  While the U.S. Supreme Court is commonly viewed as comprising a liberal bloc
and a conservative bloc, with a possible swing vote or median justice between
them, surprisingly many case decisions are not explained by this simple model.
We introduce a pair of spatial methods for conceptualizing many 5-to-4 voting
alignments that have occurred on the Court and which defy the usual
liberal/conservative dichotomy. These methods, utilizing higher order Voronoi
diagrams and halving lines (k-sets), are based on the geometry of the
two-dimensional ideal space locations obtained from applying multidimensional
scaling to voting data. We also introduce a two-dimensional metric method for
determining the crucial fifth vote in each 5-to-4 ruling and for determining
the median justice in any collection of terms within a natural court.
