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A Cosheaf Theory of Reciprocal Figures: Planar and Higher Genus Graphic Statics

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arxiv 2311.12946 v1 pith:YUAY7YV5 submitted 2023-11-21 math.AT

Zoe Cooperband , Robert Ghrist , Jakob Hansen This is my paper · ORCID
classification math.AT
keywords reciprocalconstructionsalgebradiagramsfiguresgenushigherlifts
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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.

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Cited by 5 Pith papers

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