  We consider the elastic energy of a hanging drape -- a thin elastic sheet,
pulled down by the force of gravity, with fine-scale folding at the top that
achieves approximately uniform confinement. This example of energy-driven
pattern formation in a thin elastic sheet is of particular interest because the
length scale of folding varies with height. We focus on how the minimum elastic
energy depends on the physical parameters. As the sheet thickness vanishes, the
limiting energy is due to the gravitational force and is relatively easy to
understand. Our main accomplishment is to identify the "scaling law" of the
correction due to positive thickness. We do this by (i) proving an upper bound,
by considering the energies of several constructions and taking the best; (ii)
proving an ansatz-free lower bound, which agrees with the upper bound up to a
parameter-independent prefactor. The coarsening of folds in hanging drapes has
also been considered in the recent physics literature, using a self-similar
construction whose basic cell has been called a "wrinklon." Our results
complement and extend that work, by showing that self-similar coarsening
achieves the optimal scaling law in a certain parameter regime, and by showing
that other constructions (involving lateral spreading of the sheet) do better
in other regions of parameter space. Our analysis uses a geometrically linear
F\"{o}ppl-von K\'{a}rm\'{a}n model for the elastic energy, and is restricted to
the case when Poisson's ratio is zero.
