The concept of natural tiling is applied to find primitive cages in all 194 types of the zeolite frameworks currently collected in the Zeolite Framework Database. Natural tile is considered as a new type of the zeolite building unit (natural building unit, NBU) that has a number of advantages over other polyhedral building units. (i) The choice of the NBU is unambiguous and can easily be done for the framework of any complexity by computer methods. (ii) NBUs fill the crystal space; the framework is completely composed by the NBUs. (iii) NBUs enumerate all minimal cavities in the framework; all larger cages can be obtained by gluing NBUs. (iv) The NBU faces enumerate all windows in the framework. It is shown that the natural tiling approach provides a universal classification scheme that can be used for all zeolite-type structures to find resemblances in their organization. The natural tiles and tilings as well as their topological properties are catalogued, and a number of resemblance schemes are proposed between NBUs and other types of zeolite building units. Two kinds of close similarity between zeolite frameworks are revealed, signature-equal zeolites that have the same tiling signature and tile-equal zeolites built of the same types of NBUs.

In the attempt to explain why there are so many hypothetical zeolites and so few observed, a model of assembling zeolite-type frameworks as a packing of natural building units (smallest cages) and/or essential rings (smallest windows) is proposed. The packing units have no common T atoms, hence the model takes into account the process of polycondensation of T 4+(OH)4 or [T 3+(OH)4]− complex groups resulting in oligomeric T n O m (OH) k units, eliminating water molecules, and forming T–O–T bridges. The packings were modeled for all zeolite minerals and most of the synthetic zeolite-type frameworks accounting for 163 zeolites of the 201 known. It is shown that the extra-framework cations can play a role of templates for the packing units. Application of the model to 1220 hypothetical zeolites shows that only a small set could be explained as packing of tiles, suggesting a possible ranking of feasibility that may help to unravel the zeolite conundrum.

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In terms of the Voronoi-Dirichlet partition of the crystal space, definitions are given for such concepts as ;void', ;channel' and ;migration path' for inorganic structures with three-dimensional networks of chemical bonds. A number of criteria are proposed for selecting significant voids and migration channels for alkali cations Li+-Cs+ based on the average characteristics of the Voronoi-Dirichlet polyhedra for alkali metals in oxygen-containing compounds. A general algorithm to analyze the voids in crystal structures has been developed and implemented in the computer package TOPOS. This approach was used to predict the positions of Li+ and Na+ cations and to analyze their possible migration paths in the solid superionic materials Li3M2P3O12 (M=Sc, Fe; LIPHOS) and Na1+xZr2SixP3-xO12 (NASICON), whose framework structures consist of connected M octahedra and T tetrahedra. Using this approach we determine the most probable places for charge carriers (coordinates of alkali cations) and the dimensionality of their conducting sublattice with high accuracy. The theoretically calculated coordinates of the alkali cations in MT frameworks are found to correlate to within 0.33 A with experimental data for various phases of NASICON and LIPHOS. The proposed method of computer analysis is universal and suitable for investigating fast-ion conductors with other conducting components.