Nexus 2025. Relationship between architecture and mathematics. Conference book

Between the second half of the 19th century and the 1930s, the most active mathematics institutes in European universities and polytechnic schools produced physical models of higher-order algebraic surfaces (quadrics, cubics – rifled and non-rifled, quartics). These models were primarily intended for didactic purposes but also served research needs, enabling the visualization of complex surfaces in three-dimensional space. Through physical models, it is possible to appreciate and understand the remarkable geometric and topological properties of these surfaces. These insights impacted not only the teaching of pure mathematics but also applied fields, including Descriptive and Projective Geometry, Topology, Structural Engineering, and Optics as applied to human physiology. Over time, these surfaces have inspired designers and architects, contributing to the interplay between mathematics and architecture. Today, the physical models created in the 19th century are complemented by the capabilities of parametric digital representation, which allows for the exploration of the remarkable properties of complex surfaces and the reconstruction of their geometric genesis through reverse modelling. The goal of this study is to experiment with an innovative didactic approach to complex surfaces within the educational framework of architecture and engineering students, leveraging the potential of physical models, reverse modelling, and parametric modelling (including via Visual Programming Languages, VPL).