Reconstructing Palladian Symmetry: Quotient Trees and House-and-Wing Morphospace

Abstract Palladio’s villas have long been interpreted through proportion, harmonic ratios, typology, and generative rules, yet wings and barchesse remain less developed as computational objects than the compact house plans. This article presents an inside-out graph method for constructing and comparing symmetry in sixteen villa and country-house plans traced from I quattro libri dell’architettura . Unlike justified access graphs rooted in space syntax studies, the method begins at the central sala or an identified principal symmetry root and reconstructs formal relations toward rooms, porticoes, wings, and estate-scale dependencies using quotient trees. Two measures are defined for the resulting morphospace analysis: symmetry propagation depth, which records the mean depth of quotient tree leaves, and Symmetry Reticulation Index, which records surplus relations in the collapsed quotient graph. The results separate outward extension from quotient-graph reticulation. Barbaro propagates deeply while remaining tree-like, Ragona is comparatively compact but highly reticulated, and Thiene combines maximum propagation with substantial surplus structure. The framework therefore identifies Palladian symmetry as an inside-out relational morphology whose extension and reticulation can be measured independently.