These properties are encoded in scalar and tensor fields which act on the design and give hints about material density and directionality without imposing a specific formal expression. For example, for Thomas Heatherwick’s British Pavilion at the 2010 Shanghai Expo, an interface was created for the early stages of design through which could be observed and discussed the properties and problems of different distribution methods for the pavilion’s 60,000 spikes, seen both as individual elements and as a continuum with statistical and smoothly varying aspects. TopoStruct is one application that has been developed based on the theory of topology optimisation (by Martin Philip Bends0e and Ole Sigmund), a methodology that produces optimal geometric and material distributions in space with respect to structural behaviours. A few years ago the introduction of ever more powerful computer-aided design (CAD) systems seemed to have almost eradicated mathematics from architectural practice. describe how the Optimisation Design team at Adams Kara Taylor (AKT) work with mathematical algorithms to develop interactive software applicatons that help inform structural behaviour in the early parts of the design process. This is exemplified by the project-specific software they developed for Thomas Heatherwick’s British Pavilion at the 2010 Shanghai Expo. The latter, the interpretation of the material fields, will be part of the designer’s decision-making process. This ability to operate intuitively at the borderline of the discrete and the continuum is critical for projects that are geometrically hyperfragmented and work at multiple scales. But again, the fun stops as soon as the shapes get curvy. Different combinations of boundary conditions and forces give rise to a variety of structural forms and patterns, some familiar from living organisms or typical steel structures and others more unexpected. This is easily done for planar faces and regular grids: details can be defined once and then multiplied; local changes do not induce re-evaluations of the whole structure. Architects have suddenly shown an increasing interest in mathematics1 and its abstract concepts. Abstraction

A model, by definition, is always an abstraction of reality. Hence more than any specific algorithms used, the controls and observables that make up the digital design environment constitute the ingredients for an intuitive approach to the problematic of structure. While modelling starts with gathering data, it is far more important to then throw away everything that turns out to be superficial. This was the case with the development at AKT of project-specific software for several design problems. Only the latter allowed for the fabrication of smoothly curved girders, but its definition took some help from specialists who usually work for the automotive industry. What is obvious in the workshop of a model builder sometimes gets forgotten when almost infinite digital storage space is at hand: a perfect model does not contain as much information as possible, but as little as necessary to describe the properties of an object unambiguously. With the aim of achieving a closer and yet not superdetermined relationship between design and structure, the work seeks to describe the structural aspects of design as continuous material distributions in space. In this case materiality gradients traverse the design space, carrying material information and endowing space with some structural behaviour, although of a very exotic ‘could observe and discuss’ kind. On closer inspection, however, this is a consequent progression: it turned out that the complex shapes unleashed by digital design tools did not smoothly flow down the process chain from CAD to computer-aided engineering (CAE) to computer-aided manufacturing (CAM) until automatically materialising inside some digital fabrication machines, and that the implementation of a passably seamless digital workflow requires more than the flick of a button in the designer’s CAD software. Subsequently, any 2-D plan might transport parts of the topology (how certain features are related to each other) but no reliable metrics any more (how far those features are from each other). The process involves effective abstraction of the problem, development and implementation of mathematical algorithms in order to extract a set of pre-solutions; that is, semi-determinate results that operate as design hints in the early stages of the design process rather than definite outcomes. Under this cover, even highly sophisticated mathematical concepts like non-uniform rational b-splines (NURBS) managed to sneak into architecture, understood by very few but happily applied nevertheless. Any extra bit would be meaningless for the given purpose and only impede comprehensibility. A more abstract concept of structure and material is instead introduced here where some regions of space are solid and some empty, but the boundaries between them are diffused. Images: pp 66-7, 68(tr) © Hufton + Crow; p 68(tl) © AKT; p 68(b), 69 © Sawako Kaijima / Panagiotis Michalatos

Fabian Scheurer

Rather than eradicating the need for mathematics in architectural practice, computation has intensified it. Generation of structural patterns over plates and shells, driven by material and structural properties. Such a conceptual shift can be useful in the early stages of design as it gives clues about the behaviour of material distributions and the structural patterns that emerge within them. Consistent 2-D plans can then be derived from those 3-D models on request. Intuition is considered as insights gained by practice (play) and feedback (observation) which makes possible informed decisions in the context of a specific design problem. First of all, architectural design is a process of communication. At least when looking at the curricula of architecture schools one could have the impression that this subject – anyway unloved due to its ‘uncreative’ formal rigidity – was happily replaced by CAD courses and the belief that somewhere in the background the software would take care of all the calculations. So complex designs consequently have to be modelled in three dimensions before they are flattened to drawings; the model becomes the core of communication. Recently this seems to have changed again. Design is often seen as an articulation of solid elements within an empty space or a manipulation of clear-cut boundaries in the form of surfaces. As Fabian Scheurer and Hanno Stehling of designtoproduction explain, the uneven flow of a complex design from computer-aided design (CAD) to computer-aided engineering (CAE) and through to computer-aided manufacturing (CAM) necessitates an understanding of abstract mathematical concepts that facilitate communication, precision and an accurate assessment of quality throughout the process. Shigeru Ban, Centre Pompidou, Metz, France, 2010

The roof surface as triangle mesh (left) and as NURBS surface (right). Here, the question of what to design is transformed into what are the properties of the space itself in which we embed a design artefact. The application allows those with little prior knowledge of engineering to acquire some understanding of the physics of structure. Here, rather than the distribution of material density, the quest is the abstraction of a patterning exercise to one of controlling vector fields and hence directionalities in space. Traditionally and despite all digitalisation, this is still achieved mainly by 2-D drawings. The applied research of the Optimisation Design team at Adams Kara Taylor (AKT) focuses on the development of interactive software applications that induce intuition towards specific counter-intuitive design problems, often related but not limited to the understanding of structural behaviour. In fact, after interacting with the software for some time, users tend to anticipate the results as if they have gained this intuitive understanding of the underlying principle. It is a long way from the designer’s initial idea to the built result, necessitating means to describe a design in ways that give sufficient and unambiguous instructions to the builders.