! ! The structure of spatial knowledge: Do humans learn the geometry, topology, or stable properties of the environment? ! ! ! By Jonathan D. Ericson B.S., Physiology & Neurobiology, University of Connecticut, 2006 Sc.M, Brown University, 2011 ! ! ! A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Cognitive, Linguistic, and Psychological Sciences at Brown University ! ! ! PROVIDENCE, RHODE ISLAND MAY, 2014 ! ! ! ! ! © 2014 by Jonathan D. Ericson ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! This dissertation by Jonathan D. Ericson is accepted in its present form
 by the Department of Cognitive, Linguistic and Psychological Sciences as satisfying the dissertation requirement for the degree of Doctor of Philosophy 
 ! ! Date _________________ ________________________________ William H. Warren, Advisor ! ! ! ! 
 ! ! Recommended to the Graduate Council ! ! Date _________________ ________________________________ Fulvio Domini, Reader ! Date _________________ ________________________________ Rebecca Burwell, Reader ! ! ! ! ! Approved by the Graduate Council ! Date _________________ ___________________________________ Peter Weber, Dean of the Graduate School ! ! !iii JONATHAN D. ERICSON
 Curriculum Vitae I. Personal Information Jonathan Ericson 
 Department of Cognitive, Linguistic, and Psychological Sciences
 Brown University, Box 1821
 Providence, RI 02912
 Tel: (860) 803-6521
 Fax: (401) 863-2255
 jonathan_ericson@brown.edu II. Education 2013 Ph.D., Cognitive Science, Brown University
 Primary Advisor & Mentor: Dr. William H. Warren, Jr.
 Thesis: The structure of spatial knowledge: Do humans learn the geometry, topology, or stable properties of the environment? 2011 Sc.M., Cognitive Science, Brown University
 Primary Advisor & Mentor: Dr. William H. Warren, Jr.
 Thesis: Rips & Folds in Virtual Space: Reliance on graph structure during navigation. 2006 B.S., Physiology & Neurobiology, University of Connecticut
 Magna Cum Laude, Dean’s List, New England Scholar
 Primary Advisor & Mentor: Dr. Michael T. Turvey Additional Education:
 Foreign Study Program, University of Westminster, Westminster, UK, January - June, 2005 III. Academic Positions 2007-2013 Graduate Student, Virtual Environment Navigation Laboratory,
 Brown University, Supervisor: Dr. William H. Warren, Jr. 2005 Research Assistant, Department of Experimental Psychology, 
 University of Connecticut, Supervisor: Dr. Roger Chaffin 2004 Research Assistant, Department of Behavioral Neuroscience,
 University of Connecticut, Supervisor: Dr. Etan Markus 2003 Research Assistant, Department of Physiology & Neurobiology,
 University of Connecticut, Supervisor: Dr. Andrew Moiseff !iv IV. Grants & Awards 2013 Professor Lorrin A. Riggs Graduate Student Fellowship, Brown University
 2010 NASA Rhode Island Space Grant, Brown University
 2008 Brain Science Program Graduate Fellowship, Brown University
 2007 First Year Fellowship, Brown University
 2002 Achievement Scholarship, University of Connecticut
 2004 New England Scholar, University of Connecticut
 2002 UNICO Scholarship
 2002 M. Tracy Conway Exchange Club Scholarship V. Manuscripts in Preparation Warren, W.H., Rothman, D., Schnapp, B. & Ericson, J. Wormholes in virtual space and the geometry of cognitive maps. In preparation. Ericson, J. & Warren, W.H. The influence of external landmarks, the sun, and cast shadows on learning a wormhole environment. In preparation. Ericson, J. & Warren, W.H. Ordinal violations in human spatial knowledge for navigation. In preparation. Ericson, J., Chrastil, E.R., Warren, W.H. An empirical evaluation of space syntax methods in virtual maze environments. In preparation. Ericson, J. Integrating cognitive, architectural, and robotics approaches to spatial mapping. In preparation. VI. Presentations at Meetings Ericson, J. and Warren, W. (2012, October 22-26). The Escher Museum. Demo presented at the Annual Meeting of the Human Factors and Ergonomics Society (HFES), Boston, MA. Ericson, J. and Warren, W. (2012, May 11-16). The influence of cast shadows on learning a non-Euclidean virtual hedge maze environment. Poster presented at the Vision Sciences Society Annual Meeting, Naples, FL, USA. Ericson, J. (2011, May 25-28). Integrating cognitive, artificial intelligence, and architectural approaches to spatial mapping. Talk delivered at Environmental Design & Research Association (EDRA) Annual Meeting, Chicago, IL, USA. Ericson, J. and Warren W. (2010, August 15-19). The influence of external landmarks on learning a non-Euclidean wormhole environment. Poster presented at Spatial Cognition, Mt. Hood, OR, USA. !v Ericson, J. and Warren W. (2010, May 7-12). The influence of external landmarks, the sun, and cast shadows on learning a wormhole environment. Journal of Vision, 10(7), 1057. Poster presented at the Vision Sciences Society Annual Meeting, Naples, FL, USA. Ericson, J. and Warren W. (2009, November 19-22). Ordinal violations in human spatial knowledge for navigation. Poster presented at the Annual Meeting of the Psychonomic Society, Boston, USA. Ericson, J. and Warren W. (2009, May 7-12). “Rips” and “Folds” in virtual space: Ordinal violations in human spatial knowledge. Journal of Vision, 9(8), 1143a. Poster presented at the Vision Sciences Society Annual Meeting, Naples, FL, USA. Ericson, J. and Warren W. (2008). Rips and folds in virtual space: Reliance on graph structure during navigation. Abstracts of the Psychonomic Society, 13, 1022. Poster presented at the Annual Meeting of the Psychonomic Society, Chicago, IL, USA.VI. VII. Teaching a) Certifications 2009 Sheridan Center Teaching Certificate II: Classroom Tools Seminar
 Brown University 2008 Sheridan Center Teaching Certificate I: Sheridan Teaching Seminar
 Brown University b) Teaching Assistant 2010 Perception, Illusion, and the Visual Arts
 Instructor: Dr. William H. Warren, Jr.
