  A thin, elastic sheet floating on the surface of a liquid bath wrinkles when
poked at its centre. We study the onset of wrinkling as well as the evolution
of the pattern as indentation progresses far beyond the wrinkling threshold. We
use tension field theory to describe the macroscopic properties of the deformed
film and show that the system passes through a host of different regimes, even
while the deflections and strains remain small. We show that the effect of the
finite size of the sheet ultimately plays a key role in determining the
location of the wrinkle pattern, and obtain scaling relations that characterize
the number of wrinkles at threshold and its variation as the indentation
progresses. Some of our predictions are confirmed by recent experiments of Ripp
\emph{et al.} [arxiv: 1804.02421].
