All three thermodynamic phases and their corresponding symmetries can be visualized using the structure factor : S ( q → → ) = 1 N ⟨ ⟨ ∑ ∑ i j e − − i q → → ( r → → i − − r → → j ) ⟩ ⟩ . The double sum runs over all positions of particle pairs i and j and the brackets denote an average about various configurations. The isotropic phase is characterized by concentric rings at q = 2 π π / a , if a = 1 / ρ ρ is the average particle distance calculated by the 2D particle density ρ ρ . The (closed packed) crystalline phase is characterized by six-fold symmetry based on the orientational order. Unlike in 3D, where the peaks are arbitrarily sharp ( δ δ -peaks), the 2D peaks have a finite width described with a Lorenz-curve. This is due to the fact, that the translational order is only quasi-long ranged as predicted by the Mermin-Wagner theorem. The hexatic phase is characterized by six segments, which reflect the quasi-long ranged orientational order. The structure factor of Figure 1 is calculated from the positions of a colloidal monolayer (crosses at high intensity are artefacts from the Fourier transformation due to the finite (rectangular) field of view of the ensemble).