 Dept. of Cognitive, Linguistic, & Psychological Sciences, Brown University 2009 Mind and Brain: Introduction to Cognitive Neuroscience
 Instructor: Dr. David Badre 
 Dept. of Cognitive, Linguistic, & Psychological Sciences, Brown University 2009 Laboratory in Cognitive Processes
 Instructor: Dr. Kathy Spoehr
 Dept. of Cognitive, Linguistic, & Psychological Sciences, Brown University 2008 Approaches to the Mind: Introduction to Cognitive Science
 Instructor: Dr. Sheila Blumstein
 !vi Dept. of Cognitive, Linguistic, & Psychological Sciences, Brown University VIII. Service 2010 - 2011 Brown Arts Mentoring (BAM) Program, Swearer Center for Public Service
 Brown University, Providence, RI. 2009 - 2010 Sheridan Center for Teaching and Learning Graduate Student Liaison
 Brown University, Providence, RI. 2008 - 2009 Graduate Student Liaison to the Faculty
 Brown University, Providence, RI. 2007 - 2008 Graduate Student Council Representative
 Brown University, Providence, RI. IX. Professional Memberships Vision Sciences Society (VSS)
 Spatial Intelligence and Learning Center (SILC) Spatial Network
 Human Factors and Ergonomics Society (HFES)
 Environmental Design Research Association (EDRA)
 Interaction Design Association (IxDA) X. Professional Training Brown Ethical & Responsible Conduct of Research Education Program, Jan 24, 2012
 Dynamic Field Theory Summer School, University of Iowa, June 8-12, 2009 XI. Ad Hoc Reviewing Environment & Behavior, 2013
 Human Factors & Ergonomics Society (HFES), 2012
 ! ! ! !vii Acknowledgments I would like to thank my parents, David and Caryn, for their unconditional love and support. I am especially indebted to my advisor, Bill Warren, who has been a tireless mentor, supportive advisor, and trusted friend throughout my entire graduate career. The members of my committee, Fulvio Domini, Rebecca Burwell, and Mintao Zhao have provided me with valuable feedback at all stages of my graduate career, and my work has benefited from their insights. I am also grateful for the teaching and career mentorship that I received from Sheila Blumstein, David Badre, and Kathy Spoehr. I feel very fortunate to have been serendipitously introduced to the study of visual perception by my undergraduate advisor, Michael T. Turvey, in the Fall of 2002. All of the VENLab graduate students and postdocs have provided valuable friendship and feedback over the years, and I am especially grateful to Liz Chrastil, Hugo Bruggeman, Adam Kiefer, Stephane Bonneaud, and Kevin Rio. I am also very grateful for the efforts of programmers Joost de Nijs, Reese Kuppig, and Kurt Spindler, and Neil Fulwiler. Our lab managers, Henry Harrison and Michael Fitzgerald, implemented innumerable improvements to the VENLab over the years, relentlessly filled in the weekly participant schedule, and provided critical technical support. A small army of Research Assistants who helped “wrangle” cables and prepare data also provided invaluable support. This work would not have been possible without generous support from the National Science Foundation (BCS-0214383 and BCS-0843940), the National Aeronautics and Space Administration (NASA) Rhode Island Space Grant, Brown’s Brain Science Graduate Fellowship, the Professor Lorrin A. Riggs Graduate Student Fellowship, and the !viii Department of Cognitive, Linguistic, and Psychological Sciences. Also, my participants, who boldly explored an array of non-Euclidean hedge mazes. I am grateful to all of the CLPS students, faculty, staff, as well as the close friends I was fortunate enough to make while in graduate school, especially Gideon Goldin. Finally, I would like to thank Anna Hartley for her support during my final years as a graduate student. !ix Table of Contents List of Tables! xi List of Figures! xii Chapter 1: Introduction! 1 Chapter 2: Experiment 1! 36 Chapter 3: Experiment 2! 79 Chapter 4: Experiment 3! 101 Chapter 5: Comparison of Experiments by Task! 124 Chapter 6: General Discussion! 131 References! 141 Appendices! 156 x List of Tables Table 1: Procedure for Experiment 1 .......................................... 44 Table 2: Estimated just noticeable difference (JND) analysis. .......................... 66 xi List of Figures Figure 1: Examples of graph structures ......................................... 6 Figure 2: Path integration .................................................. 8 Figure 3: The Klein hierarchy of geometries ...................................... 11 Figure 4: Euclidean, elliptic, and hyperbolic geometries .............................. 13 Figure 5: The Euclidean metric postulates ....................................... 14 Figure 6: Environmental geometry ............................................ 15 Figure 7: The triangle inequality ............................................. 21 Figure 8: Full experimental design ............................................ 34 Figure 9: The Virtual Environment Navigation Laboratory (VENLab) .................... 38 Figure 10: Design of Experiment 1 ........................................... 40 Figure 11: Mazes and displays used in Experiment 1 ................................ 41 Figure 12: First person view of shortcuts by task .................................. 46 Figure 13: Predictions for probe and control trials in Experiment 1 ...................... 48 Figure 14: Gate locations .................................................. 51 Figure 15: Test phase dependent measures ....................................... 53 Figure 16: Classification method for endpoints analysis .............................. 56 Figure 17: Classification method for path choice analysis ............................. 57 Figure 18: Experiment 1: free exploration path traces ............................... 