Using a combination of scanning tunneling microscopy (STM) in ultrahigh vacuum (UHV) and a systematic theoretical method based oil considering all possible hydrogen bond connections between molecules with subsequent density functional theory (DFT) calculations, we studied supramolecular assemblies of highly symmetrical rectangular PTCDI molecules oil the Au(111) surface. We show, using a systematic prediction procedure followed by ab initio density functional calculations, that just over 10 monolayer structures are possible assuming two molecules in the primitive cell, sonic of which Would appear indistinguishable in the STM images. By breaking down these Structures into distinct assemblies, we predict six possible phases. Two of these had been observed previously: a canted phase seen oil a number of surfaces including Au(111) and a brick wall phase seen so far only on the NaCl(001) surface. Using STM imaging of PTCDI molecules on the Au(111) surface in ultrahigh vacuum, we discovered a completely new "domino" phase, also predicted by Our theory, in which molecules attach to each other rather like dominoes, to form squares repeated periodically across the surface. The interaction of the molecules with the gold Surface Seems to influence the orientation of the phases but not necessarily their stability.