This study was inspired by a model of a triangulated cylindrical shell made by C. R. Calladine during an investigation of the mechanics of biological structures. The model consisted of identical triangular panels on a helical strip and had a small-displacement internal inextensional mechanism. It is shown that many triangulated cylinders broadly similar to Calladine’s model can be folded down to a compact stack of plates: only small strains, whose magnitude can be made arbitrarily small by the choice of suitable design parameters, are imposed during folding. A general geometric formulation of the problem is presented and then, assuming that the folding process is uniform, the folding properties of any triangulated cylinder of this generic type are discussed.