Upon increasing the temperature, a lyotropic mixture consisting of sodium dodecyl sulfate (SDS), butanol, toluene, and brine solution, with oil and water in equal volumes (midrange mixture), has been shown to order spontaneously into two kinds of cubic crystalline arrays exhibiting Fd3m and Pm3n symmetries. Cubic structures, as determined from Rietveld refinements of X-ray powder diffraction profiles, freeze-fracture transmission electron microscopy observations, and NMR self-diffusion measurements, are consistent with structures in which an aqueous film compartmentalizes the oil nanophase into polyhedral cells. These cellular networks can be described by space-filling combinations of pentagonal dodecahedra with tetrakaidecahedra in one case (Pm3n network) and with hexakaidecahedra in the other case (Fd3m network). There is a close analogy between both oil-in-water cellular structures and tetrahedrally close-packed (tcp) networks previously proposed as model foam packings. This midrange system provides a striking example of cellular fluid organizing spontaneously into three-dimensionally ordered polyhedral-foam-like structures.