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.

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.

The structure hierarchy hypothesis states that structures may be ordered hierarchically according to the polymerization of coordination polyhedra of higher bond valence. A mathematical hierarchy is an ordered set of elements where the ordering reflects a natural hierarchical relation between (or arrangement of) the elements. Here, I review the structure hierarchies for the borate, uranyl oxide, phosphate, sulfate, beryllate and oxide-centred Cu, Pb and Hg minerals (plus synthetics where appropriate). Structure hierarchies have two functions: (1) they serve to organize our knowledge of minerals (crystal structures) in a coherent manner; (2) if the basis of the classification involves factors that are related to the mechanistic details of the stability and behaviour of minerals, then the physical, chemical and paragenetic characteristics of minerals should arise as natural consequences of their crystal structures and the interaction of those structures with the environment in which they occur. We may justify the structure hierarchy hypothesis by considering a hypothetical structure-building process whereby higher bond-valence polyhedra polymerize to form the structural unit. The clusters constituting the FBBs (fundamental building blocks) may polymerize to form the following types of structural unit: (1) isolated polyhedra; (2) clusters; (3) chains and ribbons; (4) sheets; and (5) frameworks. The major advantage of this approach to structure hierarchy is the fact that the hypothetical structure-building process outlined above resembles (our ideas of) crystallization from an aqueous solution, whereby complexes in aqueous and hydrothermal solutions condense to form crystal structures, or fragments of linked polyhedra in a magma condense to form a crystal. Although our knowledge of these processes is rather vague from a mechanistic perspective, the foundations of the structure hypothesis give us a framework within which to think about the processes of crystallization and dissolution.