Coarse-grained protein structures have the unusual property of manifesting a greater regularity than is evident when all atoms are considered. Here, we follow proteins at the level of one point per residue. We confirm that lattices with large coordination numbers provide better fits to protein structures. But, underlying these protein structures, there is an intrinsic geometry that closely resembles the face-centered-cubic (fcc) lattice, in so far as the coordination angles observed in clusters of near neighboring residues are concerned. While the fcc lattice has 12 neighbors, the coordination number about any given residue in a protein is usually smaller; however, the neighbors are not distributed in a uniform, less dense way, but rather in a clustered dense way, occupying positions closely approximating those of a distorted fcc packing. This packing geometry is a direct manifestation of the hydrophobic effect. Surprisingly, specific residues are clustered with similar angular geometry, whether on the interior or on the exterior of a protein.