Fig. 1.

Folding of curved shells along a crease. (A) Creasing a shell involves thinning the shell locally along a curve that lies on the surface to form a “trench.” A local coordinate system $\left\{s,v\right\}$ on the crease is indicated. (B) Natural and folded states of a creased shell, denoted as the “+“ (unfolded) and ”−” (folded) conformations. Tangent and normal vectors to each surface “$\pm$” are given by ${\widehat{\mathbf{u}}}^{\pm }$ and ${\widehat{\mathbf{n}}}^{\pm }$, respectively. ${\widehat{\mathbf{N}}}_{\mathbf{F}}$ is the normal to the curve, whereas $\widehat{\mathbf{t}}$ indicates the tangent to both the crease and the surface. The angle ψ between ${\widehat{\mathbf{N}}}_{\mathbf{F}}$ and $\widehat{\mathbf{u}}$ is also indicated. (C) Schematics (for +) and creased experimental samples (for −) for all three prototypical geometries: helicoid ($K<0$), cylinder ($K=0$), and spherical shell ($K>0$). (Insets) Three-dimensional printed molds with embossed ridge to realize creases on ($K\ne 0$) geometries, and a scored sheet for the ($K=0$) case. Examples of curves with ${\kappa }_N=0$ are marked on schematics in white lines, and creases are marked on experimental samples with dotted yellow curves.