Lionel March - Architecture and Mathematics Since 1960
This survey of mathematical applications in architectural design since 1960 traces personal experiences in practice and theory, and does not pretend in anyway to cover the whole field.
The introduction of computers has manifestly changed the relationship of mathematics to architecture over the past forty years.
A distinction is made between mathematics which tends towards abstract generalizations, and architecture which is concretely particular. Mathematical physics used in the service of predicting architectural performance in particular instances is set aside, although this is probably the single, most common usage. Instead, the role that mathematical thought plays in furthering generic knowledge of architectural form is given emphasis. In proposing the study of form, Karl Friedrich Froebel identified three categories: quantitative, qualitative and relational. These are used to organize this survey.
Quantitative studies measure the various material dimensions of form: length, area, volume. The quantitative relationships between building forms, heights, areas of sites, road widths, open areas are examined in early "land use and built form studies". The proposal to redevelop the national and government center in Whitehall, London, is cited. The same line of thought is extended to urban forms and questions of urban densities.
Qualitative studies of form emphasize spatial transformations, the modern group theory of symmetry, and topological relations. Distinct architectural forms may well be equivalent under symmetry, or topology. An example is given of an application of Polya's enumeration theorem.
Relational studies of form were said, by Froebel, to be "forms of life". Whatever their dimensions, or their symmetrical, or topological disposition, the forms now represent a potential design occasion. Advances in shape computation address such relations - especially, in the work of George Stiny and colleagues.