  In this paper we apply new geometric and combinatorial methods to the study
of phylogenetic mixtures. The focus of the geometric approach is to describe
the geometry of phylogenetic mixture distributions for the two state random
cluster model, which is a generalization of the two state symmetric (CFN)
model. In particular, we show that the set of mixture distributions forms a
convex polytope and we calculate its dimension; corollaries include a simple
criterion for when a mixture of branch lengths on the star tree can mimic the
site pattern frequency vector of a resolved quartet tree. Furthermore, by
computing volumes of polytopes we can clarify how ``common'' non-identifiable
mixtures are under the CFN model. We also present a new combinatorial result
which extends any identifiability result for a specific pair of trees of size
six to arbitrary pairs of trees. Next we present a positive result showing
identifiability of rates-across-sites models. Finally, we answer a question
raised in a previous paper concerning ``mixed branch repulsion'' on trees
larger than quartet trees under the CFN model.
