2023/11/21 by Zoe Cooperband, Robert Ghrist, Cooperband, Zoe +3 · 3 citations
Computer Science · Engineering · #Advanced Materials and Mechanics #Algebraic Topology (math.AT) #Computational Geometry and Mesh Generation #Dynamics and Control of Mechanical Systems #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2311.12946
openalex publication_date 2023/11/21 · openalex created_date 2023/11/24 · openalex updated_date 2026/07/28
This paper introduces cellular sheaf theory to graphical methods and reciprocal constructions in structural engineering. The elementary mechanics and statics of trusses are derived from the linear algebra of sheaves and cosheaves. Further, the homological algebra of these mathematical constructions cleanly and concisely describes the formation of 2D reciprocal diagrams and 3D polyhedral lifts. Additional relationships between geometric quantities of these dual diagrams are developed, including systems of impossible edge rotations. These constructions generalize to non-planar graphs. When a truss embedded in a torus or higher genus surface has a sufficient degree of axial self stress, we show non-trivial reciprocal figures and non-simply connected polyhedral lifts are guaranteed to exist.