The geometrical characteristics of the critical state are studied, using twodimensional Discrete Element Method simulations. Various simulations have been performed in order to study the effect on the critical-state fabric tensor of interparticle friction and of the type of loading. The results for the fabric tensor, from simulations with different material properties and different loading conditions, collapse to a single curve, where high coordination number corresponds to low fabric anisotropy. This suggests that a limiting fabric state exists that has a geometrical origin. Since high confining pressure leads to high coordination number, this then means that the fabric anisotropy is low. As it is well-known that fabric anisotropy is a main factor contributing to shear strength of granular materials, this geometrical effect explains the (weak) decrease of shear strength with increasing pressure. The contact network determines loops of contacts. Based on simplified loop shapes, two theoretical relations are developed for their geometrical description. These two theories are based on orientational exclusion of contacts and constant-volume deformation of the loops, respectively. These theoretical results bracket loop fabric anisotropies that are obtained from the results of the Discrete Element Method simulations.