An approach to structure in architectural and urban design. 1. Introduction and mathematical theory

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Abstract
This is the first of three papers to be published consecutively and which, taken together, provide an account of recent researches into the question of structure—how that concept can be defined and how it can be used in the context of social science in general and of urban design in particular. The broad aim of the work has been to develop a mathematically based language of structure, dependent in practice on computer programs, and to illustrate that language by various studies in the fields of architecture, urban design, and community studies. This mathematical language is essentially combinatorial in nature and is centred on the study of relations between finite sets. Any such relation gives rise to a simplicial complex, and this in turn may be represented either as a specific geometrical structure in a multidimensional space or as an algebraic subsystem in a certain exterior product space. In either event a method of representing an holistic view of an urban environment is achieved. By pursuing both the global and local properties of the various simplicial complexes, it is possible to introduce concepts of connectivity and of structural forces which strongly suggest the intuitive experience of such an environment. Superimposed on these ideas is that of an hierarchical system (which is defined in terms of mathematical cover sets), and this hierarchy effectively introduces a filtration into the structures of the relevant simplicial complexes. In paper I there is a discussion of the background to the question of structure, appealing chiefly to the work of Alexander, which is followed by the introduction of some of the basic mathematical concepts associated with the global analysis referred to above. These concepts include the ideas of ⁁-connectivity, structure vector Q, obstruction vector Q̂, as well as the idea of a pattern IT on a complex. These are all relevant at any one hierarchical level, N, and represent properties of a geometrical structure S(N) which is termed the static backcloth. This exists in a multidimensional space and, for example, in an urban environment it is the structure which must carry all the dynamics, all the ‘action’. This latter notion is compared with the situation in Newtonian and Einsteinian physical theories.

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