Emergent two dimensional patterns in images rotated in depth

Journal of Experimental Psychology: Human Perception and Performance 1980, Vol 6, No 2, 244-264 Emergent Two-Dimensional Patterns in Images Rotated in Depth Steven Pinker Ronald A Finke Harvard University Massachusetts Institute of Technology Once a person has observed a three-dimensional scene, how accurately can he or she then imagine the appearance of that scene from different viewing angles? In a series of experiments addressed to this question, subjects formed mental images of a set of objects hanging in a clear cylinder and mentally rotated their images as they physically rotated the cylinder by various amounts They were asked to perform four tasks, each demanding the ability to "see" the two-dimensional patterns that should emerge in their images if the images depicted the new perspective view accurately—(a) Subjects described the two-dimensional geometric shape that the imagined objects formed in an image rotated 90°; (b) they "scanned" horizontally from one imagined object to another in a rotated image; (c) they physically rotated the empty cylinder together with their image until two of the imagined objects were vertically aligned; and (d) they adjusted a marker to line up with a single object in a rotated image The experimental results converged to suggest that subjects' images accurately displayed the twodimensional patterns emerging from a rotation in depth However, the amount by which they rotated their image differed systematically from the amount specified by the experimenter Results are discussed in the context of a model of the mental representation of physical space that incorporates two types of structures, one representing the three-dimensional layout of a scene, and the other representing the two-dimensional perspective view oi the scene from a given vantage point The mental representation of threedimensional space and its relation to visual imagery is a relatively unexplored area of cognitive psychology There is reason to believe that information about the threedimensional layout of a scene is preserved in some way in visual images of the scene For example, Shepard and Metzler (1971) , j , U i U i i A tshowed that when people are asked to decide whether or not two line drawings depict three-dimensional objects having the „, , , , XT • , r This research was supported by National Science Foundation Grant BNS-77-21782 awarded to S Kosslyn; the first author was supported by a Natural Science and Engineering Research Council Canada Postgraduate Scholarship We are grateful to Nancy Etcoff, Reid Hastie, Frank Restle, and an anonymous reviewer for same shape, their decision time increases j through which one ° , °, , linearl J wkh th constructive comments on the manuscript; to Stephen Kosslyn and Mary Potter for advice on the design and interpretation of the experiments; and to object has to be rotated m depth to bring it into correspondence with the other Pinker and Kosslyn (1978) and Pinker (in press) David Birdsong and Fabio Idrobo for assistance jater ^Re^e'sTs for reprints should be sent to either Steven Pinker, who is now at the Center for Cognitive Science 20D-105, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, or to Ronald Finke, who is now at the Department of Psychology, Uris Hall, Cornell University, Ithaca, New York 14850 found that the time su bjects scan the s a ce " " P , between two imagined objects increases linearly with the distance between the objects in three dimensions In addition, Attneave and his colleagues (A t j n p 1Q7?- Attnpavp & Farrar 1977(Attneave, 19/2 , Attneave & farrar, 19 / / , Attneave & Pierce, 1978) found that sub- Copyright 1980 by the American Psychological Association, Inc 0096-1S23/80/0602-0244$00.7S 244 take to TWO-DIMENSIONAL PATTERNS IN ROTATED IMAGES 245 jects are extremely accurate in mentally Pinker (in press) has pointed out that extrapolating a visible straight line segment a "three-dimensional scale model" theory behind their heads Such findings have led cannot easily explain the existence of some theorists to propose that scenes are these perspective properties of images, represented internally in a three-dimen- since perspective properties arise only sional "space," in which an object is repre- when a three-dimensional scene is prosented by a "filled-in" region of the space, jected onto a two-dimensional surface in the same way that a scale model is iso- If the representational medium underlying morphic in shape to the object it represents imagery is simply a three-dimensional (Attneave, 1972, 1974; Metzler & Shepard, coordinate system, there is no obvious 1974; Pinker & Kosslyn, 1978) reason why only the "visible surfaces" of Other recent studies, however, suggest imaged