  The edge of torn elastic sheets and growing leaves often form a hierarchical
buckling pattern. Within non-Euclidean plate theory this complex morphology can
be understood as low bending energy isometric immersions of hyperbolic
Riemannian metrics. With this motivation we study the isometric immersion
problem in strip and disk geometries. By finding explicit piecewise smooth
solutions of hyperbolic Monge-Ampere equations on a strip, we show there exist
periodic isometric immersions of hyperbolic surfaces in the small slope regime.
We extend these solutions to exact isometric immersions through resummation of
a formal asymptotic expansion. In the disc geometry we construct self-similar
fractal-like isometric immersions for disks with constant negative curvature.
The solutions in both the strip and disc geometry qualitatively resemble the
patterns observed experimentally and numerically in torn elastic sheets, leaves
and swelling hydrogels. For hyperbolic non-Euclidean sheets, complex wrinkling
patterns are thus possible within the class of finite bending energy isometric
immersions. Further, our results identify the key role of the degree of
differentiability (regularity) of the isometric immersion in determining the
global structure of a non-Euclidean elastic sheet in 3-space.
