There are 339 combinatorial types of generic metrics on six points. They correspond to the 339 regular triangulations of the second hypersimplex ∆(6, 2), which also has 14 non-regular triangulations. 1. The Metric Fan We consider the cone of all metrics on the finite set {1, 2, . . . , n}: Cn = { d ∈ R( n 2 ) : dij ≥ 0 and dij + djk ≥ dik for all 1 ≤ i, j, k ≤ n } . This is a closed convex poin...