Architected materials (metamaterials) achieve new properties from the way their structures are engineered, rather than their composition. They usually have only one fixed geometry, although a recent paper from Katia Bertoldi and colleagues describes a reconfigurable three-dimensional (3D) metamaterial. In the present paper, the same group reports a general design strategy that leads to an entire class of reconfigurable metamaterials. The authors begin with space-filling polyhedra, which they separate and generate connecting faces from the edges of one polyhedron to others in the direction normal to the surfaces (these are said to be prismatic). The researchers then create these structures and identify those that are reconfigurable. They develop a numerical algorithm to identify the parameters that allow the structures to be reconfigurable, and use it to predict mobility and deformation modes in all 28 polyhedral tilings of 3D space. The original unit cells of all 13 reconfigurable architectures contain prisms. The authors further increase the number of reconfigurable architectures by reducing the number of extrusions from the polyhedral surfaces, and find that 10% of the structures they studied could be made reconfigurable. Finally, they identify qualitatively different reconfigurations, including shear and uniform expansion, along one or two principal directions, and internal reconfigurations that do not alter the external shape.