The orientational geometry of residue packing in proteins was studied in the past by superimposing clusters of neighboring residues with several simple lattices.1,2 In this work, instead of a lattice we use the regular polyhedron, the icosahedron, as the model to describe the orientational distribution of contacts in clusters derived from a high-resolution protein dataset (522 protein structures with high resolution < 1.5Å). We find that the order parameter (orientation function) measuring the angular overlap of directions in coordination clusters with directions of the icosahedron is 0.91, which is a significant improvement in comparison with the value 0.82 for the order parameter with the face-centered cubic (fcc) lattice. Close packing tendencies and patterns of residue packing in proteins is considered in detail and a theoretical description of these packing regularities is proposed.