  Although thin films are typically manufactured in planar sheets or rolls,
they are often forced into three-dimensional shapes, producing a plethora of
structures across multiple length-scales. Existing theoretical approaches have
made progress by separating the behaviors at different scales and limiting
their scope to one. Under large confinement, a geometric model has been
proposed to predict the gross shape of the sheet, which averages out the fine
features. However, the actual meaning of the gross shape, and how it constrains
the fine features, remains unclear. Here, we study a thin-membraned balloon as
a prototypical system that involves a doubly curved gross shape with large
amplitude undulations. By probing its profiles and cross sections, we discover
that the geometric model captures the mean behavior of the film. We then
propose a minimal model for the balloon cross sections, as independent elastic
filaments subjected to an effective pinning potential around the mean shape.
This approach allows us to combine the global and local features consistently.
Despite the simplicity of our model, it reproduces a broad range of phenomena
seen in the experiments, from how the morphology changes with pressure to the
detailed shape of the wrinkles and folds. Our results establish a new route to
understanding finite buckled structures over an enclosed surface, which could
aid the design of inflatable structures, or provide insight into biological
patterns.
