Figure 2: Non-Euclidean shells on a Euclidean substrate.

(a) A spherical shell laid atop a flat body of water may develop wrinkles as means to overcome the geometric incompatibility. We denote by D the typical size of the shell and by R its radius of curvature. (b) We choose coordinates {u, v}, which at large scales follow the Cartesian coordinates {x, y} of the projection, however, at sub-wrinkle scales are proportional to arclength coordinates on the sheet. The amplitude a(u, v) and wave vector (written as in terms of a phase field φ(u, v)) may significantly vary across the sheet, however, the wavelength has a universal preferred value λ₀. (c) Experiment: a hexagonal D=6 cm patch was cut out of a 30 μm-thick PDMS spherical shell, R=6 cm, before laid atop water. Shown is a top view schlieren image of the equilibrium configuration. (d) Simulation: a finite-element code with the physical parameters of the experiment (c). Shown is the vertical displacement from the plane. A single wrinkle (solid line) may bend, that is, change its in-plane direction, with wrinkle curvature encoded in the second derivatives of the phase field φ. In the shown example, the obtained pattern breaks into six domains, such as the lightened region on the right. Wrinkle bending focuses into domain walls (thicker solid segment of the wrinkle line). We denote by L the typical domain size. (e) The height squared profile of (d), at two different slices (horizontal and vertical), normalized by . The envelope of these curves is by definition Δ(u, v). The black curve, , plotted for R=6 cm (no fitted parameters), is anticipated for an axisymmetric pattern. (f) The full Δ(u, v) measured from (d). (g) Sketch of the phase field φ(u, v) for the pattern in (c,d). Level sets of φ indicate wrinkles.