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The Auxetic materials are structural systems with a negative Poisson’s ratio. Such materials show unexpected behavior when subjected to uni-axial compression or tension forces. For instance, they expand perpendicular to the direction of an applied compressive force. This behavior is the result of their internal structural geometry. These materials, with their unique behavior, have recently found many applications in the fields of sensors, medical devices, sport wears, and aerospace. Thus, there is a lot of relevant research in the artificial design of auxetic metamaterials and the prediction of their behavior [2]. Since the behavior of these materials heavily relies on the geometry of their internal structure, the geometry-based methods of structural design, known as graphic statics, are very well suited to derive their geometry or describe their behavior. Nevertheless, graphic statics has never been used in the design of such materials. For the first time, this paper provides an introduction to the use of graphic statics in the design and form-finding of auxetic metamaterials. The paper explains multiple equilibrium states of various auxetic structures using algebraic formulations of 2d/3d graphic statics [1, 3]. Moreover, it sheds light on the geometric behavior of auxetic materials by changing the force diagram of graphic statics. Therefore, it suggests a novel approach in predicting the changes in the geometry of the material under various loading conditions by controlling the force equilibrium geometrically. 

This paper presents procedures to generate truss topologies as an input form for polyhedral graphic statics and develops an algebraic formulation to construct their force diagrams. The study's ultimate goal is to extend the authors' previous research in 2D [1] to generate 3D strut-and-tie models and stress fields for reinforced concrete design. The recent algebraic formulation constructs reciprocal polyhedral diagrams of 3D graphic statics with either form or force as input [2]. However, the input is usually a set of polyhedrons or self-stressed networks [3]. Another implementation of polyhedral graphic statics [4] includes general truss topologies. But the starting geometry is usually the global force diagram, and based on its modification or subdivision, a form diagram is constructed. Therefore, currently, there exists no formulation to analyze a spatial truss using polyhedral graphic statics. This paper develops an algorithm to build upon the algebraic 3D graphic statics formulation and notation [2, 5] to construct the force diagram for input geometries that do not include all closed cells. The article also shows how the proper definition of the external spaces between the applied loads and reaction forces and the tetrahedral subdivision of the truss makes it possible to construct the reciprocal force diagram. This technique can be further explored to generate various truss topologies for a given volume and identify an optimized solution as the strut-and-tie model for reinforced concrete. Figure 1 illustrates an example of a spatial truss with two vertical applied loads and four vertical supports, the subdivision of the inner and outer space, the constructed force diagram, and the Minkowski sum of the dual diagrams (i.e., the geometrical summation of the form and scaled force diagram).