Moose, 'MAPPING SPATIAL COGNITION WITH COMPUTERS', Arachnet Electronic Journal on Virtual Culture v2n02 (May 16, 1994) URL = http://hegel.lib.ncsu.edu/stacks/serials/aejvc/aejvc-v2n02-moose-mapping The Arachnet Electronic Journal on Virtual Culture __________________________________________________________________ ISSN 1068-5723 May 16, 1994 Volume 2 Issue 2 MOOSE V2N2 ======================================================================== MAPPING SPATIAL COGNITION WITH COMPUTERS Phil Moose (1), Teri Stueland (2), Krista Kern (3) and Tom Gentry (4) California State University, Stanislaus (1) moose@ctc.csustan.edu (2) steuland@ctc.csustan.edu (3) krista@ctc.csustan.edu (4) gentry@ceti.csustan.edu Abstract Fractal geometry, digital video technology and research on spatial cognition are combined to create new methods for describing mental models. Pointing behaviors recorded on videotape and digitized sketch maps provide two general purpose ways to derive quantitative measures for the study of cognitive maps. Fractal dimension (D) estimates of these cognitive maps were computed for the study of individual differences and the dynamic properties of mental models. Estimates of D for cognitive maps are sensitive to an apparent asymmetry for spatial processes, but exhibited low correlations with tests of personality and intelligence. The results suggest that the type of "imagination" used in cognitive maps may not be assessed effectively by the principal factors common to psychometric measures of cognition. 1.0 Introduction In the relatively brief period since the terms "fractal" and "fractal dimension" were coined by Mandelbrot (1975) there has been a rapidly expanding literature utilizing these concepts to describe natural phenomena. Schroeder (1991) provides a very readable account of the ubiquitous applications for this mathematics, including several associated with human behavior. The National Research Council's recommendations (Steen, 1990) for "new approaches to numeracy" begins with a chapter which introduces the fractal dimension concept. And the Peitgen, Jurgens, and Saupe (1992a, 1992b) textbooks "Fractals for the Classroom" -- commissioned by the National Council of Teachers of Mathematics -- are designed for general education uses beginning in high school. This broad based dissemination of fractal geometry can be attributed to at least two circumstances: the availability of increasingly affordable computing machines, and the improved fit between natural forms and these quantitative methods. Sarraille & Gentry (1994) reviewed scientific journal databases with respect to relative application across disciplines, and noted that fractal dimension measures are largely confined to the "physical" sciences. However, this may be more a matter of the words used, since power law functions have a long history in behavioral and social research; Gentry (1994, in press) suggests that psychologists can re-interpret this literature whereby the exponents derived from log/log plots can now be viewed as descriptions of cognitive complexity. Abraham, Abraham, & Shaw, (1990) and Goertzel (1993a, 1993b) provide substantial agendas for the applications of nonlinear analysis -- including fractal geometry -- for many areas of interest to students of humanity. This report focuses on techniques being used in a research program to characterize the geometry of human imagination (Gentry, Goodman, Wakefield, & Wright, 1986; Gentry & Wakefield, 1991; Gentry, 1991; Moose, 1991; Kern, 1991, 1992; Sarraille & Gentry, 1994; Stueland, 1994). We utilize two behaviors humans use to communicate spatial information: pointing and sketching. Our primary interest in using these methods is to generate accessible representations of individual cognitive maps. However, these behaviors also are relevant to some computer-human interactions (CHI). The working hypothesis that guides our experiments is the notion that the computation of fractal dimensions may be similar to what the brain does in discriminating features contained in data streams arriving from the senses. Baldly stated, if the geometry of nature is fractal, as suggested by Mandelbrot (1975, 1977, 1982), then our cognitive maps also may be fractal. In addition, the concepts of fractals and fractal dimensions may bring resolution of a long standing suggestion that Euclidean geometry is not a satisfactory mathematics for describing human visual perception. The proposition that visual perception is inherently non- Euclidean was made explicit by Luneberg (1947; see also Blank, 1959) and was supported subsequently by empirical results from a variety of studies (Dodwell, 1982). Despite the evidence that a principle sensory input to the brain -- vision -- is inappropriately described with the geometry of Euclid, there were few options until the introduction of fractal geometry. Earlier approaches which did allow for non-Euclidean methods include the multi-dimensional scaling (MDS) algorithms derived from the paper by Abelson & Tukey (1963) and extended by Kruskal (1964a, 1964b) and Shepard (1966). Despite several problems with the MDS techniques (detailed by Shepard (1974)), this mathematics provided the starting point for the present research and used the KYST-2A software (Kruskal, Young, & Seery, 1977) in the initial analysis of the pointing procedures described below. A primary objective in Gentry & Wakefield (1991) was to compare the MDS methods with the newer fractal geometry approach. It was suggested that the latter has advantages in characterizing the dynamics of and individual differences in cognitive maps. In short, Gentry & Wakefield hypothesized that cognitive maps are fractal. 