Human Four-Dimensional Spatial Intuition in Virtual Reality
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Psychonomic Bulletin & Review 2009, 16 (5), 818-823 doi:10.3758/PBR.16.5.818 BRIEF REPORTS Human four-dimensional spatial intuition in virtual reality MICHAEL S. AMBINDER, RANXIAO FRANCES WANG, JAMES A. CROWELL, AND GEORGE K. FRANCIS University of Illinois, Urbana-Champaign, Illinois AND PETER BRINKMANN City College, New York, New York It is a long-lasting question whether human beings, who evolved in a physical world of three dimensions, are capable of overcoming this fundamental limitation to develop an intuitive understanding of four-dimensional space. Techniques of analogy and graphical illustration have been developed with some subjective reports of suc- cess. However, there has been no objective evaluation of such achievements. Here, we show evidence that people with basic geometric knowledge can learn to make spatial judgments on the length of, and angle between, line segments embedded in four-dimensional space viewed in virtual reality with minimal exposure to the task and no feedback to their responses. Their judgments incorporated information from both the three-dimensional (3-D) projection and the fourth dimension, and the underlying representations were not algebraic in nature but based on visual imagery, although primitive and short lived. These results suggest that human spatial representations are not completely constrained by our evolution and development in a 3-D world. Illustration of the stimuli and experimental procedure (as video clips) and the instruction to participants (as a PDF file) may be downloaded from http://pbr.psychonomic-journals.org/content/supplemental. Representations of space and time are deeply rooted in (1965; Abbott, 1991) explained how a 4-D creature can human thinking, reasoning, and perception of the world enter a 3-D locked closet from the fourth dimension by (Abbott, 1991; Bork, 1964; Durrell, 1938; Gardner, 1975; describing how a 3-D creature enters a two-dimensional Kant, 1881/1896; Reichenbach, 1958; Rucker, 1984). (2-D) enclosure from above without touching its walls. However, living in a physical world of three dimensions, The second technique is to lift an observer into the higher humans have their perceptual and cognitive systems tai- dimensional space, so that he or she can directly experience lored for sensing, storing, transforming, and reasoning it perceptually (Abbott, 1991; Berger, 1965; Rucker, 1984; about three-dimensional (3-D) objects. Some thinkers Seyranian, 2001). For example, Abbott suggested that a 2-D (e.g., Kant, 1881/1896) believe that space is an innate con- creature can obtain 3-D intuition when it is taken into the cept, whereas perceptual experience simply fills it with 3-D space and views its world from above. Although this objects. Thus, humans may develop symbolic systems to approach is hypothesized to be the most powerful means of conceptualize multidimensional entities that they call high- acquiring 4-D intuition, it was not possible to implement dimensional space, but they can never overcome the innate the technique until virtual reality was available (D’Zmura, constraint and add another dimension to the 3-D mental Colantoni, & Seyranian, 2000; Francis, 2005). space that corresponds to the physical world, because they There have been informal subjective reports that can never figure out where that dimension could be put. mathematicians who actively interacted with computer- Much effort has been made to challenge this cognitive graphical simulations of 4-D geometric objects obtained limitation and to develop human four-dimensional (4-D) sudden, novel insights of higher dimensional space (e.g., intuitions (Davis, Hersh, & Marchisotto, 1995; Gardner, Davis et al., 1995). However, there has been no objective 1969; Rucker, 1984; Seyranian, 2001; Weeks, 1985). Two evaluation of such achievements. In the present study, we basic techniques were proposed to help people obtain an examined the dimensionality limitation in human spatial intuition of 4-D space. The first is by analogy to 3-D space. representations by constructing 4-D geometric objects This technique has been widely used. For example, Berger in virtual reality and measuring judgments of distance R. F. Wang, [email protected] © 2009 The Psychonomic Society, Inc. 818 FOUR-DIMENSIONAL SPATIAL INTUITION 819 (a property of one-dimensional [1-D] space) and angle observation chamber, a small cube (a marker) with a correspond- (a property of 2-D space), which are spatial properties of ing texture on its faces appeared at the location of its entrance. The the lowest dimensions and serve as the building blocks of marker disappeared when the vertex left the observation chamber. higher dimensional spatial properties. If the participants Procedure can learn 4-D spatial relationships, their judgments should