objects should be accessible at one that two-dimensional projective properties, time, nor why the appearance of an imsuch as the alignment or concealment of agined scene should change systematically objects at different depths, the variations of with the angle and distance of a "vantage apparent size with distance, and the varia- point." Rather, in a three-dimensional tions of apparent shape with orientation, system, we might expect that the various may also be represented in images First, processes that inspect or "read" images if people must mentally rotate two objects could be applied to any three-dimensional into correspondence to verify that they region of the coordinate system, accessing have the same three-dimensional shape, the information from any or all directions at objects must have been encoded in a form once In fact, a three-dimensional representaspecific to their original viewing perspective; otherwise the representations could tion (together with the processes that act have been matched against one another upon it) would seem to describe the haptic directly, without the need to perform sense more than the visual sense When a mental rotations (as Metzler & Shepard, three-dimensional environment is explored 1974, note) Second, Kosslyn (1978) has by touch, distant objects not appear found that the linear relation in vision "smaller," nor are the backs of objects between an object's distance and projective "occluded," nor the shapes of objects size also holds for imaged objects—the seem to change with relative orientation larger an object, the farther away it must Likewise, we would not expect to find these be imagined to subtend a constant "visual perspective properties when an internal angle." Third, several experiments have three-dimensional "environment" is inshown that people are likely to remember spected ; but nevertheless, they seem to the details of an imagined scene that were exist Alternatively, perhaps people store two "visible" from their imagined "vantage point," and to forget those details that different traces of a visual scene: a threewere "concealed" or "out of view" (Abel- dimensional structure representing its son, 1976; Fiske, Taylor, Etcoff, & Laufer, spatial layout and a two-dimensional in press; Keenan & Moore, 1979) Finally, structure (perhaps a copy of the retinal Pinker (in press) found that when sub- image), in which the original perspective jects are asked to scan from one imagined properties are preserved However, this object to another in a three-dimensional model is also inadequate Pinker (in press) scene by "sighting" the objects through also had subjects study a scene and imagine an imaginary rifle sight, the subjects' that the scene was rotated so that they scan times mirror the apparent two- were looking at it from above or from dimensional separations of the objects This is barring the unlikely possibility that the in the frontal plane This suggests that representational system underlying imagery has images preserve interpoint distances in the the functional equivalent of reflected light rays, planar projection specific to a vantage which are projected onto the "retina" of the mind's eye! point 246 STEVEN PINKER AND RONALD A FINKE the side As before, they scanned across left-right distances, to represent the effects the resulting image by "sighting" ob- of a shift to the side; or front-back distances jects through an imaginary rifle sight to top-bottom distances, to represent the The time required to scan from one effects of a shift to a bird's eye view imagined object to another increased lin- However, if people can also accurately early with increasing distance in two imagine changes that result from interdimensions between the objects as they mediate rotations or perspective shifts, the would appear in the new perspective view process of transforming an image would Since the subjects had never seen the display have to be more complex than the dimenfrom these vantage points, they could not sion-relabeling procedure suggested earlier merely have scanned a trace of their retinal In the present investigation, the visual image, but must have constructed images "scene" consisted of four small objects depicting the appearance of the objects suspended in different positions inside a from the new viewing angles, using their clear upright cylinder, which could be knowledge about the three-dimensional rotated about its vertical axis After the structure of the scene subject studied the display, the objects This result would have pleased Hermann were removed, and the subject was asked von Helmholtz, who in 1894 conjectured to imagine each object in its former position that "without it being necessary, or even inside the cylinder Then the subject was possible to describe [an