The chapter expanded on the definition, meaning, and application of Geometric Degrees of Freedom (GDoF) of the form and force diagrams in Polyhedral Graphic Statics. The number of independent edges in the network is the GDoF of a network of closed polygons/polyhedrons. An independent edge is an edge whose length can be independently chosen in constructing a network. These edges then determine the lengths of the rest of the edges in the network.

The geometric degree of freedom has different applications in the form and force diagrams. The GDoF of the force diagram in 2D and 3D allows exploring the static degrees of indeterminacies in the funicular form; the GDoF of the global force polygon/polyhedron corresponds to the external degrees of static indeterminacies in the funicular form.

Controlling the GDoF of a force diagram allows for removing reaction forces or applied loads in the boundary condition of a funicular form. This control can redistribute the internal forces in the structure’s members, e.g., from compression to tension, without changing the geometry of the funicular form and lead to exciting design options.

Like the force diagram, the form diagram also has geometric degrees of freedom. Using geometric degrees of freedom to manipulate the form diagram does change the form’s configuration but does not change the force magnitude and the equilibrium of forces in the system. Meanwhile, it opens the door to various funicular solutions ranging from compression-only to compression-and-tension combined systems.

Identifying the exact number of GDoF for both form and force diagrams in 3D is not a trivial task. It requires an algebraic formulation of the problem.

Chapter One introduces the geometric principles of equilibrium for concurrent systems of forces both in 2D and Polyhedral Graphic Statics. It starts with a simple example of a tensile or a compression-only node and explains constructing a closed force diagram for the given geometry in 2D and 3D. The chapter describes the geometrical and topological properties of a single force polygon in 2D and a polyhedron in 3D and its relationship with the static equilibrium of forces in a single node as the form diagram. In addition, the static determinacy and indeterminacy of forces are explained by the number of independent edges in the force diagram, which is referred to as the Geometric Degrees of Freedom. Furthermore, the construction of an indeterminate force polyhedron for a general node with multiple members or applied loads is discussed by explaining the Extended Gaussian Image of a polyhedron. The final examples of this chapter show how the knowledge of this chapter can be applied in the conceptual design of simple bridges with both tension and compression members.