Each participant was given a brief lecture before the experiment correlate with the actual distances and angles. on the structure of both 4-D space and 4-D objects. Analogies were drawn to studying 3-D objects through a 2-D viewing window. A METHOD summary of the lecture can be found online at http://pbr.psychonomic -journals.org/content/supplemental. After we had ensured that the Participants participants understood the nature of 4-D space, the participants Four participants were recruited from the Department of Psychol- were tested in two tasks, in one session for each task. In each ses- ogy at the University of Illinois at Urbana-Champaign. Half of them sion, they had a 15-min practice followed by approximately 20 ex- were female. Three were graduate students and naive to the design perimental trials. of the study. In each trial, the participants held a gamepad with a tracker at- tached and were instructed to walk forward and backward along Apparatus the z-axis toward an object in order to change their position in the The experiment was conducted in the six-sided, fully immersive w-direction (Figure 2B). The simulated position of the observer Beckman Institute Virtual Reality Cube1 at the University of Illinois along the w-axis was determined by the z coordinates of the game- at Urbana-Champaign. Each surface of the Cube is a rear-projection pad. That is, a 1-m displacement of the gamepad in the z direction screen. Stereovision was achieved through a Stereographics LCD shut- led to a 1-m displacement of the observation chamber in the w di- ter glass system synchronized with the projector and computer graph- rection. On average, the 3-D slices were located 1.5 m from the ob- ics. A wireless Ascension MotionStar tracking system determined the server.2 The participants studied the 3-D slices of a target object and six degrees of freedom positions and the orientation of the participant’s were asked to mentally construct the 4-D object. head and of a gamepad in order to render the appropriate images. After a free viewing study period of a maximum of 2 min, the ob- ject disappeared and the response stimuli appeared. In the distance- Stimuli judgment task, a horizontal bar appeared with two patterned boxes The stimuli were randomly shaped hypertetrahedrons (four on each end. The participant adjusted the length of the line using simplexes). Each hypertetrahedron was constructed with five 4-D the gamepad to match the Euclidean distance between the two points as the vertices. All coordinates were randomly chosen be- corresponding targets in 4-D space (Figure 2C). In the relative- tween 0.6 m and 0.6 m with certain constraints (see the Compo- orientation-judgment task, a “V” dial appeared with three markers. nent Analysis section below). Then all points were connected to all The participants judged the markers’ relative orientation by adjust- others, and all triangular faces were filled. ing the top line segments to match the angular relationship between The 4-D object was displayed using a slicing technique (Seyra- the two arms in 4-D space (Figure 2D). The initial length/angle was nian, 2001). Figures 1A and 1B show an analogy of the technique determined by the orientation of the gamepad at the moment of the in three dimensions. In principle, a 2-D creature can build a mental appearance of the stimulus and was thus random across trials. The representation of the 3-D object by integrating the 2-D slices that it actual distance in the distance-judgment task varied from 0.3 to sees over time, although it does not know where that third dimen- 1.6 m. The actual angles in 4-D space were between 10º and 150º. sion is in physical space (Abbott, 1991). Similarly, our participants The only training that the participants received was a 15-min prac- studied the 4-D objects by walking along the w-axis, which revealed tice on each task with the same type of stimuli to familiarize them- its 3-D sections successively. The 3-D cross-sections of these hyper- selves with the apparatus and the procedure. No feedback of any tetrahedrons were wire-frame objects that might take the shape of a kind was given in any of the trials. Thus, unlike those in previous re- single point, a tetrahedron, or a triangular prism (see Figure 2A). search (Seyranian, 2001), the participants could not simply learn the For each object, three of the five vertices were randomly selected relationship between the pattern of stimuli and the correct responses as the target vertices. Whenever a target vertex entered the 3-D through trial and error. Instead, they had to grasp the 4-D structure ABC A B A C B Figure 1. A three-dimensional (3-D) analogy of the slicing technique and the distance and relative-orientation judgments used in this study. (A) A two- dimensional (2-D) observation space (dark gray plane) slicing through a 3-D object (light gray egg) along a direction orthogonal to the plane.