1.1 Cognitive Maps The meaning of "cognitive maps" is the topic of a long term debate in psychology; the use of these terms by CHI researchers has expanded the range of interested participants. Both "cognitive" and "maps" elicit a diversity of interpretations and disputes. For example, Downs and Stea (1977) define cognitive mapping as, "an abstraction covering those cognitive or mental abilities that enable us to collect, organize, store, recall and manipulate information about the spatial environment." Although we shall review some of the interpretations of "cognitive map" briefly, our objective in this article is to describe some practical methods for studying human behaviors relevant to interactions with computers. Laszlo & Masulli (1993) begin a conference proceedings on this topic with Our use of the term 'cognitive maps' refers to the process by which an organism makes representations of its environment in its brain, an activity which most contemporary brain scientists seem to agree is one of the brain's main functions. This idea, in various forms, can be traced back to Hippocrates. The same collection ends with Eco's (1993) observation that: ... this conference is about the problem of maps, and who better than a student of semiotics or linguistics problems in general can be aware of the fact that the universe is a confederation of different maps, each of them representing the world in a different way? The problem today is whether these maps are commensurable, comparable, translatable among themselves, and whether there exists a metalanguage which will enable them to be described even when they are contradictory .... (Eco, 1993, pp. 281-2) One working assumption of the research summarized here is that a new "metalanguage" is available to translate among Eco's "confederation of different maps". This metalanguage arises from a cognitive change in our bedrock beliefs about the concepts of "dimension" and "space" itself. The advent of fractal geometry is more than a new mathematical tool to describe the natural world better, although it is proving to be very useful in that regard. Its more fundamental change is that the concept of fractal dimension has begun to revise our thinking about the very fabric of experience per se -- what gestalt psychologists called the "ground," but more commonly labeled the nature of "extent" or "space." We have traced elsewhere (Gentry & Wakefield, 1991; Gentry, 1994, in press) the arguments from Mach (1886) to Jaynes (1976) that our sense of "space" is the point of departure for all conscious awareness, and that with new ideas about "space" we are confronted with potential changes in how we think about everything. Mandelbrot's (1975, 1977, 1982) introduction of the formalisms needed to think in terms of "fractional dimensions" and "fractals" has broken the mindset that dimensionality comes in whole integer units. We now can think in terms of measuring spatial complexities to any degree of precision warranted by the available techniques. As reported below, it has become possible to detect subtle differences in human abilities and behaviors with a precision not available using previous methods. For example, Westheimer (1991) reported that the classical "just noticeable difference" threshold for changes in the fractal dimension of an irregular contour can be measured in the few parts per thousand range. He found that his observers could discriminate a change as small as 0.0085 in a contour with a starting fractal dimension of D = 1.15. In practical terms, for people developing artificial visual systems this type of result provides benchmark estimates for the sensitivity needed in a system to equal or exceed the discrimination capacity of humans to detect changes in complex visual tasks. Our work has suggested that similar sensitivities can be achieved using the fractal dimension to characterize motor behaviors that reflect cognitive maps. Edward Toleman (1948) was one of the first experimenters who investigated and published on the concept of cognitive mapping. In his research, he observed the ability of rats to get from a starting position to a goal by a route never used before. To do this, Toleman first trained rats in a maze that had a path which took the animals in a direction opposite from the goal. In the test situation, the rat was confronted with a fan of maze arms such that the frequency of