Mathematical structure of Alexander’s A Pattern Language: Analysis of the role of invariant patterns/

In 1977 Christopher Alexander and his colleagues from the Centre for Environmental Structure published A Pattern Language, an innovative design guide aimed at restoring life and beauty to the built environment. Since then, A Pattern Language has become one of the most widely read architectural treatises ever published. However, despite its popularity, the structure of A Pattern Language remains poorly understood. In response to this situation, this paper uses graph theory to examine Alexander’s language including the entire set of 253 patterns and over 1800 relationships between them. Through this mathematical analysis the paper tests two hypotheses about the ‘invariant patterns’ Alexander was most confident in. The first hypothesis tests whether invariant patterns occupy more prominent positions within the language, and the second tests whether invariant patterns form a core structure within the language that supports less developed patterns. Through this process the paper illuminates several previously unconsidered aspects of the structure of A Pattern Language while providing the first graphic representation of the entire underlying structure that unites the individual patterns into a coherent language.