Abstract: We present a unifying framework to represent all topologically distinct shapes in 3D, from solids to surfaces and curves. This framework can be used to build a universal and modular system for the visualization, design, and construction of shapes, amenable to a broad range of science, engineering, architecture, and design applications. Our unifying framework uses 3-space immersions of higher-dimensional-manifolds, which facilitate our goal of topological robustness. We demonstrate that a specific type of orientable 2-manifold mesh, which we call a CMM-pattern coverable mesh, can be used to represent structures in higher-dimensional manifolds, which we call block meshes. Moreover, the framework includes a set of operations that can preserve CMM-pattern coverability. In this sense, CMM-pattern-coverable meshes provide an algebraization of shape processing that (1) supports a generalized mesh representation for blocks that may not necessarily be solids, and (2) requires a minimal set of operations that transform CMM-pattern-coverable meshes to CMM-pattern-coverable meshes.