The rigidity of a planar graph in the plane has a direct application in the design of a gridshell structure in 3D that is a lift of that graph. In a gridshell, the vertical loads at the vertices are resisted by the edge bars. For reasons of cost and efficient material usage, it is desirable for the gridshell faces be planar, the loads resisted by axial forces in the 3D edge bars and the graph to be quad-dominant (the graph is composed of mostly quadrilateral polygons with a limited number of triangular or other polygons). Maximizing the number of planar lifts (the space of internal self-stresses) will help to achieve these goals. This leads to the following questions:

For a given quad-dominant graph, what vertex positions in the plane achieve the maximum number of planar lifts under the conditions of non-degeneracy (no faces collapsed to lines, no lines collapsed to points…)?

How would one design a quad-dominate graph that maximizes the number of lifts for a given convex, polygonal boundary that is also held planar in the lift?

In addition to presenting the above questions, the talk will present the background of the issues from the point of view of rigidity and describe the desirable attributes of the graph and the resulting structural benefits.

Joint work with: Cam Millar, Arek Mazurek, Toby Mitchell and Allan McRobie