A Sense of Purpose Mathematics and Performance in Environmental Design

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Specialist Modelling Group, Foster + Partners. Close-up of the angular variation of the louvres, as seen from different viewpoints

Mathematics in design is most often associated with its visual manifestation in geometrical surfaces and elements. The finely tuned ambient qualities of a space, necessary for environmental performance, may not be so apparent, but can involve the application of many branches of mathematics. Martha Tsigkari, Adam Davis and Francis Aish of Foster + Partners’ Specialist Modelling Group bring this to the fore by describing how at Al Raha Beach development in Abu Dhabi and the City of Justice in Madrid environmental considerations were interpreted through analytical numerical data.

Discussions of mathematics in architecture often concern the perception of numeric and geometric relationships as embodied in patterns of structure and material. A similar dialogue within environmental design suggests maths is the means of logically elucidating our perception of what we unconsciously sense as optimum performance, rather than a driving force directly and visibly discernible to us through the form of things.

Wheere design follows environmental considerations, perception becomes a very small part of sensible interaction with architecture. Building occupants are seldom aware of the various complex factors affecting their comfort and what they might do to improve or otherwise manipulate it. Of all the ambient qualities which affect our occupation of architecture – temperature, air flow, lighting – only some rise to the level of conscious awareness. When these qualities are given priority in design, the performance of architecture is less perceptible because it is operating more directly on our senses; this is also the way mathematics is experienced in the context of the natural environment. Patterns and forms found in the natural world are understood more in terms of experience than in the logic or principles by which they were created.Beauty in nature is largely driven by mathematics that do not necessarily have a direct visual manifestation in the form itself, but rather in the way that form performs within its context.

The manipulation of the environment through design involves many branches of mathematics: the projective geometry of light transmission, the chaotic and probabilistic maths of weather patterns, and the statistical algorithms required to make analysis legible and obtain discrete building components from continuous distributions. Such an approach always favours pragmatism over mathematical ‘purity’. Yet the results thus produced have a profound effect on the experience of architecture precisely because of the fundamental sensory experiences concerned.

This indirect employment of mathematical analysis in design promotes a spatial experience in which any visually recognisable effects of an analytical process are subordinate to an unconscious perception of comfort. Where entasis in the classical orders employs geometry in a subtle way to make things appear as they should be, environmental design employs mathematics to produce a similar sense of fitness for purpose. Both of these applications rely on commonalities of perception. However, where the former is an analytic approach to visual aptness, the latter – because of contingencies of site, climate and culture – is necessarily differentiated and synthetic. While the visceral sense of  comfort produced through this approach will be similar between designs, the visual and physical manifestations may vary considerably. Situated along the continuum between pure and applied mathematics, this approach could be conceived as the application of analytical mathematical processes as an exact science, distorted to provide a responsive synthetic solution. The selection of significant weather data, for instance, may be a probabilistic function, while the embodiment of an environmental mediator based on this data may take the form of an optimisation algorithm, approximating a continuous distribution through discrete iterations.

The Al Raha Beach development in Abu Dhabi is a characteristic example of a building driven by environmental considerations which, interpreted through analytical numerical data, inform its shape. The interaction of the elements (sun and wind) with the parametric model began a form-finding exercise that sought to balance performance-driven optimisation with more intuitive aesthetic criteria. The building’s undulating louvre system is an example of this process and is designed to minimise solar gain in response to facade orientation, while maximising views out. An optimisation problem at its core, the exercise varied the slope angle of constantlength louvres around the facade to allow no more than a maximum amount of radiation to hit the building. The derived script sorted the resulting angles based on set theory and morphed the shading system accordingly.

Foster + Partners, City of Justice, Madrid, 2007 The facades of both court buildings bear shading devices optimised to balance solar performance with views to the exterior.

A study for the facades of the City of Justice in Madrid produced a dramatically different appearance to the Al Raha Beach development, yet was based on very similar underlying principles. As with that project, the facades of the court buildings minimise incident solar radiation on the facade, above a threshold value based on historical weather data. Instead of a single-dimensional optimisation, the City of Justice facades have a threedimensional solution space: edges of the pyramidal shading device were tested for possible degrees of aperture along three independent axes. The optimal shading device would offer the best ratio of shading performance to the area of opening, in order to promote views out from the judicial offices behind.

Despite the obvious differences of appearance between the two projects, each responds to the continuity of the sun’s path over the days and seasons as continuity of transition across neighbouring bays of the facade. The numerical relationships governing the cycle of the sun’s travel and the intensity of its rays underlie the sensible experience of the buildings.

Text © 2011 John Wiley & Sons Ltd. Images © Foster + Partners  Sample points around the facade evaluate the range of different sun directions and intensities throughout the year and use the results to drive the louvres’ slope angle. far left: Each orientation requires a different shading configuration, according to the optimisation algorithm used. The west-facing and east-facing shades are asymmetrical due to the intended hours of occupation and the differences between morning and evening solar radiation, as obtained from historical weather data.

Shading is optimised for each bay in a three-step process. First, sample points on the facade directly behind each shade are projected onto a virtual pyramidal shade across each month, day and hour as required by occupancy. Historical weather data is used, associating each point of solar incidence with a radiation value. Second, different apertures are tested along three axes. Each aperture’s fitness is determined by the total radiation blocked, factored against the visible area through the aperture. Finally, the aperture which provides the optimal ratio of radiation blocked to obscured area is selected as the louvre for that bay.

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