Multi-stable design of triangulated origami structures on cones of revolution

Abstract: It is well-known that the Kresling pattern of congruent triangles can be arranged either circularly on a cylinder of revolution or in a helical way. In both cases the resulting cylindrical structures are multi-stable. We generalize these arrangements with respect to cones of revolution, where our approach allows to construct structures, which snap between conical realizations whose apex angles serve as design parameters. In this context we also figure out shaky realizations, intervals for self-intersection free realizations and an interesting property related to the cross sectional area. Finally, we analyze these origami structures with respect to their capability to snap by means of the so-called snappability index.