Owing to the fact that effective properties of low-density cellular solids heavily rely on their underlying architecture, a variety of explicit and implicit techniques exist for designing cellular geometries. However, most of these techniques fail to present a correlation among architecture, internal forces, and effective properties. This research introduces an alternative design strategy based on the static equilibrium of forces, equilibrium of polyhedral frames, and reciprocity of form and force. This novel approach reveals a geometric relationship among the truss system architecture, topological dual, and equilibrium of forces on the basis of 3D graphic statics. We adapt this technique to devise periodic strut-based cellular architectures under certain boundary conditions and manipulate them to construct shell-based (shellular) cells with a variety of mechanical properties. By treating the materialized unit cells as representative volume elements (RVE), multiscale homogenization is used to investigate their effective linear elastic properties. Validated by experimental tests on 3D printed cellular polyhedral materials, we show that by manipulating the RVE topology using the proposed methodology, alternative strut materialization schemes, and rational addition of bracing struts, cellular mechanical metamaterials can be systematically architected to demonstrate properties ranging from bending- to stretching-dominated, realize metafluidic behavior, or create novel hybrid shellulars.