Developability

Let us extend this experiment a little further. Were the roof to be clad with copper over a hyperbolic paraboloid substrate the advantages of developability are apparently lost; by definition any child of the parent geometry will have a degree of double curvature, however small. Hooke’s law (1660) states that all materials may deform elastically to a limit.3 The following century this was defined by Thomas Young (1773-1829) as the ‘modulus of elasticity’ which, when passed, means that the metal will not spring back to its original state once the applied load has been removed.4 The yield point in a ductile material such as copper allows it to be formed easily compared with steel. Developable surfaces present fascinating opportunities for architects and engineers interested in ‘affordable complexity of form’. Steel for making car bodies, which is far less malleable compared with copper, can also be ‘stretched’ into shape by stamping sheet steel over formers, the cost of which is eye-wateringly significant but afforded by the large number of repeats that the vehicle industry strives for. This is not a practical arrangement for the building industry, but let us not forget the intrinsic flexibility of rigid sheets of material, including steel, within the limits of its molecular structure. Figure 14. From the parent template, child templates of individual pieces can be appropriately sized to match practical copper-sheet size or maximum size for practical on-site manipulation by the plumber, whichever is the major constraint. The advantage of developability is that subsections of the whole retain the geometrical qualities of the parent. The designer would no doubt appreciate being able to try this in real time, but this has yet to be put to test as the computation aspects are still to be resolved in terms of performance. If the underlying geometry is mathematical such as a hyperbolic paraboloid, there is a greater possibility for a tessellation to be derived that looks more organised than would be the case were this approach to be applied to an extravagantly expressive freeform surface, especially one with folds and creases. Whereas our dry sheet would not conform even with uniformly applied pressure and the aid of a relatively unexaggerated hyperbolic paraboloid former (a ‘mould’ to press a material against to make it change geometrical form) as it would crease, in contrast the wet cardboard sheet would ‘relax’ into the former. The unexpected flexibility of glass is first witnessed in the school laboratory when a long glass rod is shown to be bendable to a surprising degree, and if left to droop between two supports eventually will assume the catenary curve. Figure 13. Figure 12. Needless to say, our copper cladding example needs to be refined a little, as copper, like all metals, is malleable, and in fact is one of the most malleable of all – hence its traditional use for making cooking vessels. Taken to its limit the sheet will eventually fold diagonally across to form two triangles hinged along the crease. To take advantage of the malleability, a former has to be provided, and is a major cost if metal is to be stretched into a new form. Trying to make a hyperbolic paraboloid from the same sheet requires two sets of hands. The reason for the difference is obvious, of
course, as the soaking allows the cellulose fibres to stretch as well as displace spatially (to a limited degree). Any given material has a modulus of elasticity and a limit. If the flexibility of a sheet metal of a particular thickness and its elastic limits are built in as a parameter of flexion over length, algorithmically, a surface can be tiled with pieces of the chosen cladding that will conform to double curvature without creasing or snapping. This material quality of flexibility is available in all sheet materials, even the most brittle when applied carefully. Mathematically this is a demanding optimisation task. If Actor A passively supports a sheet of card by holding it by diametrically opposite corners, while Actor B pulls down on both the opposite corners the sheet is curved cylindrically. To make the computation task more demanding still, criteria such as searching for a shape as regularly polygonal as possible can be added along with a nesting requirement that minimises wastage. The aesthetic would therefore be the difference between a tortoiseshell pattern organised principally around the hexagon, for example, conforming to a doubly curved ruled surface, and the ordered chaos of a Voronoi diagram which would most likely result from tackling a freeform in this way. Felix Candela, L’Oceanografic, Valencia, Spain, 1998
A rich mixture of constructional logic, extraordinary structural performance and : :a compelling aesthetic that seems to work at any scale. Vladimir Shukhov, Shabololvka (Shukhov) Tower, Moscow, 1922
Interior view showing a series of hyperboloids of revolution stacked on top of each other. If Actor A pulls their corners up while Actor B pulls theirs down with equal force, the stiffness of the sheet prevents the double curvature from taking place; the sheet stays resolutely planar. Imagine that the sheet of cardboard used in our experiment above was soaked in water before attempting to form a hyperbolic paraboloid. This can be appreciated by holding a piece of cardboard in our hands. A hyperbolic paraboloid with straight edges (directrices) and rulings (generatrices).

Updated: 30.10.2014 — 19:22