High surface area, high porosity, nanometric polygonal silica foams with hierarchically connected and uniformly sized pore systems are reported here. We observe a remarkable increase in foam cell sizes from mesoscopic to macroscopic dimensions upon swelling the self-assembled template with oil. The resultant structures resemble classical macroscopic soap foams and display, among other features, Plateau borders and volume fractions approaching the dry limit of 100%. In well-developed foams of this kind, dimensionally isometric polyhedral cells are connected by relatively short, flat cylindrical mesopores through polyhedral faces and micropores through the walls. For one sample, with approximately 75 nm diameter primary foam cells, we infer three separate sets of cell-connecting mesopores puncturing tetragonal, pentagonal, and hexagonal faces of the component polyhedra. A multiple step model of foam formation is discussed where an organic silica precursor progressively hydrolyzes and condenses as a growing flexible shell from the core-corona interface of oil-swollen triblock copolymer micelles or microemulsion droplets, inducing a clouding phenomena in the otherwise stabilizing poly(ethylene oxide) chains, leading to aggregation, deformation, and jamming to high volume fractions.