Towards Homological Methods in Graphic Statics

• Cooperband, Zoe ^[1] ; Ghrist, Robert ^[1]
  1. [1] University of Pennsylvania

     University of Pennsylvania

     City of Philadelphia, Estados Unidos

     
• Localización: Journal of the International Association for Shell and Spatial Structures, ISSN-e 1996-9015, ISSN 1028-365X, Vol. 64, Nº. 4, 2023, págs. 266-277
• Idioma: inglés
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• Resumen

  • Recent developments in applied algebraic topology can simplify and extend results in graphic statics – the analysis of equilibrium forces, dual diagrams, and more. The techniques introduced here are inspired by recent developments in cellular cosheaves and their homology. While the general theory has a few technical prerequisites (including homology and exact sequences), an elementary introduction based on little more than linear algebra is possible. A few classical results, such as Maxwell's Rule and 2D graphic statics duality, are quickly derived from core ideas in algebraic topology. Contributions include: (1) a reformulation of statics and planar graphic statics in terms of cosheaves and their homology; (2) a new proof of Maxwell's Rule in arbitrary dimensions using Euler characteristic; and (3) derivation of a novel relationship between mechanisms of the form diagram and obstructions to the generation of force diagrams. This last contribution presages deeper results beyond planar graphic statics.