60 Figure 19: Experiment 1: shortcut data, shortcut task .............................. 62 Figure 20: Experiment 1: shortcut data, neighborhood shortcut task ..................... 63 Figure 21: Experiment 1: shortcut data, route task ................................. 64 Figure 22: Experiment 1: final angular error ...................................... 65 Figure 23: Experiment 1: percentages of endpoints falling in neighborhoods ................. 72 Figure 24: Experiment 2: virtual environments .................................... 81 Figure 25: Design of Experiment 2 ............................................ 82 Figure 26: Experiment 2: free exploration path traces ................................ 84 Figure 27: Experiment 2: shortcut data, neighborhood shortcut task ..................... 86 xii Figure 28: Experiment 2: shortcut data, route task.................................. 87 Figure 29. Experiment 2: final angular error. ...................................... 88 Figure 30: Experiment 2: cluster analysis of angular errors, neighborhood shortcut task ......... 90 Figure 31: Experiment 2: cluster analysis of angular errors, route task ..................... 91 Figure 32: Experiment 2: percentages of endpoints falling in neighborhoods ................ 93 Figure 33: Experiment 3: virtual environments .................................... 103 Figure 34: Design of Experiment 3 ............................................ 104 Figure 35: Experiment 3: free exploration path traces ................................ 106 Figure 36: Experiment 3: shortcut data, neighborhood shortcut task ....................... 109 Figure 37: Experiment 3: shortcut data, route task ................................. 110 Figure 38. Experiment 3: final angular error. ...................................... 111 Figure 39: Experiment 3: percentages of endpoints falling in neighborhoods ................. 113 Figure 40: Experiment 3: path choices in the Swap Maze .............................. 115 Figure 41: Comparison of experiments by task .................................... 125 xiii Chapter 1: Introduction The course of our daily lives is significantly shaped by the spatial relations present in our ecological niche. At the relatively small scale, adaptive spatial behavior involves avoiding visible obstacles, walking to visible goals, and learning configurations of visible objects. At a larger scale, it involves exploring neighborhoods, towns, or entire cities, and using the knowledge acquired to guide wayfinding. Adaptive spatial behavior at the larger scale therefore requires integrating over many spatially and temporally extended encounters with the environment to learn about the spatial relations among environmental features (e.g., places, landmarks, routes) not currently in view. The knowledge acquired through this spatiotemporal integration has been traditionally described as a “cognitive map,” implying that humans learn as cartographers, building a metric, globally consistent, Euclidean survey map of the environment from these local encounters. Topological spatial knowledge is the most accurate, enabling novel shortcuts, and supporting derivations of weaker geometric relations. More recent studies suggest that although human navigation behavior is often consistent with this kind of survey map knowledge, it is often equally consistent with primarily topological spatial knowledge. Collectively, these studies suggest that humans do not learn as cartographers, continually embedding the coordinates of environmental features in a globally consistent reference frame. Rather, successful navigation may be better explained by spatial knowledge that primarily preserves topological relationships (e.g. neighborhood relationships or the connectivity of places), and that is supplemented with coarse, local information about distances and angles. This form of spatial 1 knowledge is more robust than a Euclidean survey map because it is less vulnerable to noise and error, and the topology of the environment is preserved when metric properties are not accurately acquired. Although most experimental work on human spatial knowledge has focused on demonstrating that humans are able to acquire either Euclidean metric structure or topological structure, a third possibility is that human spatial knowledge is flexible and adaptive, and preserves whatever geometric properties remain stable during learning. This dissertation critically examines three hypotheses concerning the geometric structure of “cognitive maps” by selectively destabilizing particular geometric properties during learning, and probing the spatial knowledge acquired by measuring navigation behavior. The Euclidean Hypothesis posits that spatial knowledge has a primarily constant Euclidean metric structure. Primarily Euclidean spatial knowledge is the most accurate, enables novel shortcuts, and other geometric relations are derived from it. If this hypothesis is correct, when metric structure is unstable during learning derived geometric relationships will also be degraded, so navigation performance on neighborhood and Route Tasks should deteriorate. The Topological Hypothesis posits that spatial knowledge has a primarily topological structure. In this case, when metric structure is destabilized during learning, navigators should still perform well on neighborhood and Route Tasks because these structures are not derived from Euclidean knowledge. The Stability Hypothesis posits that spatial knowledge reflects the specific geometric properties that remain stable during learning. If this hypothesis is correct, performance should always be consistent with preserved geometric structure when other 2 environmental properties are unstable. For example, when metric and graph structure are unstable but neighborhoods are preserved, performance