A Visual Path into Differential Geometry for Architects: From Curves to Freeform Surfaces

Abstract

Today, architectural creativity is increasingly supported by algorithmic design and artificial intelligence tools, yet without a deep understanding of differential geometry, designers often remain passive software users. This paper presents a visual pedagogical approach to teaching differential geometry specifically tailored for architectural engineering students. The methodology balances mathematical precision with architectural applicability through visualization, avoiding self-serving abstraction. Built upon the Van Hiele model of geometric thinking and Cognitive Load Theory, the course utilizes GeoGebra 3D as its primary didactic tool. By focusing on parametric form-finding and form recognition instead of abstract proofs, students are guided toward higher levels of geometric rigor. The article details a specialized syllabus – ranging from surfaces of revolution to freeform geometries. Student projects demonstrate that mastering mathematical threshold concepts significantly increases creative freedom and enables the conscious manipulation of complex forms. These findings suggest that bridging the gap between mathematical theory and architectural practice is essential for educating critical-thinking designers in the digital age.