World population is growing approximately linearly at about 80 million per year. Thus, as time goes by, there is necessarily less space per person and this is particularly acute in cities. Perhaps this is why the engineering community seems to be obsessed with folding things. We develop a mathematical approach to “rigid folding” based on the use of piecewise rigid isometric mappings that have a group structure. The ideas are inspired by the way atomistic structures form naturally. Their characteristic features in molecular science imply highly desirable features for macroscopic structures, particularly 4D structures that deform. We illustrate these by constructing some “objective origami” structures. A key mathematical problem is rigidity. Owing to the group property, the resulting structures can exhibit constructive/destructive interference with solutions of Maxwell’s equations and therefore provide interesting new models of photonic metamaterials.