should reflect acquisition of neighborhoods. Cognitive Maps The term “cognitive map” was introduced by Tolman (1948), who observed that rats sometimes take direct (i.e., “as the crow flies”) novel shortcuts between familiar locations. Because these shortcuts did not always coincide with previously learned routes, Tolman interpreted this observation as evidence that rats acquire map-like knowledge of large-scale environmental geometry. Similar results were subsequently reported for dogs (Chapuis, Thinus-Blanc, & Poucet, 1983), hamsters (Chapuis, Durup, & Thinus- Blanc, 1987), birds (Gould, 1982; 1986), chimpanzees (Menzel, 1973), and insects (Gould, 1986; Wehner, Michel, & Antonsen, 1996). This led to the notion that the ability to take novel shortcuts is a defining characteristic of animals possessing cognitive maps (O'Keefe & Nadel, 1978; Tolman, 1948), and cemented a longstanding assumption that cognitive maps resemble two-dimensional, view-from-above “survey maps” with globally consistent, Euclidean metric structure (Gallistel, 1990; Piaget & Inhelder, 1956; Siegel & White, 1975). However, the results of these experiments were cast into doubt when Gould’s (1986) report of novel shortcuts in honeybees could not be replicated, and it was found that landmark following (Dyer, 1991), path-integration (Wehner, et al., 1996), or “snapshot matching” (Cartwright & Collett, 1983; Wehner, et al., 1996) strategies could produce equivalent homing behavior if the environment is not sufficiently controlled. These early studies of wild animals may have been confounded 3 by the possibility that supposedly novel shortcuts are not really novel, as it can be difficult to establish that an animal has not traveled along a particular route prior to testing (Bennett, 1996). Navigation and wayfinding. In these studies, “navigation” or “wayfinding” behavior is used to make inferences about the geometry of spatial knowledge. Borrowing from Levitt & Lawton (1990), Trullier et al. (1997) proposed that the problem of navigation is best characterized by three questions: "(1) “Where am I?” (2) “Where are other places relative to me?” and (3) “How do I get to other places from here?” In robotics, the problem of self-localization (1) is often solved by attempting to determine position with respect to an external coordinate frame (see Thrun, 2002 for a review). However, this can be computationally expensive and, as Truillier et al. (1997) point out, simply recognizing a previously encountered place would obviate the need for external self-localization if one can simply replay the sequence of events that previously brought one to a goal. Because it is possible to navigate without a global spatial representation, Trullier et al. (1997) also distinguish between "local navigation skill," or the ability to move around within the sensory horizon, and "wayfinding," or the ability to move through a larger- scale environment when both the goal and relevant cues lie outside the sensory horizon. They then identify four types of navigation: (1) guidance by beacons, or local navigation (2) place recognition-triggered response, or "wayfinding without planning" (3) topological navigation, which is limited to known routes, and (4) metric navigation, 4 whose hallmark is the ability to take novel shortcuts that do not coincide with previously learned routes. To illustrate the strategies, consider an animal attempting to return to a distant goal (e.g. food) location. The animal can use a strategy of first aligning with a beacon (e.g. a visual landmark) remembered to be in the vicinity of the goal, and then simply steer towards the goal when it becomes visible. This process could be repeated for many intermediate subgoals across a much larger spatial scale, such that the animal never needs to know the precise spatial location of its ultimate goal—it can simply alternate between beacon aiming, and place recognition-triggered response, or "place-goal-action" associations. However, this strategy will fail as soon as the animal is unable to recognize its current location (e.g. due to a detour around a new obstacle) with respect to the next subgoal, and the animal will need to wander around until it finds a familiar place. Route knowledge. An animal could scale up this strategy of navigating by beacons and place recognition-triggered response to develop route knowledge ("place-goal-action- place" associations), and could merge these routes into a topological representation that would permit determination of paths and sub-paths to the goal (Trullier et al., 1997). One way of representing this topological knowledge is a place graph in which nodes correspond to familiar places, and edges represent the adjacency or connectivity relations among places. Graphs (Figure 1) can be undirected (if the edges have no orientation) or directed (if the relationship between nodes is asymmetric). Edges and nodes can be weighted with information (e.g., cost, distance, or similarity) about the relationships between neighboring edges and nodes. 