Interpenetration of homogeneous sphere packings and of two-periodic layers of spheres.

All systems of interpenetrating sphere packings that occur with highest symmetry in the cubic, hexagonal or tetragonal crystal family are tabulated. Homogeneous sphere packings belonging to 49 different types may be intertwined to systems of interpenetrating sphere packings belonging to 74 types. For all compatible lattice complexes, the coordinate and lattice parameters are given. The corresponding patterns of interpenetration are analysed. For the interpenetration of two, three, four, five and eight sphere packings, eleven, three, five, one and two different patterns, respectively, are distinguished. In addition, four types of interpenetrating layers of spheres were found. Each such sphere configuration splits into two or three subsets of parallel sphere layers with an angle of 90 degrees or of 120 degrees , respectively, between the directions of the normals of the layers. A single sphere layer corresponds either to a honeycomb net or to a net built up from quadrangles and octagons.