object] in asked to rotate his or her image of the words we can clearly imagine all the suspended objects by various amounts and perspective images which we may expect to perform several tasks that require upon viewing from this or that side" "seeing" the two-dimensional spatial (Warren & Warren, 1968, pp 252-254) properties and relations that should emerge In the present investigation, we put this if the image was rotated accurately These conjecture to further experimental test tasks consisted of (a) naming the twoIn particular, we wished to determine how dimensional geometric figure that the accurately images can represent the two- imagined objects formed when "viewed" dimensional perspective properties that mentally from a new angle, (b) scanning emerge when a scene is rotated with respect mentally in a horizontal direction from one object to another in the image of the to a vantage point Existing evidence on this issue is equiv- rotated scene, (c) rotating their image ocal Consider, for example, the "three- until two of the imagined objects were mountain problem" (Huttenlocher & Pres- vertically aligned, and (d) judging the son, 1973; Piaget & Inhelder, 1956) In this horizontal displacement of a single object task, subjects observe a three-dimensional that was imagined to have revolved around2 display consisting of three mountains and the axis of the cylinder by various amounts must select from a group of pictures the one that represents the appearance of the Experiment display from the side That adults can solve this problem, however, merely shows It is often suggested that a mental image that they know how to substitute one can serve as a surrogate percept, allowing linear ordering for another—in this case, a front-to-back ordering becomes a left2 To control, or in some cases, measure, the angle to-right ordering Although Pinker's (in of rotation of the image, we had subjects physically press) scanning experiments seem to inrotate the empty cylinder by the same amount as dicate that metric, and not just ordinal, they mentally rotated its contents Thus, the mental information is present in images result- rotation process we studied is not identical in all ing from a shift in perspective, they respects to the one Shepard and Metzler studied, only examined the effects of 90° shifts since in our experiments the subjects could align their image with a frame of reference at the same Hence, subjects in these experiments could time that they shifted portions of their image by have transformed front-back distances to predetermined amounts TWO-DIMENSIONAL PATTERNS IN ROTATED IMAGES people to detect some pattern or property in a remembered scene that they did not encode explicitly when they saw the scene initially However, attempts to demonstrate this ability experimentally have been disappointing (see Reed, 1974; Reed & Johnsen, 1975) In the first experiment, we hoped to demonstrate that people can encode a three-dimensional scene, mentally rotate it in depth, and "see" in their new image a two-dimensional shape that they did not notice in the original display Subjects studied a configuration of objects which, unknown to them, had been arranged so that the configuration defined a particular two-dimensional shape when viewed from the front and a different shape when viewed from the side After mentally rotating the configuration, the subjects were asked to identify the second, emergent figure To rule out alternative explanations for performance in this task, we compared the shapes subjects named (a) with those they used to describe the two-dimensional figure when imagining the configuration from the front, (b) with those named by a second group of subjects who actually saw the figure from the side, and (c) with those named by a third group of subjects who never saw the display Method Subjects Eighteen students and employees of Harvard University were paid to participate in an "imagery" condition Two could not carry out the task instructions and did not complete the experiment Sixteen additional members of the Harvard community participated in a "perception" condition Another 72 participated in a "control" condition by filling out a brief questionnaire Apparatus A clear Plexiglas cylinder, 27 cm high X 20 cm in diameter, was mounted on a turntable on a wooden platform and could be rotated 360° about its vertical axis The front of the cylinder was marked by a thin black line running vertically down the length of the cylinder; the rear was marked by a line running down the top and bottom thirds of its length An angular scale was located on the bottom of the rim of the turntable and was visible to the experimenter in a small mirror mounted under the 247 Figure The shape formed by the objects when the display is rotated 