5 Figure 1. Examples of graph structures. Graphs (Tutte, 1984) consist of nodes connected by edges. In an undirected graph, the relations between nodes are symmetric, whereas in a directed graph, nodes are asymmetric, and the edges are said to have an orientation. In a weighted graph, weights are attached to edges (e.g. distances). In a labeled graph, edges, nodes (vertices), or both may be labeled with metric information. For example, edges may be labeled (weighted) with metric information about the distances between nodes, while the nodes themselves may be labeled with metric information about the angles between adjacent edges. In a labeled and directed graph, labels may contain metrically inconsistent information (e.g. they may violate the triangle inequality, may include distance asymmetries, and may not be globally or locally consistent). A graph is said to be mixed if it contains both directed and undirected edges. The most limited form of knowledge would be pure route knowledge, consisting of place-action associations. This form of knowledge should be distinguished from graph knowledge which would permit the integration of paths intersecting at the same place. For example, assume that an animal has traveled through the following sequence of places (Figure 1, Undirected Graph): F-D-B-A-D-E. Next, the animal is transported from the endpoint of the route (E) back to the starting location (F), but is deprived (as in many 6 pigeon homing experiments) of any sensory input specifying an edge connecting F to E. Now suppose the animal must travel from F to E. On the basis of pure route knowledge, the animal would be unable to integrate over this self-intersecting sequence to determine that F-D-E is the most efficient route because two different actions are associated with D (D-B and D-E). In contrast, graph knowledge would enable F-D-E because multiple actions may be associated with a given node (D). The experimental results from the animal literature discussed in the foregoing sections demonstrate that in addition to integrating over previously traveled routes (to determine that F-D-E is a viable route from F to E), many animals are able to determine novel shortcuts (F-E) even after a single exposure to a given route. Thus, many animals, including humans, appear to acquire something beyond pure, unintegrated, goal- dependent route knowledge. While novel shortcuts could be supported by a global metric embedding—a metric cognitive map that embeds local information into a globally consistent coordinate frame (Hübner & Mallot, 2007; Thrun, 2008)—they do not necessarily imply one. Assuming an ability to integrate (e.g. through path integration) distance and angle information, novel shortcuts could also be supported by a labeled graph in which edges are labeled with information about distances between nodes, and the nodes are themselves labeled with information about the angular relationships among adjacent edges (Figure 1, Labeled & Directed Graph). Thus, path integration could support novel shortcuts via estimates of the local distances and angles in a graph, without a global metric embedding. 7 Path integration. In principle, path integration (Figure 2) could be used to build a not only a graph, but a comprehensive and globally consistent Euclidean cognitive map that accurately encodes metric distance and angle information (Gallistel, 1990; Gallistel & Cramer, 1996; Loomis, et al., 1999). Path integration refers to an animal’s ability to update their spatial position by tracking distances traveled and angles turned via temporal Figure 2. Path integration. Path integration refers to an animal’s ability to update their spatial position by tracking distances traveled and angles turned via temporal integration of sensory information about velocity or acceleration. Path integration could be used to update a homing vector to a starting location or compute the distance and direction between two locations, enabling novel shortcuts via vector subtraction. integration of sensory information about velocity or acceleration. Path integration could be used to update a homing vector to a starting location or compute the distance and direction between two locations, thereby enabling a novel shortcut via vector subtraction (Figure 2). For example, repeated experience traveling legs two legs (A and B) could support accurate determination of the third leg of the triangle (A-B). Desert ants have a highly developed ability to execute these homing vectors to their nest after traveling complex outbound routes to distant food sources (Muller & Wehner, 1988; Collett et al, 8 2001; Collett et al., 2003; Müller & Wehner, 2010), as do honey bees (Cartwright & Collett, 1983; Gould, 1986; Wehner et al., 1996), and homing pigeons (Wiltschko & Wiltschko, 1978). In contrast, human experiments using triangle completion (Kearns, Warren, Duchon, & Tarr, 2002; Kearns, 2003), blindfolded path integration (Loomis et al., 1993), and novel shortcut paradigms (Foo et al. 2005) suggest that the distance and angle information available through human path integration may be too coarse to use in building an accurate cognitive map. Suppose that a participant has repeatedly walked two legs of a triangle (Figure 2, A & B); in principle, path integration provides information that can support a novel shortcut between A and B via vector subtraction (A- B). Kearns et al. (2002) investigated the contributions of optic flow and idiothetic (path integration; motor, inertial, and proprioceptive cues) information to path integration using triangle completion, and found that optic flow can be used to perform coarse path integration given sufficient visual structure; however, information about self-motion (available through vestibular and proprioceptive information) tends to dominate homing behavior. Despite the availability of visual and idiothetic information, path integration exhibited large constant (M ≈ 58-63º) and variable (AD ≈ 24-31º) angular errors. Foo et al. (2005) employed a novel shortcut paradigm in which participants repeatedly traveled legs A and B, and again found large constant and variable angular errors in a desert environment (M ≈ 13-24º, AD ≈ 30-32º). Novel shortcuts (A-B) were more accurate (M ≈ 0-4º) when participants were provided with visual structure in the form of landmarks, while variable errors remained high (AD ≈ 18-30º). If the landmarks were shifted during 9 testing, shortcuts tended to shift with the landmarks unless landmark displacements were large, in which case participants