  Predicting the large-amplitude deformations of thin elastic sheets is
difficult due to the complications of self-contact, geometric nonlinearities,
and a multitude of low-lying energy states. We study a simple two-dimensional
setting where an annular polymer sheet floating on an air-water interface is
subjected to different tensions on the inner and outer rims. The sheet folds
and wrinkles into many distinct morphologies that break axisymmetry. These
states can be understood within a recent geometric approach for determining the
gross shape of extremely bendable yet inextensible sheets by extremizing an
appropriate area functional. Our analysis explains the remarkable feature that
the observed buckling transitions between wrinkled and folded shapes are
insensitive to the bending rigidity of the sheet.