90° counterclockwise rim The platform rested on a table 35 cm away from a chinrest The chinrest was adjusted so that when the subject was seated at the table, he or she would be gazing into the center of the cylinder Four small plastic animal toys were suspended by clear nylon thread from brass rods (30 cm X cm X cm) that lay across the top of the cylinder The animals, between 2- and 4-cm long, consisted of a red bug, a black bear, a yellow fish, and a green frog An animal could be positioned anywhere in the cylinder by sliding its rod to a given location and by winding or unwinding the thread around the rod to raise or lower the animal At the start of the experiment, the animals were positioned so that when viewed from the front, the two-dimensional shape they defined (i.e., the plane figure whose vertices corresponded to the animals) could be described either as a triangle or a quadrilateral, as three of the animals were roughly collinear However, when seen from the left side, the shape they defined corresponded to a tilted parallelogram (see Figure 1) The experimenter positioned the animals by placing over the cylinder a clear plate that was marked with the correct positions of the rods in the horizontal plane and by inserting into the cylinder a "dipstick" on which the correct heights of the animals were indicated Procedure Imagery condition The experimenter, reading from a script, told the subjects that they were participating in a study on visual imagery and visual memory, and asked them to sit at the table with chin in chinrest and to study the display, trying to form an accurate mental image of it After several minutes the subject was asked to study one particular object Then that object was lowered to the bottom of the cylinder by allowing the thread STEVEN PINKER AND RONALD A FINKE u j: s J ~* 0-1 CM vO O So ~ Sc00 Ov •o a ,_Sj oo oo O fo' *~~* ' 10 o tj ^ O a fti rt 3s •^ O O 1 "So 33 $CJ r\ q *ã * "ãKĐ ^ tU a& J3 VI 0! e fe o "s Ê 'Đ a a Ui q f; I *ôãằ ^ i 5s ^ q ^ ^ oo" (^5 o ^, ^ V) ^ir> cs *^ CN o ^o o 10 (S —.JO CN 1O 2^—< t^» Ov CS O vo '^-^ 3"^ 1—t -* 00 v_^ ui s»^ — "2 •M CN 00 ~ X5 00 •ato ^-"^^ ^ PO ^s *~~f *•—' S3 V_^ S_^ fS CO (N — a flj TWO-DIMENSIONAL PATTERNS IN ROTATED IMAGES These data weaken the contention that subjects in these experiments simply overestimate the extent of the horizontal displacement of the imagined object for a given angle, (e.g., if they focused on a marking on the surface of a cylinder instead of an imagined object inside it), which would have yielded a sine function with a greater amplitude but with no leftward or rightward shift as compared to the correct function Similarly, the results show that when people rotate an image, the movement of an imagined object in two dimensions is not simply a linear function of the amount of rotation, but is closer to the sine function dictated by the laws of geometry.6 These results are consistent with those of Experiments and 3, in which the inaccuracies in the subjects' performance were attributable to their rotating their images faster than the cylinder To assess whether the overrotation of images in the present experiment is of the same approximate magnitude as it was in the previous experiment, we estimated the correct angle of rotation that would correspond to each of the mean marker settings These estimations were made by dividing each mean by the amount of the largest mean and calculating the arcsine of that value To compare the two experiments directly, we selected the first eight angles used in the present experiment (corresponding to the range of rotations employed in Experiments and 3), calculated the arcsines separately for the clockwise and counterclockwise directions, and plotted the eight actual cylinder angles used in this experiment against the eight arcsines, which estimate the angles through which the images were rotated Thus, cylinder rotations are plotted against corresponding image rotations, as they were in Figure As before, the two sets of angles correlate highly (.99) and fall along a line with a slope less than unity (.87; the intercept is —5°) The two estimates, 77 and 87, seem to be in reasonable agreement, given the differences between the two methods used to estimate them Why should the marker settings for the three largest angles of rotation in the fourth 261 quadrant fall below the correct values when the trend of the preceding angles would lead us to expect that they should fall above them? Perhaps the presence of the lines on the cylinder, which were at the sides at the beginning of rotation, cued the subjects after a certain amount of rotation that their image of the object was quickly approaching its original position in the cylinder As a result, they may have deliberately slowed down