This chapter introduced algebraic formulations as an innovative approach to 2DGS and 3DGS. These formulations capture the essence of the reciprocal relationship, representing the dual diagram algebraically. While requiring deeper insight into reciprocity, this method surpasses traditional approaches in its potency. With the aid of modern computational tools, we can efficiently navigate the solution space of these equations. The resultant algebraic framework promises increased efficiency, flexibility, and precision, enhancing our understanding of the relationship between form and force. Such advancements prove invaluable not only for academic pursuits but also for practical applications.

The method of Algebraic graphic statics comes with certain limitations. Primarily, it’s suited for linear static issues and linear constraints. Handling more intricate problems, such as the previously mentioned constraints on face areas, might not be straightforward. Such challenges may require quadratic functions, increasing the computational effort and cost for solutions. In terms of the implementation, although the data structure allows self-intersecting faces and cells to exist through later manipulation, the initial construction of the data structure cannot accept self-intersecting surfaces, i.e., all cells in the starting primal diagram need to be convex; otherwise, the topology cannot be detected and defined properly.

We invite the interested reader to visit other related topics under graphic statics, including vector-based 3D graphic statics \cite{Cremona1890,EPFL-CONF-218827,D'Acunto_2019}, and approaches using projections of polyhedral systems to higher dimensions and back by \cite{Konstantatou} and \cite{Baranyai2021}. Furthermore, \cite{Lee2018} proposed a method called \textit{Disjointed Force Polyhedra} where the equilibrium of the system was computed by constructing a single convex polyhedron for each node using Extended Gaussian Image algorithm, and then areas of the adjacent faces have matching areas but are allowed to have different shapes \cite{Lee2018,Lee2018phd}. In \cite{mcrobie_2016_minkowski}, a methodology was introduced for the computation and visualization of the Maxwell (in two dimensions) and Rankine (in three dimensions) diagrams that are reciprocal to a truss subjected to load. The formulation of these reciprocals is grounded in the Minkowski summation of the polyhedral stress functions inherent to both the original and the reciprocal diagrams. This approach aligns seamlessly with 3D graphic statics that utilize polyhedral reciprocal diagrams.

This chapter introduces compression-only form finding in both 2D and 3D. The properties of primitive constrained funicular forms in 2D and 3D are explained. The step-by-step procedures for constrained form finding for a single applied load in 2D and 3D are provided. The form-finding process is then extended from a single applied load to a non-concurrent system of loads, for instance, a series of parallel applied loads. We show how the form and force diagrams can be combined using the Minkowski sum to check the correctness of funicular construction. The relationship between the applied loads and the number of non-concurrent loads in polyhedral systems is explained. A design technique by subdividing the applied loads is introduced that yields a variety of load networks and, thus, shells with various topologies. Besides, the effect of support locations in 3D is illustrated using multiple examples. We provide examples to show an approach to controlling the width of funicular forms in 3D and constraining a funicular form’s height. The topic of the three-hinged arch in 2D and 3D is introduced, and the construction of overlaying funicular geometries for asymmetric loading scenarios is then explained.

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.

In this chapter, we showed the broader application of Polyhedral Graphic statistics in other fields of science and briefly introduced research directions and topics that go beyond the polyhedral limitations of this method. Particularly, we show a research project in which graphical methods were used to analyze the structural pattern of a dragonfly wing. The result was then combined with machine learning methods to generate the structure of a wing of an airplane with enhanced out-of-plane performance. We also visited applications in the design of strut-and-tie structures for referenced concrete and its further application in designing multi-material structural components where the direction of the deposition of material is adjusted with respect to the internal force flow to maximize mechanical performance. The application of Polyhedral Graphic Statics was shown in the design of cellular solids and briefly discussed how particular subdividing of the force diagram can control the stress distribution in the system and the overall behavior of the structure from bending dominant to stretching dominant system. We also showed the application of the structures designed using Polyhedral Graphic Statics in self-healing structural components and 3D-printed structural systems with maximized surface area and minimized mass. Another important topic was the extension of the methods of Polyhedral Graphic Statics to non-polyhedral systems using disjointed force polyhedra. In the end, advanced topics related to completeness, being, and kinematics in Polyhedral Graphic Statics were discussed, which opened the door to many further research directions in this field.

This chapter introduced the geometric steps of construction to find the equilibrium for concurrent and non-concurrent systems of forces in both 2D and 3D. It showed that constructing the force polygon in 2D and the closed force polygon in 3D as a force diagram ensures that the sum of the forces in the system is zero. In addition, we learned that constructing a closed funicular polygon in 2D and a closed funicular polyhedron in 3D in the form diagram guarantees moment equilibrium in the system. We need both constructions to have a complete equilibrium. Various configurations of forces in 3D were introduced, and the process of finding the resultant by constructing the closed force polyhedron and funicular polyhedron was discussed. Ultimately, we showed how it is possible to design various hanging sculptures similar to those of Alexander Calder using geometric construction of forces and finding resultants in 2D. The design of such sculptures in 3D was pursued by subdividing the force diagram and locating equilibrating forces in the form diagram. With this introduction to equilibrium and funicular construction, we are ready to move to the next chapter, where the funicular construction for specific boundary conditions is discussed, and the principles of compression-only or tension-only structural form finding for architectural structures will be explained.

This chapter covers various topics on the design of funicular forms with an emphasis on particular features in the structural forms. It starts by introducing the topological and geometrical relationship between the form and force diagrams in 2D and 3D and describes nodal and global equilibrium in the context of both 2D and polyhedral graphic statics. It explains how to find closed polyhedral cells from a group of faces and the computational approach that could be used for this purpose. It argues the necessity of implementing computational tools to explore the realm of intricate funicular forms with pronounced design features. To give instances of how computational implementation can help in design, various design examples are introduced, and their related design topics are explained. Note that the design domain in this approach is the force diagram. Thus all these design features are explained with respect to techniques of design incorporated in the force diagram that gives the desired feature in the funicular form. In the end, the translation of the cellular to shellular systems is explained and discussed as an interesting design approach that results in a more resilient structural form by creating anticlastic curvature in three-dimensional space.

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.

Masoud Akbarzadeh is an Associate Professor of Architecture in Structures and Building Technologies and the Director of the Polyhedral Structures Laboratory. In 2020, he received the National Science Foundation CAREER Award for extending the methods of 3D/Polyhedral Graphic Statics for Education, Design, and Optimization of High-Performance Structures.

In this chapter, we started with a graphical analysis of a simple truss system to explain the properties of force diagrams and the geometry of the equilibrium of systems with both tensile and compressive members. We showed the procedures to analyze polyhedral truss systems and explained the conditions of overhangs and their effect on the force diagram. We continued by visiting systems with self-intersecting nodal force polygons in both 2D and 3D.

We also showed examples of structures with static degrees of indeterminacy and their reciprocal force diagram. The geometric degrees of freedom in the force diagram of a polyhedral truss show a minimum number of states of self-stress in a system. We then introduced design techniques such as subdividing the applied load network and the cut and extrude method that can manipulate and carve the force diagram of a simple truss and derive articulated funicular systems with combined tensile and compressive members in 2D and 3D.

We also introduced the topic of force optimization by containing the force diagram to particular geometries. Finally, we showed an approach to realizing a funicular network for real-world application by embedding the thrust in anticlastic periodic surfaces using Volumetric Modeling and Signed Distance Functions to enhance the performance of such systems under asymmetric loading.