STRUCTURE-GENERATING FUNCTIONS, POLYSOMATIC SERIES AND NON-POLYSOMATIC SERIES

In minerals that are related by a building algorithm, bond topologies and chemical compositions may be incorporated into a general formula by writing the local topological details of each cation and anion, along with their stoichiometry, as a general expression called a structure-generating function. Consider T-O-T (N = 1; biopyriboles) and H-O H structures (N = 2; TS-block minerals, astrophyllite-group minerals, micas): For {T_2nΘ_m} to be a chain or ribbon, we may write m as 5n + N, where N is an integer. Within this unit, we may recognize three types of anion vertices: (1) bridging anions, Θ^br, bonded to two T cations; (2) apical anions, Θ^ap, that link to cations out of the plane of the bridging anions; and (3) linking anions, Θ^l, that link to non-T cations in the plane of the Θ^br anions. We may use the handshaking dilemma of graph theory to examine the interaction between T, H and O layers, and write a Structure-Generating Function, S_n, that gives the stoichiometry and aspects of the bond topology of the structures:

S_(N;n) = X_i[M_(3n+2N-3)C_2(n+N-2)(D_2(N-1) Φ^ap_2(N-1){T_2n Θ^br_(3n-N) Θ^ap_2nΘ^l_2N}₂)Φ^t_0-2(N-1)]

Thus N = 2, n = (2,7) gives the chemical composition and structure of veblenite:

KNa(H₂O)₃ [(Fe²⁺₅Fe³⁺₄Mn₆Ca□)(OH)₁₀(Nb₄O₄{Si₂O₇}₂{Si₈O₂₂}₂)O₂].

Many sheet-silicate minerals are based on 3-conected and mixed 2-, 3- and 4-connected plane nets that may be derived from 3-connected nets by topological (oikodoméic) operations. Associating up- and down-pointing tetrahedra with net vertices results in single-, double- and triple-layer silicate sheets. We write a formula-generating function that expresses stoichiometry in terms of tetrahedron connectivities, and a structure-generating function that associates the formula-generating function with specific two-dimensional nets. The net and the local topological details of each anion relate exactly to the stoichiometry, and in turn, the stoichiometry points to specific topological details of the sheet of tetrahedra. Thus insertion of 2-connected vertices on all edges of the 6³ net gives a [T₅O₁₄] net which corresponds to the silicate sheet in zeophyllite: Ca₁₃[Si₅O₁₄]₂F₈(OH)₂ (H₂O)₆.