Hypothetical binodal zeolitic structures (structures containing two kinds of tetrahedral sites) were systematically enumerated using tiling theory and characterized by computational chemistry methods. Each of the 109 refineable topologies based on "simple tilings" was converted into a silica polymorph and its energy minimized using the GULP program with the Sanders-Catlow silica potential. Optimized structural parameters, framework energies relative to alpha-quartz and volumes accessible to sorption have been calculated. Eleven of the 30 known binodal topologies listed in the Atlas of Zeolite Framework Types were found, leaving 98 topologies that were unknown previously. The chemical feasibility of each structure as a zeolite was evaluated by means of a feasibility factor derived from the correlation between lattice energy and framework density. Structures are divided into 15 families, based on common structural features. Many "feasible" structures contain only small pores. Several very open structures were also enumerated, although they contain three-membered rings which are thermodynamically disfavoured and not found in conventional zeolites. We believe that such topologies may be realizable as framework materials, but with different elemental compositions to those normally associated with zeolites.