The structure hierarchy hypothesis states that structures may be ordered hierarchically according to the polymerisation of coordination polyhedra of higher bond-valence. A hierarchical structural classification is developed for sheet-silicate minerals based on the connectedness of the two-dimensional polymerisations of (TO4) tetrahedra, where T = Si4+ plus As5+, Al3+, Fe3+, B3+, Be2+, Zn2+ and Mg2+. Two-dimensional nets and oikodoméic operations are used to generate the silicate (sensu lato) structural units of single-layer, double-layer and higher-layer sheet-silicate minerals, and the interstitial complexes (cation identity, coordination number and ligancy, and the types and amounts of interstitial (H2O) groups) are recorded. Key aspects of the silicate structural unit include: (1) the type of plane net on which the sheet (or parent sheet) is based; (2) the u (up) and d (down) directions of the constituent tetrahedra relative to the plane of the sheet; (3) the planar or folded nature of the sheet; (4) the layer multiplicity of the sheet (single, double or higher); and (5) the details of the oikodoméic operations for multiple-layer sheets. Simple 3-connected plane nets (such as 63, 4.82 and 4.6.12) have the stoichiometry (T2O5)n (Si:O = 1:2.5) and are the basis of most of the common rock-forming sheet-silicate minerals as well as many less-common species. Oikodoméic operations, e.g. insertion of 2- or 4-connected vertices into 3-connected plane nets, formation of double-layer sheet-structures by (topological) reflection or rotation operations, affect the connectedness of the resulting sheets and lead to both positive and negative deviations from Si:O = 1:2.5 stoichiometry. Following description of the structural units in all sheet-silicate minerals, the minerals are arranged into decreasing Si:O ratio from 3.0 to 2.0, an arrangement that reflects their increasing structural connectivity. Considering the silicate component of minerals, the range of composition of the sheet silicates completely overlaps the compositional ranges of framework silicates and most of the chain-ribbon-tube silicates.

When crustal abundance (A, measured in atomic parts per million) of a chemical element is plotted vs. number of mineral species in which that element is an essential constituent (S), a significantly positive correlation is obtained, but with considerable scatter. Repeated exclusion of outliers at the 90% confidence level and re-fitting leads, after the sixth iteration, to a steady state in which 40 of the 70 elements initially considered define a trend with log S = 1.828 + 0.255 log a (r = 0.96), significantly steeper than the original. Three other methods for reducing the effect of outliers independently reproduce this steeper trend. The 'diversity index' D of an element is defined as the ratio of observed mineral species to those predicted from this trend. D separates elements into three groups. More than half of the elements (40 of 70) have D = 0.5–2.0. Apart from these 'typical' elements, a group of 15 elements (Sc, Cr, Ga, Br, Rb, In, Cs, La, Nd, Sm, Gd, Yb, Hf, Re and Th) form few species of their own due to being dispersed as minor solid solution constituents, and a hitherto unrecognized group of 15 elements are essential components in unusually large numbers of minerals. The anomalously diverse group consists of H, S, Cu, As, Se, Pd, Ag, Sb, Te, Pt, Au, Hg, Pb, Bi and U, with Te and Bi by far the most mineralogically diverse elements (D = 22 and 19, respectively). Possible causes and inhibitors of diversity are discussed, with reference to atomic size, electronegativity and Pearson softness, and particularly outer electronic configurations that result in distinctive stereochemistry. The principal factors encouraging mineral diversity are: (1) Particular outer electronic configurations that lead to a preference for unique coordination geometries, enhancing an element's ability to form distinctive chemical compounds and decreasing its ability to participate in solid solutions. This is particularly noteworthy for elements possessing geometrically flexible 'lone-pair cations' with an s2 outer electronic configuration. (2) Siderophilic or chalcophilic geochemical behaviour and intermediate electronegativity, allowing elements to form minerals that are not oxycompounds or halides. (3) Access to a wide range of oxidation states. The most diverse elements can occur as anions, native elements and in more than one cationic valence state.

