The Homological Frame

Graphic statics — representing structural equilibrium through geometric diagrams — works beautifully for trusses: forces correspond to polygonal faces, equilibrium to closed polyhedra. But rigid-jointed frames carry bending moments and torsion alongside axial forces, and the spaces between bars need not be flat polygons. The classical method breaks down.

CW-complexes from algebraic topology fix it (arXiv:2603.12093). Replace flat polygons with general closed loops — circuits of bars forming space curves — and use cellular homology to track the algebraic relationships between bars, loops, and enclosed regions. The decomposition into loops is the homology basis; the equilibrium conditions are the boundary maps.

The structural insight: the limitation of classical graphic statics was not physical but topological. The method assumed the structure lived in a cell complex of flat faces. Real 3D frames don’t. Replacing geometric polygons with topological CW-cells — which can have any shape — extends the method to arbitrary frame geometries while naturally incorporating shear, bending, and torsion. The moments that were invisible to graphic statics become visible once the topology is right. The tool knew the physics; it just couldn’t see the geometry until the geometry was generalized.