their rotation as the object passed into the fourth quadrant, perhaps to the extent of causing it now to lag behind the cylinder Given the presence of a clear frame of reference on the cylinder, it is not unreasonable to expect that subjects might eventually have noticed the distortions introduced by the rotation process and taken corrective measures of some sort Overall, the results of the last three experiments agree in suggesting that at least for rotations up to 130°, subjects tend to rotate their images approximately 15% to 35% farther than they rotate the cylinder These results are compatible, in a weaker sense, with those of Experiment as well, inasmuch as consistent overestimation of the amount of rotation might cause the imagined configuration to differ from the target configuration General Discussion The present series of experiments provides convergent evidence that people can localize mental images of objects in threedimensional visual space, that they can mentally rotate the configuration of objects in depth, and that they can detect twodimensional perspective properties that emerge from that rotation These properties include the two-dimensional geometric shape formed by a set of objects, the horizontal separations between objects, the vertical alignment of objects, and the Examination of the marker settings of individual subjects confirms that the sinusoidal shape of the aggregate function is not an averaging artifact Only one of the eight subjects seemed to have moved the marker linearly with increasing rotation and did so only in the first two quadrants 262 STEVEN PINKER AND RONALD A FINKE position of an object's projection onto the frontal plane The procedures that transform the positions of imagined objects introduce a degree of error into the resulting image, primarily by failing to coordinate a rotation in imagined space with the corresponding rotation in visuomotor space However, this source of error should not obscure the fact that subjects' images were indeed accurate when their performance is considered within the range of possible responses in each experiment First, the two-dimensional shape actually formed by the rotated display was the shape subjects most often reported "seeing" in their images, and the other shapes subjects named were similar to that shape Second, the time subjects required to scan horizontally between objects in a rotated image reflected the horizontal separations between the objects as seen from the new angle (although, in one case, that angle was overestimated) Third, subjects rotated the cylinder by an amount that was linearly related to the degree of rotation that would align two objects, were they visible in the display Fourth, the judged position of the frontal projection of an imagined object varied sinusoidally with the angle through which it had revolved about the vertical axis What these results say about the mechanisms underlying the mental representation of visual space? At the very least, we need a representation in which the twodimensional perspective properties specific to a vantage point are represented and also a representation that preserves the threedimensional spatial structure of a scene Furthermore, the present results (as well as those of Shepard & Metzler, 1971, and of Pinker & Kosslyn, 1978, and Pinker, in press) suggest that the latter structure might represent information in a format in which rotation or translation can be performed smoothly and continuously Recent advances in the computational study of shape recognition (Marr, 1978; Marr & Nishihara, 1978) suggest what these two sorts of representations might be like Marr and Nishihara propose that the process of recognizing an object's shape makes use of a format in which objects are represented by volumetric "shape primitives" ("generalized cones" of various shapes and sizes) whose relative positions are defined with respect to a coordinate system centered on the object Something like this objectcentered representation, or 3-D model, is plausible as a long-term memory representation for objects and their absolute locations in a scene Marr and Nishihara also propose a second format, in which the visible regions of a scene, as well as their depth and tangent-plane orientations with respect to the viewer, are represented in a coordinate system centered on the viewer's vantage point Something like this viewercentered representation, or "2f-D sketch," is plausible as a short-term memory representation of the appearance of a set of objects, since the two-dimensional properties and relations visible from a given vantage point are represented perspicuously Marr and Nishihara propose that during perception, the information in the retinal images is converted into a 2J-D sketch, which is then