  Split networks are a popular tool for the analysis and visualization of
complex evolutionary histories. Every collection of splits (bipartitions) of a
finite set can be represented by a split network. Here we characterize which
collection of splits can be represented using a planar split network. Our main
theorem links these collections of splits with oriented matroids and
arrangements of lines separating points in the plane. As a consequence of our
main theorem, we establish a particularly simple characterization of maximal
collections of these splits.
