness of a good shell ≈ 400–1000 (an egg: ≈ 100)
Span the whole depth of the building with the structure so that each floor is a single open plate, adaptable to uses that cannot be foreseen.
Pompidou: 48 m clear span between column lines, 13 bays of 12.8 m
Define a form by a few parameters and a rule, and let every element vary continuously with them, instead of repeating identical elements.
floor i: rotation = i × θ, scale = 1 − (1 − τ)·(i ⁄ n)^1.3
Place a point (or curve) in the design and let a property of every element (size, rotation, opening) be a function of its distance from that point.
r(panel) = r_min + (r_max − r_min) · d ⁄ d_max
In a framed building the columns and beams carry every load; the outer wall can be reduced to a light cladding hung on the frame, floor by floor, and need grow no thicker as the building rises.
load-bearing wall: thickness grows with height · frame: cladding thickness constant
Build the house as a standard skeleton of flat concrete slabs, a few columns set back from the slab edges, and a stair; everything else (walls, windows, partitions) is infill that can be placed freely.
3 slabs + 6 columns + 1 stair
Cite this page Chris Paton, “The hyperbolic paraboloid is made of straight lines”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/hypar-straight-lines. Text, drawings and models licensed under CC BY-NC 4.0.