A structure hierarchy is developed for chain-, ribbon- and tube-silicate based on the connectedness of one-dimensional polymerisations of (TO4)n− tetrahedra, where T = Si4+ plus P5+, V5+, As5+, Al3+, Fe3+, B3+, Be2+, Zn2+ and Mg2+. Such polymerisations are described by a geometrical repeat unit (with ng tetrahedra) and a topological repeat unit (or graph) (with nt vertices). The connectivity of the tetrahedra (vertices) in the geometrical (topological) repeat units is denoted by the expression cTr (cVr) where c is the connectivity (degree) of the tetrahedron (vertex) and r is the number of tetrahedra (vertices) of connectivity (degree) c in the repeat unit. Thus cTr = 1Tr12Tr23Tr34Tr4 (cVr = 1Vr12Vr23Vr34Vr4) represents all possible connectivities (degrees) of tetrahedra (vertices) in the geometrical (topological) repeat units of such one-dimensional polymerisations. We may generate all possible cTr (cVr) expressions for chains (graphs) with tetrahedron (vertex) connectivities (degrees) c = 1 to 4 where r = 1 to n by sequentially increasing the values of c and r, and by ranking them accordingly. The silicate (sensu lato) units of chain-, ribbon- and tube-silicate minerals are identified and associated with the relevant cTr (cVr) symbols. Following description and association with the relevant cTr (cVr) symbols of the silicate units in all chain-, ribbon- and tube-silicate minerals, the minerals are arranged into decreasing O:T ratio from 3.0 to 2.5, an arrangement that reflects their increasing structural connectivity. Considering only the silicate component, the compositional range of the chain-, ribbon- and tube-silicate minerals strongly overlaps that of the sheet-silicate minerals. Of the chain-, ribbon- and tube-silicates and sheet silicates with the same O:T ratio, some have the same cVr symbols (vertex connectivities) but the tetrahedra link to each other in different ways and are topologically different. The abundance of chain-, ribbon- and tube-silicate minerals decreases as O:T decreases from 3.0 to 2.5 whereas the abundance of sheet-silicate minerals increases from O:T = 3.0 to 2.5 and decreases again to O:T = 2.0. Some of the chain-, ribbon- and tube-silicate minerals have more than one distinct silicate unit: (1) vinogradovite, revdite, lintisite (punkaruaivite) and charoite have mixed chains, ribbons and/or tubes; (2) veblenite, yuksporite, miserite and okenite have clusters or sheets in addition to chains, ribbons and tubes. It is apparent that some chain-ribbon-tube topologies are favoured over others as of the ~450 inosilicate minerals, ~375 correspond to only four topologically unique graphs, the other ~75 minerals correspond to ~46 topologically unique graphs.

Structural complexity of minerals is characterized using information contents of their crystal structures calculated according to the modified Shannon formula. The crystal structure is considered as a message consisting of atoms classified into equivalence classes according to their distribution over crystallographic orbits (Wyckoff sites). The proposed complexity measures combine both size- and symmetry-sensitive aspects of crystal structures. Information-based complexity parameters have been calculated for 3949 structure reports on minerals extracted from the Inorganic Crystal Structure Database. According to the total structural information content, IG, total, mineral structures can be classified into very simple (0–20 bits), simple (20–100 bits), intermediate (100–500 bits), complex (500–1000 bits), and very complex (> 1000 bits). The average information content for mineral structures is calculated as 228(6) bits per structure and 3.23(2) bits per atom. Twenty most complex mineral structures are (IG, total in bits): paulingite (6766.998), fantappieite (5948.330), sacrofanite (5317.353), mendeleevite-(Ce) (3398.878), bouazzerite (3035.201), megacyclite (2950.928), vandendriesscheite (2835.307), giuseppetite (2723.097), stilpnomelane (2483.819), stavelotite-(La) (2411.498), rogermitchellite (2320.653), parsettensite (2309.820), apjohnite (2305.361), antigorite (m = 17 polysome) (2250.397), tounkite (2187.799), tschoertnerite (2132.228), farneseite (2094.012), kircherite (2052.539), bannisterite (2031.017), and mutinaite (2025.067). The following complexity-generating mechanisms have been recognized: modularity, misfit relationships between structure elements, and presence of nanoscale units (clusters or tubules). Structural complexity should be distiguished from topological complexity. Structural complexity increases with decreasing temperature and increasing pressure, though at ultra-high pressures, the situation may be different. Quantitative complexity measures can be used to investigate evolution of information in the course of global and local geological processes involving formation and transformation of crystalline phases. The information-based complexity measures can also be used to estimate the 'ease of crystallization' from the viewpoint of simplexity principle proposed by J.R. Goldsmith (1953) for understanding of formation of simple and complex mineral phases under both natural and laboratory conditions. According to the proposed quantitative approach, the crystal structure can be viewed as a reservoir of information encoded in its complexity. Complex structures store more information than simple ones. As erasure of information is always associated with dissipation of energy, information stored in crystal structures of minerals must have an important influence upon natural processes. As every process can be viewed as a communication channel, the mineralogical history of our planet on any scale is a story of accumulation, storage, transmission and processing of structural information.

