A Pattern Language by Christopher Alexander is renowned for providing simple, conveniently formatted, humanist solutions to complex design problems ranging in scale from urban planning through to interior design. This text is also believed to be the most widely read architectural treatise ever published. Despite this, there is also little acknowledgement in its popular reception that it is only one part of a trilogy of works documenting Alexander’s ‘second theory’ of architecture. Thus, while A Pattern Language is widely referenced in architectural scholarship, most of these references simply acknowledge its existence and fail to engage with its content. Furthermore, the literature that does critically engage with Alexander’s theory, challenging its ideas and assumptions, is often difficult to find, and the criticisms are diverse and complex. The intent of this paper is to facilitate a deeper understanding of these criticisms and the relationships between them. The 28 criticisms identified in past research are organised hierarchically in this paper into three tiers representing those associated with the: (i) conceptualisation, (ii) development and documentation and, (iii) implementation and outcomes of Alexander’s theory. The relationships between these criticisms are then mapped diagrammatically thereby forming the basis for thematic groupings within each hierarchical tier. This organisation reveals that only two criticisms relate to the concept of pattern languages in isolation, while the remainder arise, directly or indirectly, from Alexander’s idiosyncratic ontological and epistemological positions. The conclusion analyses the relationships between the criticisms to develop a holistic and understanding of where the problems in Alexander’s theory might lie.

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.

In a famous architectural discussion, Colin Rowe links the geometric properties of two sixteenth century villas by Andrea Palladio and two twentieth century villas by the architect Le Corbusier. Rowe observed that different structural systems produced heightened geometric complexity in cross sections through Palladio’s villas and in Le Corbusier’s plans. Rowe also described a particular type of geometric scaling in portions of the four villas which he partially explains as a type of mathematical ‘natural beauty’ akin to the golden ratio and Fibonacci sequence. In his writings, Rowe refers to several geometric properties that encapsulate complex, scaled structures, but he lacked a mathematical system to rigorously describe and test his ideas. The present paper utilises the box-counting method for calculating fractal dimensions to analyse 100 images, consisting of architectural plans, sections, and elevations of the four villas and two Fibonacci sequences, to test Rowe’s ideas. Ultimately, the results of this research do not support the majority of Rowe’s claims about geometric complexity in the villas of Palladio and Le Corbusier, but they do provide insights into Rowe’s discussion of geometric scaling and the properties of four famous houses.

Purpose: In the 1947 article, The Mathematics of the Ideal Villa, Colin Rowe famously compared the spatial and geometric properties of buildings by two architects: Palladio and Le Corbusier. Many of Rowe's observations in this article have since been extensively debated but not rigorously tested. This paper examines Rowe's proposition that Palladio's villa plans possess greater intelligibility and cellularity than Le Corbusier's villa plans.Design/methodology/approach: Two established computational techniques, axial line analysis and isovist analysis, are adopted in this paper to quantify and compare the properties of intelligibility and cellularity in the four villas that Rowe used to construct his argument: Malcontenta, Rotunda, Stein, and Savoye.Findings: While acknowledging methodological limitations, the results of this paper do not support Rowe's claims, but they do lead to a unique quantitative examination of spatial configurations and properties of four famous villa plans.Originality/value: This is the first paper to quantitatively examine Rowe's claims that Palladian villas possess greater intelligibility and cellularity than Le Corbusier's villas.