the initial direction selected would indicate whether or not the rat merely had learned a serial sequence of stimulus-response behaviors (or, actually had formed a generalized cognitive map -- so that it could choose the general direction to the goal). Toleman's results (Downs and Stea, 1977, pp. 31-36; Toleman, 1948) support the view that organisms such as the rat can develop cognitive maps that enable them to take novel routes to previously visited geographic locations. This type of internal model of the environment now is considered common to many species. However, debates over how this is accomplished continue (Alcock, 1989; Ellen & Thinus-Blanc, 1987; Lieblich, 1982; McFarland, 1985; Olton, 1982). Cognitive maps may entail abstract representations of real environmental spaces. Some of our work in developing geographic information systems (GIS) has been directed towards this meaning (Gentry & Wakefield, 1991). However, our studies also have included the mapping of "people" and "things" with the goal of deriving nonlinear geometric models for general cognition. Exploring what we mean by "imagination" provides the focus for our research. Some investigators make distinctions between mental maps based on perception versus those based on cognition (e.g., Bedford, 1993). Carley & Palmquist (1992) utilized a linguistic analysis approach to construct maps, by using computer assisted analysis of spoken and textual comparisons of terms, concepts and information structures. Other uses of the term "mapping" appear in the development of software to support decision analysis (Zhang, Chen, & Bezdek, 1989), neural networks intended to aid or simulate cognition (Eberts, Villegas, Phillips & Eberts, 1992; Holyoak & Thagard, 1989) and the transfer of training from one computer command structure to another (Schumacher & Gentner, 1988; Schmidt, Fischer, Heydemann & Hoffmann, 1991). The historical and expanding diversity of meanings for cognitive or mental maps makes Eco's concerns (cited above; Eco, 1993) a central agenda for those interested in the theme of "computerized tools as intermediaries in the communication of mental maps." A common spatial language and quantitative system for describing "maps" of all types may remain only an ideal, but the emergence of fractal geometry is a new opportunity for developing a unified theory of spatial cognition and associated technologies. 1.2 Fractal Geometry and Space It is likely that many readers of this journal know of Mandelbrot's contributions, via the elaborate computer graphics that are associated with the set named after him. Creating complex images with the iteration of small equations, such as the Mandelbrot set, has become popular "screen-saving" technique. Mandelbrot was explicit in his claim that the concept of a "fractal dimension" represented both an improvement in our understanding of "dimension" and in the significance of the observer in anything -- even mathematics! It is the second point that has been the most distasteful for some, since it continues a theme which opened the 20th Century with the birth of relativistic physics. On the first point, Mandelbrot (1982, p. 12) states that "the loose notion of dimension splits into several distinct components"; on the latter, he describes his notion of "effective dimension" as "the relation between mathematical sets and natural objects. ... In other words, effective dimension inevitably has a subjective basis. It is a matter of approximation and therefore of degree of resolution." (p. 17). Using iterated equations for the generation of patterns that model natural phenomena is increasingly popular in the field of computer graphics, but it is running this process in reverse that provides the significant achievement which makes fractal geometry a powerful analytical method. Barnsley (1988) provides a good introduction to the methods by which it now is possible to go from images of complex forms to numbers that characterize the fractal dimensionality of natural patterns. Barnsley (1988, p. 3) defines the fractal dimension of a set [as] a number which tells how densely the set occupies the metric space in which it lies. It is invariant under various streachings and squeezings of the underlying space. This makes the fractal dimension meaningful as an experimental observable; it possesses a certain robustness, and is independent of the measurement units. We use the term fractal dimension (D) to denote what is more formally called the "capacity" or Hausdorff dimension. Our computations of D which utilized the Sarraille-DiFalco computer program also provided "correlation" and "information" dimension estimates, but in this research they have been essentially the same as the capacity dimensions. However, the potential utility for different types of fractal dimension calculations has been described by Schroeder (1991). 