The linear-zigzag transition is observed and studied in dielectric barrier discharge with rectangular frames for the first time by two photomultipliers, an intensified charge-coupled device, and a high-speed video camera. The unstable linear spot pattern transforms into a stable zigzag superlattice pattern with increasing voltage. The zigzag superlattice pattern is made up of dim spots at each corner, light spots between dim spots, and a zigzag line which is composed of moving spots and zigzag halos. All the spots in the linear spot pattern discharge simultaneously, and they have equal electric quantities, while the discharge sequence in the zigzag superlattice pattern is light spots, dim spots, halos, moving spots, and electric quantities of light spots are more than that of dim spots. The difference in the electric quantities leads to the formation of zigzag halos. In a word, the zigzag superlattice pattern results from unequal wall charge quantities of different sublattices and statistical self-organization of moving spots.