Four-connected three-dimensional nets generated from the 1,3-stellated cube: topological analysis and distance-least-squares refinement

The four-connected three-dimensional (three-dimensional) nets obtained by sharing the vertices, edges and faces of the 1,3-stellated cube (called the bru polyhedral unit) were enumerated by Alberti [Am. Mineral. (1979), 64, 1188–1198]. This enumeration has been re-examined and the topological properties of the simpler nets have been determined. Among the 45 selected three-dimensional nets, there are seven polyhedral subunits, eight three-connected two-dimensional (two-dimensional) nets and 16 types of one-dimensional (one-dimensional) subunit including chains, columns and tubes. Only four nets are represented by actual materials: brewsterite, heulandite, scapolite and stilbite. The only common subunit to these four nets is bru. Two nets, including scapolite, have tetragonal symmetry. Ten out of 45 3D nets were chosen for distance-least-squares refinement to obtain refined coordinates of T atoms and cell dimensions. This type of topological analysis provides useful information for the classification of framework structures. The concepts used here are yielding many additional three-dimensional nets from other polyhedral units.