1.3 Pointing Behavior As Data The antiquity of using pointing arms and fingers to convey information is necessarily speculative, but it is not difficult to believe that it predates both the emergence of writing (about 6,000 years before the present, ybp) and symbolic drawings (about 30,000 ybp). If one considers the many examples of non-human intraspecies communication involving the orientation of body parts, pointing must have been one of the original geographic information systems used by our ancestors. Indeed, it still is commonly used whenever the auditory language channel is inappropriate -- such as in noisy environments, or where silence is important. The ability of humans to point in the direction of real or imagined targets has been employed in several types of research on spatial abilities. Clinical uses of pointing have been reported from the middle of the 19th Century (von Graefe, 1854), and remain common tests for a range of neurological disorders (e.g., Riddoch, 1917; Bock & Kommerel, 1986). Ott, Eckmiller, and Bock (1987) constructed a head-mounted, computer controlled device to record pointing towards 80 light emitting diodes contained within the head-gear. The subjects cannot see their pointing behavior, which is important for clinical testing. In the treatment of autistic children, pointing behaviors have been used to replace "autistic leading" and as an intermediary step in the acquisition of verbal behavior (Carr & Kemp, 1989). The clinical studies and use of pointing behaviors to improve autistic dysfunctions are consistent with the view that pointing represents a very early manifestation of an individual's cognitive mapping abilities. As with all gross motor movements, the researcher is confronted with the problem of response specification and measurement. The literature contains a range of methods that vary in complexity, cost and precision. Duhamel, Pinek, & Brouchon (1986) measured pointing towards auditory targets with blindfolded subjects by rating whether the index finger of the pointing hand was either to the left or right of a sound source. Using this relatively simple and inexpensive method, they found significant differences between left- versus right-handed subjects (right handers were more accurate), and between which arm/hand was used, irrespective of handedness (left hand/arms were more accurate). Presson, Delange, and Hazelrigg (1989) devised a hand-held circular dial with a pointer that could be moved by subjects to indicate directions, in experiments on spatial memory for maps and paths. Their objective was to determine the effects of scale. They found that small scale displays were coded in an orientation specific way, while very large scale displays were remembered "in a more orientation-free manner." They cite their work as support for the "view that there are distinct spatial representations, one more perceptual and episodic and one more integrated and model-like, that have developed to meet different demands faced by mobile organisms." The advent of the microcomputer has been accompanied by pointing devices that offer new opportunities for the assessment of individual differences in spatial abilities. Jones (1989) has used mouse, joystick, and track ball devices to manipulate the monitor cursor, in studies in which subjects "point" to targets displayed on the screen. He found that estimates of processing rates (bits/sec.) based on the method of Fitts were substantially slower for the computer pointing tasks that for problems of similar difficulty measured with direct pointing (Fitts 1954, Fitts & Peterson, 1964; see also Langolf, Chaffin, & Foulke, 1976). Sholl (1987) used a modified joystick in a "point-to-unseen- targets" test of cognitive maps for geographic regions of different size. Her results suggest that orienting schemata direct orientation with respect to local environments, but that orientation with respect to large geographical regions is supported by a different type of cognitive structure. This finding is similar to that in the study by Presson, Delange, and Hazelrigg (1989) cited above. Gentry and Wakefield (1991) review the literature on using pointing as a behavioral measure and describe a procedure for characterizing cognitive maps using pointing behavior recorded on videotape. These videotapes are processed in order to create a composite plot of the subject's pointing toward imagined targets. The resulting "cloud-of-points" (COP) plots then are used to compute the fractal dimensions of the subject's cognitive map. The present experiments were conducted to provide a more extensive examination of the method described by Gentry & Wakefield. 