THE FIFTH FOLD: COMPLEX SYMMETRIES IN KRESLING-ORIGAMI PATTERNS

The present study focuses on particular symmetry properties of a twist buckling pattern in thin-walled cylinders, named 'Kresling-pattern' (Stewart, 1999; Kresling, 2008 and 2012). The pattern is formed by wrapping a sheet of paper around two mandrels, coaxial with a gap between them. On twisting the mandrels in opposite directions about their common axis, a regular pattern of folds is generated and the mandrels are drawn closer together. The cylindrical wall collapses to a diaphragm. There is neither stretching nor warping, only twist by folding. The pattern is able to unfold. Five folds meet at vertices, the fifth fold lays on a generatrix of the initial cylinder and stabilizes the polygonal borders of the diaphragm. Parallel lines of transverse shear, of identical length, help to define the geometries of: - foldable cylinders and polygonal prisms; - nearly stress free foldable bellows; - a cylindrical Miura-ori; - the foldable 'skeleton' of hyperboloids of revolution; - their portions, individually foldable saddle-shaped tesselations. All five solutions had been held impossible to be realized from developable surfaces, by means of straight folds. This presentation shows the contrary.