  Curvature and mechanics are intimately connected for thin materials, and this
coupling between geometry and physical properties is readily seen in folded
structures from intestinal villi and pollen grains, to wrinkled membranes and
programmable metamaterials. While the well-known rules and mechanisms behind
folding a flat surface have been used to create deployable structures and shape
transformable materials, folding of curved shells is still not fundamentally
understood. Curved shells naturally deform by simultaneously bending and
stretching, and while this coupling gives them great stability for engineering
applications, it makes folding a surface of arbitrary curvature a non-trivial
task. Here we discuss the geometry of folding a creased shell, and demonstrate
theoretically the conditions under which it may fold continuously. When these
conditions are violated we show, using experiments and simulations, that shells
undergo rapid snapping motion to fold from one stable configuration to another.
Although material asymmetry is a proven mechanism for creating this bifurcation
of stability, for the case of a creased shell, the inherent geometry itself
serves as a barrier to folding. We discuss here how two fundamental geometric
concepts, creases and curvature, combine to allow rapid transitions from one
stable state to another. Independent of material system and length scale, the
design rule that we introduce here explains how to generate snapping
transitions in arbitrary surfaces, thus facilitating the creation of
programmable multi-stable materials with fast actuation capabilities.
