  The art and science of folding intricate three-dimensional structures out of
paper has occupied artists, designers, engineers, and mathematicians for
decades, culminating in the design of deployable structures and mechanical
metamaterials. Here we investigate the axial compressibility of origami
cylinders, i.e., cylindrical structures folded from rectangular sheets of
paper. We prove, using geometric arguments, that a general fold pattern only
allows for a finite number of \emph{isometric} cylindrical embeddings.
Therefore, compressibility of such structures requires either stretching the
material or deforming the folds. Our result considerably restricts the space of
constructions that must be searched when designing new types of origami-based
rigid-foldable deployable structures and metamaterials.
