Mathematical beauty and Palladian architecture: Measuring and comparing visual complexity and diversity

Palladio's design principles, including proportion and harmony, have often been associated with mathematical definitions of beauty. However, the geometric and semantic properties of his façades have rarely been analysed in a holistic manner. There is little evidence of how mathematical beauty may be embodied in his architecture. This research investigates complexity (fractal dimension or D) and diversity (perplexity or PP) as aesthetic indices, aiming to capture distinct characteristics of Palladian architecture. The D and PP values of 22 Palladian villa façades are measured and analysed, before being compared with those of three Renaissance facades by Sebastiano Serlio. The combination of D and PP captures the geometric and semantic aesthetic qualities of architectural compositions. Importantly, the developed scatter plot of D and PP results supports the identification of four distinct aesthetic types of Palladian façade designs. The novel combination of D and PP measures contributes to a better understanding of one definition of the mathematical beauty of architecture, wherein “the whole is other than the sum of the parts” in compositional terms (a famous Gestalt principle). Specifically, this research provides new mathematical insights into the visual character of Palladian architecture and compares two important measures thereof.