Three experiments examined parameters affecting human observers' ability to detect the global three-dimensional (3D) organization of a random-dot display corresponding to the polar projection of a rotating sphere. Results indicate that the detection of 3D structure is critically dependent on the detectability of motion, is disrupted by increased redundant information specifying the two-dimensional location of points in the display, and undergoes a rapidly increasing resistance to the disruptive effects of noise with increasing numbers of frames. These results, in conjunction with earlier findings, are inconsistent with existing theories concerning the perception of three-dimensionality in moving displays, in that they indicate a high degree of visual sensitivity to stimulus organizations with unique topological representations.