fell back on path integration to perform the task. The Geometry of Spatial Knowledge Thus, the term “cognitive map” has accumulated considerable baggage, leading some (e.g. Bennett, 1996) to propose that it should be discarded from the literature to avoid bias and confusion. In its place, the term “spatial knowledge” is often used to convey the notion that there is a mapping between the “true” (Euclidean) environmental geometry and the spatial knowledge acquired through experience. This more neutral term acknowledges that the mapping may not necessarily preserve globally consistent, metric, Euclidean relations; rather, cognitive maps are frequently distorted, and not map-like in many respects. Non-Euclidean geometry. A global metric map is only one possible structure for spatial knowledge; a "cognitive map" may preserve geometrical relations weaker than Euclidean distance and angle. Felix Klein’s (1872; Coxeter, 1961; Suppes et al, 1989) hierarchy of geometries provides a framework for studying the geometry of spatial knowledge by arranging geometries along a continuum from strong to weak, with Euclidean geometry (the strongest) at one end, and pure topology (the weakest) at the other (Figure 3). Klein (1872) defined geometries in terms of transformations and the properties (curvature, distance, area, angle) those transformations leave invariant. Euclidean geometry is the strongest because the distance and angle relations among all points in the space remain invariant over the class of Euclidean transformations (translations and rotations). Progressively weaker geometries include similarity geometry, 10 Figure 3. The Klein hierarchy of geometries. Felix Klein’s (1872; Coxeter, 1961; Suppes et al, 1989) hierarchy of geometries provides a framework for studying the geometry of spatial knowledge by arranging geometries along a continuum from strong to weak, with Euclidean geometry (the strongest) at one end, and pure topology (the weakest) at the other. which preserves angles but allows size to vary; affine geometry, which preserves parallelism but alters distances and angles; projective geometry, which preserves collinearity but alters distances and angles; and topology, the weakest geometry in the hierarchy, which preserves only connectivity and adjacency relations among neighboring sets of points (Coxeter, 1989). Topology can be thought of as the class of “rubber sheet” transformations (stretching, bending, shearing) of a Euclidean plane that do not introduce discontinuities (e.g. holes or folds). A protracted historical failure to prove Euclid’s fifth (or parallel) postulate from his first four postulates led to the gradual discovery of non-Euclidean geometries, 11 including the more familiar elliptic and hyperbolic geometries (both of which Klein classified under projective geometry; Klein 1872; Greenberg, 2007). Euclid’s first 5 postulates are (quoted from Greenberg, 2007): (1) For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q. (2) For every segment AB and for every segment CD there exists a unique point E on line AB such that B is between A and E and segment CD is congruent to segment BE. (3) For every point O and every point A not equal to O, there exists a circle with center O and radius OA. (4) All right angles are congruent to one another. (5) For every line l and for every point P that does not lie on l, there exists a unique line m through P that is parallel to l. The parallel postulate (5) differs radically from postulates (1-4). While postulates (1-4) are readily demonstrated (for plane geometry) through pencil and paper constructions, numerous historical attempts to prove the parallel postulate (for plane geometry) failed because the postulate resists empirical validation: one can only draw finite line segments, and thus the behavior of infinitely extended line segments cannot be empirically confirmed (Greenberg, 2007). Because logical consistency in a mathematical system can be established independent of empirical validation, it is possible to negate the parallel postulate and derive other logically consistent geometries (Figure 4). Negating the parallel postulate to allow for an infinite number of lines through P that are parallel to l yields hyperbolic geometry; negating the parallel postulate such that all lines through P intersect l yields elliptic geometry; omitting the parallel postulate entirely yields absolute geometry, which reveals the properties shared by Euclidean, elliptic, and hyperbolic geometries (Coxeter, 1961; Greenberg, 2007). Imposing a metric on a (Euclidean or non-Euclidean) space permits measures of distance relations among the elements (e.g., points) of the set (Greenberg, 2007). The 12 Figure 4. Euclidean, elliptic, and hyperbolic geometries. Negating the parallel postulate to allow for an infinite number of lines through P that are parallel to l yields hyperbolic geometry; negating the parallel postulate such that all lines through P intersect l yields elliptic geometry; omitting the parallel postulate entirely yields absolute geometry, which reveals the properties shared by Euclidean, elliptic, and hyperbolic geometries (Coxeter, 1961; Greenberg, 2007). Euclidean metric postulates include (Figure 5; McNamara & Diwadkar, 1997; Beals, Krantz, & Tversky, 1968): 1. Positivity: the distance from any point to itself is zero and the distance between any two distinct points is greater than zero. 2. Symmetry: the distance between any two points is independent of the direction in which the distance is measured. 3. The Triangle Inequality: the distance between two points can be no greater than the sum of the distance between each of these points and a third point. 