Ecological constraints on internal representation: resonant kinematics of perceiving, imagining, thinking, and dreaming.

ness of Internalized Perceptual Constraints The two-dimensional case of Chasles's theorem provides the simplest illustration of the abstractness of the internalized constraints. From considerations of physical dynamics, one might guess that two planar figures alternately presented in positions that differ arbitrarily (and hence by both a translation and a rotation, as in Panel f of Figure 2) would give rise to an apparent motion in which the center of mass of the apparently moving body traverses the shortest, straight line between its two terminal positions. Because the two views also differ by a rotation, such a motion would have to be accompanied by an additional, apparent rotational transformation, as illustrated for two rectangles in Figure 3, Part a. Instead of such a double transformation, however, Foster (1975) found Figure 3. Intermediate positions of a rectangle (drawn in thin lines) between the same two rectangles (drawn in heavy lines), which differ arbitrarily in both position and orientation, along a path consisting of a combined rectilinear translation and a rotation (Part a), and the path (which Foster, 1975, found to be preferred in apparent motion) consisting of a rotation only (Part b). (From Mental Images and Their Transformations by R. N. Shepard and L. A. Cooper, 1982, p. 316. Copyright 1982 by The Massachusetts Institute of Technology. Adapted by permission.) that the motion is generally experienced over a curved path. By having observers adjust the variable intermediate rectangle (indicated in Figure 3 by thinner lines) so that it appeared to fall on the path of motion, he found that (under conducive conditions) the motion tended to be experienced over that unique circular path that rigidly carries the one figure into the other by a single rotation about a fixed point, P, in the plane, as shown in Figure 3, Part b. It seems that here, as in the case of the moire pattern of Glass (1969; an example of which is shown in Figure 4, Part b), the visual system picks out the fixed point implied by the two presented positions of a rigid configuration in the plane and, hence, identifies the two configurations with each other by means of a simple rotation. (See Foster, 1975, 1978; Shepard, 1981b; and for a review and theoretical discussion, Shepard & Cooper, 4 In an investigation of apparent motion motivated by similar objectives, Warren (1977) reported that alternation between two-dimensional shapes differing by an affine transformation did not yield rigid apparent motion. However his allegedly affine pair (Panel g) was not affine, and his instructions and resulting subjective reports are open to questions of interpretation, choice of criterion, and effects of perceptual set or expectancy. 426 ROGER N. SHEPARD Figure 4. Moire pattern described by Glass (1969), in which two identical transparencies of a random texture (Part a), when superimposed in an arbitrary misalignment, give rise to the appearance of concentric circles (Part b). (As one transparency is shifted with respect to the other, the center of the concentric circles moves in an orthogonal direction.) 1982.) Incidentally, the visual system also extracts fixed points in the case of nonrigid transformations, as has been demonstrated by Johansson (1950, 1973), Wallach (1965/ 1976), and most extensively by Cutting and his associates (see Cutting, 1981; Cutting & Proffitt, 1982). There are good reasons why the automatic operations of the perceptual system should be guided more by general principles of kinematic geometry than by specific principles governing the different probable behaviors of particular objects. Chasles's theorem constrains the motion of each semirigid part of a body, during each moment of time, to a simple, six-degrees-of-freedom twisting motion, including the limiting cases of pure rotations or translations. By contrast, the more protracted motions of particular objects (a falling leaf, floating stick, diving bird, or pouncing cat) have vastly more degrees of freedom that respond quite differently to many unknowable factors (breezes, currents, memories, or intentions). Moreover, relative to a rapidly moving observer, the spatial transformations of even nonrigid, insubstantial, or transient objects (snakes, bushes, waves, clouds, or wisps of smoke) behave like the transformations of rigid objects (Shepard & Cooper, 1982). It is not surprising then that the automatic perceptual impletion that is revealed in apparent motion does not attempt either the impossible prediction or the arbitrary selection of one natural motion out of the many appropriate to the particular object. Rather, it simply instantiates the continuing existence of the object by means of the unique, simplest rigid motion that will carry the one view into the other, and it does so in a way that is compatible with a movement either of the observer or of the object observed. Possibly some pervasive principles of physical dynamics (such as a principle of momentum), in addition to the more abstract principles of purely kinematic geometry, have been internalized to the extent that they influence apparent motion (Foster & Gravano, 1982; Freyd, 1983a,. 1983c, 1983d, 1983e; Freyd & Finke, 1984; Ramachandran & Anstis, 1983). But there evidently is little or no effect of the particular object presented. The motion we involuntarily experience when a picture of an object is presented first in one place and then in another, whether the picture is of a leaf or of a cat, is neither a fluttering drift nor a pounce; it is, in both cases, the same simplest, rigid displacement. True, we may imagine a leaf fluttering down or a cat pouncing, but in doing so we voluntarily undertake a more complex simulation (just as we might in imagining a leaf pouncing or a cat fluttering down). Such mental simulations may be guided by internalizations of more specific principles of physical dynamics and even perhaps of animal behavior. Pervasive Constraints of Time and Distance I have taken the sources of the perceptual constraints considered so far to be corresponding constraints in the world, for example, the 24-hour diurnal cycle and principles of kinematic geometry and perhaps of physical dynamics. However, there are other highly orderly perceptual regularities that may not be reflections of constraints that happened to prevail in our world so much as manifestations of constraints that are unavoidable in any system that could exist in this world. Thus, much as the velocity of light limits the speed of communication between distant bodies, the necessarily finite velocity of signal ECOLOGICAL CONSTRAINTS ON INTERNAL REPRESENTATION 427