In The Mathematics of the Ideal VillaMathematics of the Ideal Villa (The), Colin Rowe identifies two distinct formal properties in Palladio’s and Le Corbusier’s designs. Rowe describes the first of these properties, ‘natural beauty’, as arising from a symbiotic geometric system wherein distinct shapes or proportions are repeated at multiple scales. The second property, called ‘expressive freedom’ in this chapter, refers to the ways particular construction systems—stone and masonry for Palladio and concrete and steel for Le Corbusier—have different innate capacities for formal and spatial modelling. Rowe argues that the technologies available to Palladio and Le Corbusier, coupled with their interests in particular mathematical ideas, led them to create distinct patterns of formal richness and complexity in their designs. Rowe connects these same patterns to beliefs about timeless beauty, which is why this chapter explores both natural beauty and expressive freedom. To investigate Rowe’s arguments, this chapter measures the formal complexity of six works: Palladio’s villas Malcontenta and Rotonda, Le Corbusier’s villas Stein and Savoye and Eisenman’s houses I and VI. The first four are the designs Rowe used to construct his arguments, and the last two reflect Rowe’s later critical response to neo-classical and modernist architecture. In addition to examining these works, the chapter discusses three further designs to assist in interpreting the results: Palladio’s Villa EmoVilla Emo, Le Corbusier’s Villa PlaneixVilla Planeix and Eisenman’s House IVHouse IV. In this chapter, Rowe’s arguments about natural beauty and expressive freedom are reframed as eight hypotheses, which are tested using fractal dimensionFractal dimension analysis of plans, sections and elevations of the six villas. Importantly, the goal is neither to prove nor disprove Rowe’s theories. It is to facilitate a greater understanding of the properties of Palladio’s, Le Corbusier’s and Eisenman’s architecture.