  We study the harmonic polytope, which arose in Ardila, Denham, and Huh's work
on the Lagrangian geometry of matroids. We describe its combinatorial
structure, showing that it is a $(2n-2)$-dimensional polytope with
$(n!)^2(1+\frac12+\cdots+\frac1n)$ vertices and $3^n-3$ facets. We also give a
formula for its volume: it is a weighted sum of the degrees of the projective
varieties of all the toric ideals of connected bipartite graphs with $n$ edges;
or equivalently, a weighted sum of the lattice point counts of all the
corresponding trimmed generalized permutahedra.
