Towards Homological Methods in Graphic Statics

TL;DR

This paper introduces homological algebraic topology methods to analyze graphic statics, providing new proofs and relationships that extend classical results to higher dimensions and deepen understanding of force and form diagrams.

Contribution

It reformulates graphic statics using cosheaves and homology, offers a new proof of Maxwell's Rule in arbitrary dimensions, and uncovers a novel link between mechanisms and force diagram obstructions.

Findings

Reformulation of statics via cosheaves and homology

New proof of Maxwell's Rule in any dimension

Discovery of a relationship between mechanisms and force diagram obstructions

Abstract

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.