1.4 Sketch Maps As Data The step from pointing with a finger to indicate features in the environment to drawing a representative sketch in the dirt may not seem a particularly dramatic advance in cognitive abilities, since we combine the two easily when communicating with each other. But a good deal of experimental evidence suggests that the sketch map is a much more complex task, and has greater sensitivity to experiential variables and individual differences. Evans (1980) reviewed the use of sketch map methodologies that have shown the effects of age, gender, culture, and type of environment selected for the research. Reports of gender differences in cognition generally attract attention; sketch maps have been used in several such experiments (Boardman, 1990; Grieve & Van Staden, 1988; Holding & Holding, 1989; Pearce, 1977). In general, the sex differences appear to be related more to what is emphasized or to style. Also, these differences are not apparent in younger children. Sketch maps are sensitive particularly to the age variable, and consequently are utilized in developmental studies (Biel, 1986; Blades, 1990; Golledge, 1985; Grieve & Van Staden, 1988). Other variables that reportedly influence the construction of sketch maps are (1) spatial abilities, as measured by some psychometric tests (Moore, 1975), and (2) measures of visualization, orientation, and sense-of-direction (Rovine & Weisman, 1989). The report by Blades (1990) concerning the reliability of data collected from sketch maps is central to the development of the analysis method described in this article. Blades utilized the common "two independent judges" method to obtain reliability estimates, but the work of Lewin & Wakefield (1979) and Wakefield (1980) demonstrated that unexpectedly high concordance between two observers is needed to achieve significant correlations. For example, if two observers detected 90 percent of the target features in a task and their inter-observer agreement was 82 percent, this would be equal to a zero (r = 0.00) correlation! The use of trained observers or "experts" to render independent scores which then are used to compute the reliability of a measurement is applied in many areas involving complex patterns (such as human hand writing or sketching). Computing fractal dimensions for these types of patterns offers new quantitative methods in the analysis of human cognitive complexity. 2.0 Method We present here new methods for studying the complexity of mental maps, methods that are very sensitive to the fractional dimensionality of spatial cognition. It is anticipated that hardcopy published versions, with associated graphics, will be available in the future. Our intention is to describe the methodologies we have developed. They may have general utility for a range of research interests that involve the analysis of motor behaviors recorded on videotape, or that involve hand movements and drawings that can be generated for a wide variety of reasons. Two different groups of subjects were used. The first group provided data for (1) a manual computation of the capacity fractal dimension of pointing behaviors and (2) for an automated analysis of sketch maps. The second experimental group provided data for a replication of the pointing behavior results using a more automated videotape analysis system and a fast algorithm (fd3) for computing the capacity, information, and correlation dimensions (Sarraille, 1991). /1/ 2.1 Pointing Experiment 1 2.1.1 Subjects The subjects were 55 university students (35 female, 20 male) enrolled in an Introductory Psychology course; they received extra credit for participating. Their ages ranged from 17 to 37, with a mean age of 21 years. Each participant completed a consent form describing the nature of the experiment and was told that all information would remain confidential. Confidentiality was assured by assigning a code number to each participant. 2.1.2 Questionnaires Three pencil and paper tests were administered during regular class periods: Eysenck's (1975) personality questionnaire (EPQ), two parts of the Differential Aptitude Test (DAT), and a test of spatial relations and verbal reasoning (Bennett, Seashore, & Wesman, 1984). 2.1.3 Apparatus A video camcorder recorded the subjects' pointing behavior during the individual video recording sessions. This procedure took place in a windowless room. A meter stick was placed on the wall directly behind the subjects, to provide a reference scale. Two audio cassette players were utilized in order to provide continuous prerecorded instructions. Five audio instruction tapes were created prior to the conduct of the experiment. The first tape explained what would transpire during the experimental session. The four remaining audio tapes contained 20 randomized landmark locations, used to elicit the pointing responses. The videotapes of the subjects were analyzed using a VCR, a 13" color television monitor, and an 8 x 11-inch (23 x 28-cm) piece of plexiglass. Acetate sheets (8 x 11-inch) and black, water resistant pens were used to record the pointing locations on the plastic overlays. 