converted into a 3-D sketch, the level at which shape recognition takes place We suggest that during imagination the inverse of this process might occur In this case, information stored in a "3-D" format, together with a specification of the position and viewing direction of the viewer's vantage point relative to the scene, would be fed into a process that computes the appropriate "2J-D" or viewer-centered representation At this level, which is probably common to both imagery and perception, two-dimensional properties like those defined in the present experiments can be "read off" the display by ignoring depth and surface orientation information The distinction we have drawn between these two types of representations receives further support from Kosslyn and Shwartz's (1977, 1978) computer simulation of twodimensional visual imagery For entirely different reasons, they, too, posited two levels of representation: At their "deep level," an object is represented as a list of two-dimensional polar coordinates, corresponding to the filled-in points in a depiction of the object, with the coordinate TWO-DIMENSIONAL PATTERNS IN ROTATED IMAGES axes centered on that object At their "surface level," the coordinates of the various objects to be imagined are mapped onto a single two-dimensional coordinate system, and the corresponding points in that coordinate system are "filled in" to display the objects Kosslyn and Shwartz's "deep" representation resembles a 3-D or "spatial" representation, with its coordinate systems each centered on the represented object Their "surface display" resembles a 2f-D or "perspective" representation, with its single coordinate system centered on the viewer's "fixation point." Finally, in both our account and theirs, there must be a process that maps the various object-centered representations onto a single viewer-centered representation, a process that in our account also computes the two-dimensional properties of the surface image determined by the laws of perspective If in fact there is a process that transforms information from a three-dimensional to a perspective-specific format, the results of the present experiments indicate that the values it can accept for the new vantage point are not restricted simply to 90° shifts relative to the original vantage point Nor are they restricted to rotations within some narrow range of angles In fact, it seems possible that the process instantiates a very general algorithm fordepicting shapes from any angle, like those found in sophisticated computer graphics programs In any case, our results suggest that this process would introduce noise into images, both in the form of random perturbations of remembered objects' positions and in the form of systematic perturbations that result when the amount of rotation has been mistakenly estimated with respect to a physical rotation in the external world A final note: The claim that the human mind is equipped with a component that computes the exact perspective appearance of a scene from any viewing angle might strike the reader as farfetched After all, children, non-Western peoples, and preRenaissance artists are notorious for their inability to depict three-dimensional scenes accurately in two-dimensional media such 263 as painting and drawings (Arnheim, 1954) However, this supports several interpretations Perhaps these people can visualize the perspective appearance of the scene, but are simply unable to coordinate their motor plans for drawing with the patterns depicted in their images A second possibility is that these people are capable of letting a perspective mental image guide their sketching, but it does not occur to them to use this strategy Instead, they may rely on their knowledge of objects' three-dimensional shapes, which is sound practice when reasoning about objects in the world, but may lead to systematic error when the objects must be depicted on a two-dimensional surface (cf Phillip, Hobbs, & Pratt, 1978), On this view, the development of artistic skill may in part be an example of "metacognitive development" (cf Flavell, 1977), whereby people learn how to exploit the special properties of their own cognitive structures and processes.7 A third possibility is that certain people may have difficulty simultaneously mapping the different underlying components of an object or scene into a unique, viewer-centered coordinate system Instead, they might mentally depict each part as it would appear from its own optimal vantage point and combine the separate views into a single sketch This would explain why a common error people make in depicting cubes is to draw each face as a square and to display more faces than can actually be seen in a single glimpse Clearly, we have few grounds for distinguishing among