Relative to its extremely low abundance in the Earth's crust, tellurium is the most mineralogically diverse chemical element, with over 160 mineral species known that contain essential Te, many of them with unique crystal structures. We review the crystal structures of 703 tellurium oxysalts for which good refinements exist, including 55 that are known to occur as minerals. The dataset is restricted to compounds where oxygen is the only ligand that is strongly bound to Te, but most of the Periodic Table is represented in the compounds that are reviewed. The dataset contains 375 structures that contain only Te4+ cations and 302 with only Te6+, with 26 of the compounds containing Te in both valence states. Te6+ was almost exclusively in rather regular octahedral coordination by oxygen ligands, with only two instances each of 4- and 5-coordination. Conversely, the lone-pair cation Te4+ displayed irregular coordination, with a broad range of coordination numbers and bond distances. A threshold was applied for Te4+–O links of ∼2.45 Å or 0.3 valence units with some flexibility, as a criterion to define strongly bound Te–O polymers and larger structural units. Using this criterion, Te4+ cations display one-sided 3-, 4- or 5-coordination by oxygen (with rare examples of coordination numbers 2 and 6). For both valence states of Te, examples are known of TemOn complexes which are monomeric (m = 1; neso), noncyclic finite oligomers (soro), rings (cyclo), infinite chains (ino), layers (phyllo) and frameworks (tecto tellurates). There is a clear analogy to the polymerization classes that are known for silicate anions, but the behaviour of Te is much richer than that of Si for several reasons: (1) the existence of two cationic valence states for Te; (2) the occurrence of multiple coordination numbers; (3) the possibility of edge-sharing by TeOn polyhedra; (4) the possibility for oxygen ligands to be 3-coordinated by Te; and (5) the occurrence of TemOn polymers that are cationic, as well as neutral or anionic. While most compounds contain only one or two symmetrically distinct types of Te atom, Pauling's Fifth Rule is frequently violated, and stoichiometrically simple compounds such as CaTeO3 can have polymorphs with up to 18 distinct Te sites. There is a tendency for local symmetry features such as the threefold axis of a TeO6 octahedron or the acentric symmetry of a Te4+On polyhedron to be inherited by the host structure; the latter in particular can lead to useful physical properties such as nonlinear optical behaviour. We develop for the first time a hierarchical taxonomy of Te-oxysalt structures, based upon (1) valence state of Te; (2) polymerization state of TemOn complexes; (3) polymerization state of larger strongly-bound structural units that include non-Te cations. Structures are readily located and compared within this classification.

Two-dimensional nets may be used to generate the stoichiometry and structure of single-layer and double-layer sheet-silicate minerals. Many sheet-silicate minerals are based on the 3-connected plane nets 63, 4.82, (4.6.8)2(6.82)1and (52.8)1(5.82)1, and some more complicated nets, e.g. (5.6.7)4(5.72)1(62.7)1, (4.122)2(42.12)1, (52.8)1(5.62)1(5.6.8)2(62.8)1,have one or two representative structures. Many complicated sheet-silicate minerals are based on sheets of 2-, 3- and 4-connected tetrahedra that may be developed from 3- and 4-connected plane nets by a series of oikodoméic operations on 3- or 4-connected nets that change the topologyof the parent net. There are three classes of oikodoméic operations: (1) insertion of 2- and 3-connected vertices into 3- and 4-connected plane nets; (2,3) replication of single-layer sheets by topological mirror or two-fold-rotation operators, and condensation of the resulting twosingle-layer sheets to form double-layer sheets. The topological aspects of these sheet structures may be described by functions that express stoichiometry in terms of tetrahedron connectivities (formula-generating functions) and functions that associate these formula-generating functionswith specific two-dimensional nets. Using these functions, we may generate formulae and structural arrangements of single-layer and double-layer silicate structures with specific local and long-range topological features.