An algorithmic definition of the axial map

The fewest-line axial map, often simply referred to as the `axial map', is one of the primary tools of space syntax. Its natural language definition has allowed researchers to draw consistent maps that present a concise description of architectural space; it has been established that graph measures obtained from the map are useful for the analysis of pedestrian movement patterns and activities related to such movement: for example, the location of services or of crime. However, the definition has proved difficult to translate into formal language by mathematicians and algorithmic implementers alike. This has meant that space syntax has been criticised for a lack of rigour in the definition of one of its fundamental representations. Here we clarify the original definition of the fewest-line axial map and show that it can be implemented algorithmically. We show that the original definition leads to maps similar to those currently drawn by hand, and we demonstrate that the differences between the two may be accounted for in terms of the detail of the algorithm used. We propose that the analytical power of the axial map in empirical studies derives from the efficient representation of key properties of the spatial configuration that it captures.  The original hand-drawn axial map of Gassin, France, vectorised from figure 28 in Hillier and Hanson (1984, page 91), with detail of `stringy' (axial) and `beady' (convex) extensions of a point, after figure 27. who examined the relationship of whole buildings in urban plans. Although these methods, and later ones (including others introduced by Hillier and Hanson themselves) are similar in that they construct a graph of spatial components, it is the axial map that has captured the imagination of many architects, designers, and urban planners, and become the mainstay of space syntax tools. Graph measures obtained from axial maps have been used to analyse successfully the effect of configuration of space on pedestrian movement in urban areas (Hillier et al, 1993; Peponis et al, 1989), traffic flows (Penn et al, 1998), crime distribution (Hillier and Shu, 2000), and land values (Desyllas, 2000), amongst many others (see de Holanda, 1999; Hanson, 2003; Peponis et al, 2001; UCL, 1997). Extensive research into axial maps has also led to their considerable usage in architectural and planning practice in the United Kingdom (and also elsewhere), particularly related to pedestrianisation as, for example, in the recent remodelling of London's Trafalgar Square (Space Syntax Limited, 2003). The axial map was introduced after observation of real systems and experimentation with generative algorithms. Hillier and Hanson (1984) noted that urban space in particular seems to comprise two fundamental elementsö`stringiness' and `beadiness'ö such that the space of the systems tends to resemble beads on a string (see inset in figure 1). They write: `̀We can define `stringiness' as being to do with the extension of space in one dimension, whereas `beadiness' is to do with the extension of space in two dimensions'' (page 91). Hence, the epistemology of their methodology involves the investigation of how space is constructed in terms of configurations of interconnected beads and strings. To this end, they develop a more formal definition of the elements in which strings become `axial lines' and beads c̀onvex spaces'. The definition they give is one that is easily understood by human researchers, but which, it has transpired, is difficult to translate into a computational approach: `̀An axial map of the open space structure of the settlement will be the least set of [axial] lines which pass through each convex space and makes all axial links'' (Hillier and Hanson, 1984, pages 91 ^ 92). The term axial line is defined as the longest line (1) that can be drawn through an arbitrary point in the spatial configuration (see inset figure 1). Similarly, the term convex space is a `fully fat' convex polygon around a point (see inset figure 1). To `make all axial links' is to ensure that all axial lines are connected together if they possibly can be. However, it has recently been pointed out that, for a computational implementation, this apparently rigorous definition contains a problem (Batty and Rana, 2004; Ratti, 2004): the set of axial lines that fulfil these criteria cannot be precisely defined. Hillier and Hanson have, in fact, become victim to their own precision. As a researcher recently remarked online: `̀ do geographers spend so long defining road centre lines [as the space syntax community spends debating axial lines]?'' The answer is, of course not. The map of the open space is a cartographer's artefact, and includes only features that he or she considers important. The basis for a road-centre line is that it should simply follow the centre of the road. If one then asks questions such as `at exactly what point is a road considered a road and not a path?' we descend into an ultimately pointless debate. Batty and Rana (2004) sidestep this debate, and suggest that the definition of an axial line be broadened to include a range of differently specified sets of lines to be studied for their own interest. However, this approach (1) Sometimes the `longest visibility line' is referred to; however, the axial line as Hillier and Hanson define it is a purely geometrical entity. 426 A Turner, A Penn, B Hillier