2.1.4 Procedure The psychometric tests were administered in three separate class periods. Prior to the videotaping, a series of steps was initiated. First, during a regular class period (conducted in a windowless room), all subjects were instructed to draw a sketch map of the campus, a map that included nineteen specified landmarks. The subjects then were directed to go outside the building and stand facing in a direction which made all target landmarks visible. Verbal instructions were given to look over the campus and to create a mental picture of the surroundings and the specified landmark locations. The subjects then were asked to point to these landmarks as each one was verbalized. Following this direct observation of the target landmarks, they returned immediately to the classroom and were instructed to draw a second sketch map. Next, subjects were taken, one at a time, into a windowless room. Masking tape on the floor directly across from the camcorder showed subjects where to stand. A ping-pong ball, drilled to make a finger sized hole, was given to each subject so that it could be placed on the fore-finger of their right (R) or left (L) hand. An L-R-R-L or R-L-L-R control pattern was used alternately across subjects, in an effort to counterbalance any order effects due to which arm was used for pointing first. The ping-pong ball made the pointing location easier to identify later, when researchers analyzed the videotapes. The video camcorder was stopped after the first and third audio tapes were completed. This provided time for subjects to change the location of the ping-pong ball from one hand to the other. Prior to the start of a session, all subjects listened to a prerecorded tape with the following instructions: During the entire procedure, please keep your eyes closed and try to relax as best you can. When I instruct you to point to a specific location please hold your hand steady until I say okay. Let's begin. Imagine yourself on the front steps of the classroom building. You will be facing Monte Vista Avenue. The reflecting pond and fountain will also be in front of you. The imagined geographic location was identical to the site where the subjects stood during the earlier practice session. It should be noted that when subjects were in the room that was used for the videotape session, they were at a 90 degree angle to the imagined location. Consequently, subjects had the task of imagining themselves rotated to an orthogonal geographic orientation relative to where they were standing, during the actual testing procedure. The behavior of importance in this task was the subjects' ability to point toward the specified campus landmarks, viewed from an imagined position. This behavior was recorded on videotape and used to extract estimates of the fractal dimension (D) for pointing to the targets. After all subjects were videotaped, the tapes were played in a VCR; each subject's pointing locations were plotted on 8 x 11- inch acetate sheets. Each sheet was taped onto the plexiglass, which had been attached to a television monitor. The piece of plexiglass provided a flat surface to mark the pointing locations. Each subject yielded two acetates, one for the right hand pointing locations and one for the left hand. The cloud-of- points (COP) data from each subject were analyzed by making 15 copies of the COP patterns. The copies were used to circle the cloud-of-points according to the procedure described by Barnsley (1988, p. 190). The circle templates were placed on one copy at a time, starting with the largest radius and going down to the smallest. In all, 15 circles were used, with radii ranging from 1/4 inch (6 mm) to 3-1/2 inches (88.9 mm). The "algorithm" used was a procedure to cover all points with a minimal number of circles of each radius. No circle was counted that did not include a data point. This was done for the 15 circle sizes, and plotted as a power function. A fractal dimension was computed for each cloud-of- points set where the logarithm of the number of circles needed to cover the points with a particular radius is plotted against the logarithm of the reciprocal of the radius for the corresponding circle. A regression line was computed through the resulting points, using the 04-StatWorks program for the Macintosh. The slope of the regression line was the estimate of the fractal dimension. (This particular estimate is known as the capacity dimension; an example using a cloud of points can be found in Barnsley (1988, p. 190)). 2.1.5 Results There were only two small significant correlations between the pointing behaviors and the subjects' psychometric test scores. The correlations between the left arm and the DAT verbal subtest was r = 0.309 (p = 0.022); the DAT spatial subtest was r = 0.276 (p = 0.044). All other correlations were smaller -- and nonsignificant. The principle result in this pointing experiment, and in the following second experiment, was the difference in the fractal dimension of the cloud-of-points generated by the left and right arms. Table 1 shows these results. Table 1 THE PAIRED T-TEST BETWEEN THE ESTIMATED FRACTAL DIMENSIONS 'D' OF THE SUBJECT'S RIGHT AND LEFT ARM POINTING COP PATTERNS ----------------------------------------------------------------- Left Arm Right Arm -------- --------- Mean 0.984 0.952 SD 0.078 0.082 ----------------------------------------------------------------- Notes: t = 4.118; df = 54; p < 0.001 2.2 Sketch Map Analysis and Results The