4. Additivity: any two points must be joined by a segment along which distances are additive. Because these postulates are specific to Euclidean metric spaces, non-Euclidean spaces can be arrived at in either of two ways: (1) by negating the parallel postulate, or (2) by relaxing any of the four Euclidean metric postulates. Non-Euclidean geometry may appear to bear little relation to the geometry of the real world: after all, geometry at the scale of human behavior is well described by three dimensional Euclidean geometry. Yet the discovery of non-Euclidean geometry was essential to fundamental innovations in physics. Einstein’s theory of special relativity and his notion of spacetime curvature rely heavily on hyperbolic geometry (Greene, 2000). For example, the paths taken by photons through space are not straight lines in the 13 Figure 5. The Euclidean metric postulates. Imposing a metric on a (Euclidean or non- Euclidean) space permits measures of distance relations among the elements (e.g., points) of the set (Greenberg, 2007). The Euclidean metric postulates include (McNamara & Diwadkar, 1997; Beals, Krantz, & Tversky, 1968) positivity, the triangle inequality, symmetry, and additivity. Because these postulates are specific to Euclidean metric spaces, non-Euclidean spaces can be arrived by negating the parallel postulate or by relaxing any of the four Euclidean metric postulates. Euclidean sense, but are influenced by the gravitational force of any object with mass. Here, the notion of straightness does not conform to our familiar notion of straight lines in a Euclidean space because the geodesics (straightest lines through the encompassing space) for photons are curved paths. As a result, the geometry of the environment at the scale of human behavior is sometimes described as “locally Euclidean” to account for the conjecture that the global geometry of the universe might be non-Euclidean (Greene, 2000; Weeks, 2002). The spherical model of elliptic geometry has many important applications in aviation: the shortest distance between two cities lies on a great circle. Projective geometry is fundamental to cartography: our two-dimensional maps of the Earth rely on projective transformations of the Earth’s spherical surface. These examples illustrate that these seemingly abstract mathematical constructs have concrete 14 applications in real world contexts, and underscore that there is—at least from a mathematical standpoint—nothing particularly special about Euclidean geometry. Environmental geometry. While the geometry of spatial knowledge need not conform to any particular geometry in Klein’s hierarchy, experimental research on spatial knowledge suggests several possibilities for understanding how spatial knowledge deviates from Euclidean metric structure. These possibilities are directly related to the three hypotheses investigated in the present experiments: (1) the Euclidean Hypothesis, Figure 6. Environmental geometry. (A) metric structure, which includes globally consistent information about distances and angles, (B) graph structure, which captures the connectivity of places, (C) neighborhood structure, which captures adjacency and inclusion relations among salient regions, and (D) circular and linear ordinal structure, which captures the order, betweenness, and sign of salient places along a route or with respect to a reference point. The present experiments focus on the acquisition of metric, neighborhood, and graph structure. which proposes that spatial knowledge is primarily Euclidean, (2) the Topological Hypothesis, which proposes that spatial knowledge is primarily topological and therefore “weaker” than Euclidean geometry in Klein’s sense, and (3) the Stability Hypothesis, which proposes that spatial knowledge does not have a constant structure at all, but preserves the geometric properties of the environment that remain stable during learning. To investigate these questions, we can apply Klein’s framework to the geometry of the environment, identifying specific subsets of geometric relations that human spatial 15 knowledge might preserve based on empirical research. We can then design experiments in which we selectively manipulate the stability of these geometric relations during learning, and probe their acquisition during testing. While spatial knowledge might preserve properties corresponding to any of the above geometries, the present studies are primarily focused on investigating whether spatial knowledge preserves Euclidean structure, topological structure, some combination of the two, or whatever structure remains stable during learning. The topological structure of naturalistic environments can be operationalized in several ways (Figure 6). Graph structure (B) captures the connectivity of salient features (e.g. places, landmarks, neighborhoods) without necessarily preserving distance and angle relations among features or along connecting routes. Neighborhood structure (C) captures adjacency and inclusion relations (e.g. neighborhood I is adjacent to II, and neighborhood II is between I and III) among salient regions. Regions may defined by the skeleton of major paths in a town, city, or a maze (Kuipers, Tecuci, & Stankiewicz, 2003), and formal descriptions of the possible spatial relations among regions have been established through Region Connection Calculus (Clarke, 1981; Randell & Cohn, 1989; Randell & Cohn, 1992). Circular ordinal structure (D) is the clockwise order in which features would be encountered as an observer turns about a fixed point in the environment, while linear ordinal structure (D) is the sequence, betweenness and sign (left or right) of features encountered along a route. Note that Euclidean metric structure (A) encompasses all of the weaker forms of geometric structure; that is, weaker structure 16 can be derived from globally consistent information about