  A toric cube is a subset of the standard cube defined by binomial
inequalities. These basic semialgebraic sets are precisely the images of
standard cubes under monomial maps. We study toric cubes from the perspective
of topological combinatorics. Explicit decompositions as CW-complexes are
constructed. Their open cells are interiors of toric cubes and their boundaries
are subcomplexes. The motivating example of a toric cube is the edge-product
space in phylogenetics, and our work generalizes results known for that space.
