  The tropical variety defined by linear equations with constant coefficients
is the Bergman fan of the corresponding matroid. Building on a self-contained
introduction to matroid polytopes, we present a geometric construction of the
Bergman fan, and we discuss its relationship with the simplicial complex of
nested sets in the lattice of flats. The Bergman complex is triangulated by the
nested set complex, and the two complexes coincide if and only if every
connected flat remains connected after contracting along any subflat. This
sharpens a result of Ardila-Klivans who showed that the Bergman complex is
triangulated by the order complex of the lattice of flats. The nested sets
specify the De Concini-Procesi compactification of the complement of a
hyperplane arrangement, while the Bergman fan specifies the tropical
compactification. These two compactifications are almost equal, and we
highlight the subtle differences.
