THE DENSITY OF 3-DIMENSIONAL NETS

Estimates for the density of periodic three-dimensional nets in Euclidean three-dimensional space (R3) are derived. The analysis assumes that the nets tile triply periodic hyperbolic surfaces that are free of self-intersections (embedded in R3). Upper and lower bounds of the net density as a function of the average ring size on the surfaces are given. These geometrical relations are compared with framework densities of a range of silicon-rich zeolites, silica clathrasils and dense four-connected silicates in order to separate the roles of geometry and chemistry in setting silicate densities. The data suggest that silica frameworks are constrained by an approximate requirement of constant area per framework vertex in addition to the impositions of Euclidean three-space and are thus hyperbolic two-dimensional (layer) structures.