these possibilities at present, but the issue may be a promising subject for the developmental and crosscultural study of visual cognition In fact, some artists are trained to represent perspective by "projecting" a mental image of a scene onto the canvas and sketching in the contours that they "see." Reference Note Spoehr, K T., & Williams, B E Retrieving distance and location information from mental maps Paper presented at the 19th annual meeting 264 STEVEN PINKER AND RONALD A FINKE of the Psychonomic Society, San Antonio, Texas, November 9-11, 1978 References Abelson, R P Script processing in attitude formation and decision making In J S Carrol & J W Payne (Eds.), Cognition and social behavior Hillsdale, N.J.: Erlbaum, 1976 Arnheim, R Art and visual perception Berkeley: University of California Press, 1954 Attneave, F Representation of physical space In A W Melton & E J Martin (Eds.), Coding processes in human memory Washington, D.C.: V H Winston, 1972 Attneave, F How you know? American Psychologist, 1974, Z9, 493-499 Attneave, F., & Farrar, P The visual world behind the head American Journal of Psychology, 1977, 90, S49-S63 Attneave, F., & Pierce, C R Accuracy of extrapolating a pointer into perceived and imagined space American Journal of Psychology, 1978, 91, 371-387 Finke, R A Levels of equivalence in imagery and perception Psychological Review, in press Fiske, S T., Taylor, S E., Etcoff, N L., & Laufer, J K Imaging, empathy, and causal attribution Journal of Experimental Social Psychology, in press Flavell, J H Cognitive development Englewood Cliffs, N.J.: Prentice-Hall, 1977 Huttenlocher, J., & Presson, C Mental rotation and the perspective problem Cognitive Psychology, 1973, 4, 277-299 Keenan, J M., & Moore, R E Memory for images of concealed objects: A re-examination of Neisser and Kerr Journal of Experimental Psychology: Human Learning and Memory, 1979, 5, 374-385 Kosslyn, S M Measuring the visual angle of the mind's eye Cognitive Psychology, 1978, 10, 356-389 Kosslyn, S M., Ball, T M., & Reiser, B J Visual images preserve metric spatial information Journal of Experimental Psychology: Human Perception and Performance, 1978, 4, 47-60 Kosslyn, S M., Pinker, S., Smith, G., & Shwartz, S P The how, what, and why of mental imagery Behavioral and Brain Sciences, in press, (a) Kosslyn, S M., Pinker, S., Smith, G., & Shwartz, S P On the demystification of mental imagery Behavioral and Brain Sciences, in press, (b) Kosslyn, S M., & Shwartz, S P A simulation of visual imagery Cognitive Science, 1977, /, 265-295 Kosslyn, S M., & Shwartz, S P Visual images as spatial representations in active memory In E M Riseman &A R Hanson (Eds.), Computer vision systems New York: Academic Press, 1978 Marr, D Representing visual information In E M Riseman & A R Hanson (Eds.), Computer vision systems New York: Academic Press, 1978 Marr, D., & Nishihara, H K Artificial intelligence and the sensorium of sight Technology Review, 1978, 81, 2-23 Metzler, J., & Shepard, R N Transformational studies of the internal representation of threedimensional space In R Solso (Ed.), Theories in cognitive psychology: The Loyola Symposium Potomac, Md.: Erlbaum, 1974 Phillip, W A., Hobbs, S B., & Pratt, F R Intellectual realism in children's drawings of cubes Cognition, 1978, 6, 15-33 Piaget, J., & Inhelder, B The child's conception of space London: Routledge & Kegan Paul, 1956 Pinker, S Mental imagery and the third dimension Journal of Experimental Psychology: General, in press Pinker, S., & Kosslyn, S M The representation and manipulation of three-dimensional space in mental images Journal of Mental Imagery, 1978, 2, 69-84 Reed, S K Structural descriptions and the limitations of visual images Memory & Cognition, 1974, 2, 329-336 Reed, S K., & Johnsen, J A Detection of parts in patterns and images Memory & Cognition, 1975, 3, 569-575 Segal, S Nonparametric statistics for the behavioral sciences New York: McGraw-Hill, 1956 Shepard, R N., & Metzler, J Mental rotation of three-dimensional objects Science, 1971, 171, 701-703 Warren, R M., & Warren, R P Helmholtz on perception: Its physiology and development New York: Wiley, 1968 Received April 2, 1979 • ... suspended in their exact positions and to rotate the cylinder slowly in a clockwise direction, TWO- DIMENSIONAL PATTERNS IN ROTATED IMAGES maintaining an image of the objects rotating with the cylinder.. .TWO- DIMENSIONAL PATTERNS IN ROTATED IMAGES 245 jects are extremely accurate in mentally Pinker (in press) has pointed out that extrapolating a visible straight line segment a "three -dimensional. .. down to two In the TWO- DIMENSIONAL PATTERNS IN ROTATED IMAGES following experiments, we attempt to distinguish between these two possibilities and to replicate and extend the results obtained so