The vertices of the lattice fall into 3 classes numbered 1, 2, and 3, given by the 3 different ways to fill space with hard hexagons. There are 3 local densities ρ 1, ρ 2, ρ 3, corresponding to the 3 classes of sites. When the activity is large the system approximates one of these 3 packings, so the local densities differ, but when the activity is below a critical point the three local densities are the same. The critical point separating the low-activity homogeneous phase from the high-activity ordered phase is z c = ( 11 + 5 5 ) / 2 = ϕ ϕ 5 = 11.09017.... with golden ratio φ. Above the critical point the local densities differ and in the phase where most hexagons are on sites of type 1 can be expanded as