Syntactical and Mathematical Measures of Spatial Configuration

Syntactical and Mathematical Measures of Spatial Configuration

DOI: 10.4018/978-1-7998-1698-0.ch005

Abstract

This chapter presents a detailed explanation of the construction and analysis of a justified plan graph (JPG) of a building plan. It introduces the classic syntactical and mathematical measures derived from a JPG and discusses their interpretation in terms of the original architectural plan the results are derived from. Thereafter, an alternative weighted and directed JPG is introduced which uses four measures: centrality, degree centrality, centrality closeness, and betweenness. The mathematical measures introduced in this chapter are applied in Section 3 of this book to examine two syntactical and grammatical applications. Throughout the present chapter, three “Palladian” villas—Villa Saraceno, Villa Sepulveda, and Villa Poiana—are used as examples to explain and demonstrate the concepts.
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Introduction

As discussed in the previous chapter, Space Syntax techniques typically commence with the abstraction of the spatial properties of a plan (or section) into a set of nodes and edges (or links), which collectively form a graph. This abstraction process relies on a conceptual shift in our understanding of architecture. Instead of seeing architecture in terms of formal, dimensional or “geographic” properties, the abstraction that is at the heart of Space Syntax views architecture solely in terms of spatial, relational or “topological” properties. This shift occurs in the process of translating architecturally defined space (or convex space) and the connections between spaces, into the nodes and edges of a graph. Once converted into a graph, the shape-based, dimensional and geographic properties are stripped away from the plan, and all that remains is topology, which can be mathematically analysed. This chapter explains the syntactical and mathematical measures that can be derived from spatial configurations in architecture. More specifically, this chapter describes the justified plan graph (JPG) technique in detail, and its use to analyse or explore enclosed or defined programmatic spaces (nodes) with the connections between them (links).

Graph theory is conventionally regarded as originating in the seventeenth century paradox of the Bridges of Königsberg, a mathematical puzzle about seven bridges separating four landmasses and a Knight’s desire to cross each bridge only once while moving in a continuous sequence (Harary, 1960; Hopkins & Wilson, 2004). In mathematics, this problem became known as the “Euler characteristic’, as the Swiss mathematician Leonhard Euler used a graphical method (comprising nodes and connections) to prove that it was not possible to complete the Knight’s desired journey for the particular set of spatial conditions. While isolated examples of graph theory may be traced in nineteenth century mathematics, it was not until the 1960s and early 1970s that there was a growth in interest in graph theory and its capacity to explain a variety of geographic and spatial phenomena (Harary, 1960). Furthermore, by the 1970s, graph theorists had begun to apply simple mathematical calculations to their node and line (or vertex and edge) diagrams to calculate the relative depths of these structures (Seppänen & Moore, 1970; Taaffe & Gauthier, 1973). These same formulas provided the mathematical basis for Space Syntax research a decade later, and in the intervening years they were responsible for encouraging a range of mathematical applications in architecture. For example, Christopher Alexander (1964) developed a variation of graph theory to explain urban connectivity before combining graph theory and a rule-based grammar to define a pattern-based approach to design (Alexander, Ishikawa, & Silverstein, 1977). However, within a few years of this publication Alexander rejected the mathematics of graph theory, preferring instead to seek geometric or relational systems in graphs. It was Lionel March and Philip Steadman (1971) who collaboratively developed the early stages of a syntactical model of architectural form that drew on graph theory, before later separately extrapolating this idea in different ways (March, 1976; Steadman, 1983). Their work at this time was largely focused on architectural form, and despite their efforts, this application of graph theory was never developed in a productive way. It wasn’t until the 1980s that Hillier, Hanson, and their colleagues suggested to architectural researchers that graph theory was more useful in spatial analysis.

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