sketch maps described in the Method Section were used for a subsequent analysis that involved comparing estimates of the fractal dimension between the drawings made before the subjects viewed the campus landmarks to those made after they had returned to the classroom to draw the second sketch maps. These drawings were converted to "islands" by connecting the landmarks with straight lines, using the rule that the connecting lines cannot cross. This method yields an irregular perimeter polygon for each drawing. The resulting outlined sketch-maps then were digitized using a Targa8 framegrabber and a microcomputer running Jandel Scientific's JAVA image analysis program. The edge tracking feature of JAVA yielded the XY coordinates for the perimeter of these sketch map "islands." The data sets then were analyzed by the DiFalco-Sarraille algorithm to compute D. A t-test between the D values derived from the first sketch maps and the second set of sketch maps indicated that the brief direct observation of the landmarks resulted in a significant increase in the fractal dimension of the post-observation drawings, as indicated in Table 2. Table 2 THE PAIRED T-TEST BETWEEN THE ESTIMATED FRACTAL DIMENSIONS (D) OF THE PRE-OBSERVATION AND POST-OBSERVATION SKETCH MAPS ----------------------------------------------------------------- Pre-observation D Post-observation D ----------------- ------------------ Mean 1.177 1.212 SD 0.072 0.075 ----------------------------------------------------------------- Note: t = 2.905; df = 53; p < 0.005 One also could ask how correlated the two methods were in estimating the fractal dimension of the internal cognitive maps (e.g., D derived from sketch maps versus derived from pointing behavior). Table 3 provides the correlation matrix of these estimates of D. Table 3 CORRELATIONS BETWEEN THE FRACTAL DIMENSION ESTIMATES (D) FOR THE SKETCH MAPS AND POINTING METHODS ----------------------------------------------------------------- D D D D (Pre- (Post- Left Right sketch) sketch) Arm Arm Pre-sketch D 1.000 0.255 0.231 0.162 Post-sketch D 1.000 0.046 -0.006 Left Arm D 1.000 0.745* Right Arm D 1.000 ----------------------------------------------------------------- Note: *p < 0.001 The only significant correlation was the association between the two arms in the pointing procedure (r = 0.745, p < .001). Even though pointing with the left arm exhibited significantly higher values of D (see Table 2), the fractal dimension of the cloud-of- points for the two arms in the same individual is highly correlated. The correlations in Table 3 suggest that sketch maps tap a cognitive process that is sufficiently different from the pointing behavior to warrant separate consideration. At this juncture, it was decided to focus on the asymmetry in the pointing behavior, since it was difficult to believe that such a small effect could reach the p < 0.001 level of significance. The work by Westheimer (1991) was consistent with Barnsley's (1988) claims for the fractal dimension's sensitivity, so we conducted a second pointing experiment to test the robustness of our initial main effect. The subjects were different, the experimenter was different, the targets for pointing were a very different mixture of people, places and things, the pyschometric battery was expanded, and the video analysis method was changed from the manual overlays and hand drawn circles to a videotape rear projection digitizing tablet system that yielded the coordinates for the pointing. 2.3 The Second Pointing Experiment 2.3.1 Subjects The subjects for the second pointing experiment consisted of 42 university students (13 males and 29 females) enrolled in an introductory psychology class. The mean age of the males and females was 20.08 and 20.48 years, respectively. 2.3.2 Questionnaires The participants completed four psychometric tests: the Minnesota Multiphasic Personality Inventory-2 (MMPI-2; Hathaway & McKinley, 1989), the Eysenck Personality Questionnaire (EPQ; Eysenck, 1975), and two sections of the Differential Aptitude Tests (DAT) --verbal reasoning and spatial relations (Bennett, Seashore, & Wesman, 1984). 2.3.3 Apparatus A video camera recorded the subjects' pointing behaviors during taping sessions that occurred in a windowless room. Masking tape placed on the floor indicated where subjects were to stand. A ping-pong ball with a hole drilled in it was placed on the forefinger of each subject to facilitate future analysis of the videotaped pointing behaviors. 2.3.4 Stimulus Materials Two audio tapes, one for each pointing session, provided instructions for the participants. The directions were as follows: During the entire procedure, please relax and try to keep your eyes closed. When I instruct you to point to a specific person, location, or object, please point and hold your arm steady until I say 'okay.' You can hold your arm as close to or far from your body as you like, but please hold your arm steady until I say 'okay.' Let's begin. After providing the procedural instructions, the audio tapes gave a list of the targets to which the subjects were to point. During each session, the names of 18 cities, 18 people, and 18 common objects were given. The tapes provided 10 seconds for each subject to imagine the target and to decide where to point. The order of the targets remained the same across subjects and sessions. 