There is a confluence between GIS and space syntax, which is based on the theory of topology and well known in the research field of Computer-Aided Architectural Design (CAAD). Although many GIS software have been employed to develop analytical tools in accordance with the theory of space syntax, there is still a need to explore its mathematical foundation for better understanding and interpretation of the results produced by the programs. This research attempts to provide more profound mathematical definitions about some basic concepts of space syntax. Furthermore, we will delve into the mathematical definition to discover some embedded characteristics that have not yet been mentioned by space researchers. Some conjectures are also proposed and tested using two traditional Taiwanese settlements. The concept of saturation, suggested by this research, is a new possibility to identify the centrality, peripheral regions, private place, key gateways, and subregions in settlements. This study also shows that this quantitative approach can well supplement the qualitative approach used by most researches nowadays.
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