Beyond two dimensions: Architecture through three-dimensional visibility graph analysis

Architecture consists of spatial relations that accommodate functions, afford social relations and create visual interest. Through openings and walls, architects manipulate continuities and discontinuities of visual fields in two and three dimensions. Analytical diagrams and models of these fields have been offered by space syntax, especially through visibility graph analysis (VGA), graphing visual relations in two dimensions. This paper introduces a new approach to VGA that departs from planar restrictions. We show how a graph can be generated of inter-visible locations on a planar surface that incorporates relations among elements in three dimensions. Using this method, we extend the current space syntax analysis of architectural space to a new methodology for diagramming and modelling three-dimensional visual relationships in architecture. The paper is structured in three parts. The first section provides an overview of the principles of visibility analysis using graphs, and explains the method by which visibility relations of ‘accessible’ and ‘inaccessible’ space in two and three dimensions are computed. This leads to a graph representation, which uses a mix of ‘directed’ and ‘undirected’ visibility connections, and a new multi-variant spatial categorisation analysis that informs the properties of multi-directional graphs. The second part of the paper tests the three-dimensional visibility model through the analysis of hypothetical and real spatial environments. The third part analyses Giuseppe Terragni’s Casa del Fascio, describing architectural characteristics that are not captured by two-dimensional analysis, and allowing a comparative understanding of spatial configuration in two and three dimensions. The paper concludes with a discussion about the significance of this new model as an analytical and architectural tool.

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