2.3.5 Design and Procedure The EPQ and the two sections of the DAT were administered during regular class periods. Due to the large amount of time required to complete the MMPI-2, this measure was given as a homework assignment. Two video recording sessions were conducted. During Session 1, subjects first pointed to all the targets with their right hand. They then were asked to point toward the same targets with the left hand. The subjects began pointing with the left hand in the second session. 2.3.6 Data Reduction and Analysis Videotapes of the pointing behaviors were analyzed by using the computer program SigmaScan (Jandel Scientific, 1991). The projected videotaped image was reflected off of a mirror under a rear projection digitizing glass tablet. A mouse with gun-sight cross-hairs was used to mark the target locations indicated by the subjects. The computer program then recorded the X,Y coordinates of the data points onto a spreadsheet. The fractal dimension of the X,Y coordinates of these points was computed by using the fd3 program. Separate fractal dimensions were calculated for the three different kinds of targets. Intercorrelations were computed (using SPSS) between (1) the fractal dimensions of the pointing toward the people, locations, and objects and (2) the scores on the MMPI-2, EPQ, and DAT. 2.3.7 Results A 2X3 factorial ANOVA was calculated to determine if either the hand with which the subjects pointed or the type of target significantly affected the fractal dimensions of the pointing behaviors. A significant main effect for the hand with which the subjects pointed was found, F (1, 246) = 12.671, p <.001. The fractal dimension of pointing with the left hand (M = 0.998, SD = 0.143) was significantly higher than that of the right hand (M = 0.899, SD = 0.156). The type of target toward which the subject pointed (city, person, or object) had no significant main effect, p >.4. No significant interaction was found between the hand used for pointing and type of target, p > .8. Correlation coefficients were also calculated between the fractal dimensions of pointing toward the different targets, and scores on the psychometric tests. No significant correlations were found between the psychometric test scores and the fractal dimensions of the pointing behaviors. 3.0 Discussion The major findings of this research suggest that the pointing behavior used to study an individual's cognitive map is consistent with the large literature on the asymmetry of human brain function. Namely, the dimensionality of pointing with the left hand/arm motor system is greater than that for pointing with the right arm, as measured by the fractal dimension. This is consistent with the results of Duhamel, Pinek, & Brouchon (1986) that accuracy for auditory localization was better for the left arm irrespective of "handedness", and the observation with the so called "split brain" right handed subjects, who rendered more realistic drawings with their left hand after the corpus callosum was severed (Ornstein, 1985, p. 153). The failure to find any consistent relationships between the psychometric tests used and the apparently robust pointing asymmetry supports a growing suspicion that some important forms of cognition are not being assessed with the commonly used psychometric tests. The work of West (1991, 1992a, 1992b) that describes visual thinkers and gifted people with learning difficulties as some of "the ironies of creativity" suggests that the spatial behavior studied in these experiments may be better understood in the context of psychometric measures that reflect the "imagination" used in pointing towards the imagined targets. We noted in the introduction that the multidimensional scaling methods developed by Shepard, Kruskal and others had been used in the initial analysis of distance measures derived from the pointing behaviors reported in Gentry & Wakefield (1991). Recently, Shepard (1987) has described a "probabilistic geometry" for generalization in learning that includes exponential decay functions which may resolve his earlier concerns (Shepard, 1974) about the MDS methods, and may be compatible with the proposition that power functions (e.g. fractal dimensions) are appropriate, robust, and sensitive measures for the dynamics of human behavior and cognition. Footnotes 1. Copies of the "fd3" program can be obtained via the ftp archive at lyapunov.ucsd.edu in the directory pub/cal-state- stan. Acknowledgments The authors are especially indebted to the long term encouragement and support provided by Frederick Abraham, James Goodwin, Sally Goerner, Julie Gorman, John Sarraille, and James Wakefield. 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