distances and angles (Thrun & Bücken, 1996). Violations of the metric postulates in spatial knowledge Although some researchers have proposed that survey knowledge develops with experience (Shemyakin, 1961; Siegel & White, 1975), many studies suggest that spatial knowledge does not preserve Euclidean metric structure. For example, the large majority of humans are not able to accurately integrate separately learned routes even after repeated exposure(Ishikawa & Montello, 2006). Violations of the metric postulates of symmetry (Holyoak & Mah, 1982; McNamara & Diwadkar, 1997; Steck & Mallot, 2000), the triangle inequality (Cohen, Baldwin, & Sherman, 1978; Kosslyn, Pick, & Fariello, 1974; Shemyakin, 1961; Wilton, 1979), and additivity (Byrne, 1979; Sadalla, Burroughs & Staplin, 1980; Sadalla & Staplin, 1980) have been found in studies of human spatial cognition. It is important to note that these studies used a variety of paradigms to obtain estimates, including map drawing (Shemyakin, 1961), paper and pencil judgments among remembered targets (Holyoak & Mah, 1982; McNamara & Diwadkar, 1997; Wilton, 1979), distance judgments among locations within a highly familiar area (e.g. a town; Kosslyn, Pick & Fariello, 1974; Sadalla & Staplin, 1980; Sadalla, Burroughs & Staplin, 1980) or along frequently traveled and familiar routes (Byrne, 1979). As a result, some results may not generalize to active human navigation tasks. Other research using navigation tasks in desktop and immersive VR provides evidence that apparently metric behavior may be based on topological navigation strategies (Byrne, 1979; Harrison & Warren, unpublished manuscript; Ishikawa & 17 Montello, 2006; Weiner & Mallot 2003, 2004), and studies of navigation in non- Euclidean environments demonstrate that successful navigation does not depend on Euclidean structure (Ericson & Warren, 2010, 2012; Rothman & Warren, 2006; Schnapp & Warren, 2007; Seyranian & D'Zmura, 2001; Warren et al., unpublished manuscript). This growing body of empirical evidence suggests that human spatial knowledge frequently violates the properties of Euclidean metric spaces. Human navigation may rely, instead, on weaker forms of geometric structure such as the place graph, ordinal structure, neighborhood structure. For example, apparently Euclidean navigation (e.g. novel shortcuts) could be supported by a strategy based on weaker spatial knowledge and a primarily topological strategy; for example, navigating to a target neighborhood and aligning one’s view to correspond to the remembered view of landmarks in the vicinity of the goal. Topological maps and strategies have proven successful in the context of robot navigation. For example, Thrun & Bücken (1996) compared nearly 24 million simulated routes using grid-based and topological maps in mobile robots. Route lengths generated by the topological approach were longer than those generated by the grid map by only 0.29 meters (1.82% of ideal path length) on average. In addition, planning based on a metric map was approximately 4.9 x 103 times more computationally expensive than planning with the topological map; planning on the topological level increases the efficiency by more than three orders of magnitude, while inducing a performance loss of only 1.82% (Thrun & Bücken, 1996). Although these shortcuts resemble routes taken through a maze rather than novel shortcuts, their results provide preliminary evidence a topological strategy can produce apparently metric behavior in mobile robot navigation. 18 Several violations of the metric symmetry postulate (Figure 5) can be found in the literature (McNamara & Diwadkar, 1997; Steck & Mallot, 2000). McNamara & Diwadkar (1997) used a paper and pencil paradigm in which participants memorized labeled points that had been organized into rough groups on a two dimensional map to investigate whether remembered distances satisfy the metric symmetry postulate and found that distance estimates between landmarks (place names printed in capital letters) and neighboring non-landmarks (names printed in lowercase letters) are asymmetric. They also found that the direction of the asymmetry depends on which of these two environmental features was used to establish the context for the distance estimate; specifically, estimates made from landmarks to non-landmarks were larger than estimates made from non-landmarks to landmarks if the landmark and non-landmarks probed in the estimate are located within the same region. Additionally, more place names were recalled when participants were prompted with landmark cues than when prompted with non-landmark cues, indicating an asymmetry in contextual recall of places. Although this study did not involve actual locomotion or path integration, these results suggest that spatial memory for layouts falling entirely within the sensory horizon is susceptible to considerable context-driven distortion. Although not a strict violation of the metric symmetry postulate, humans show asymmetries in their use of salient environmental features such as landmarks when learning and navigating a novel environment. Steck and Mallot (2000) used a paradigm in which participants explored a virtual town environment by making movement decisions that would result in virtual self-motion through the town. They found that when local and 19 global landmarks were placed in conflict, nearly all participants were able to make use of either type of landmark to support their navigational decisions, but participants generally preferred to use one or their other type. Some participants showed an asymmetry in their use of the two types of landmark based on their location in the environment, flexibly switching between local and global landmark strategies depending on their location. Violations of the triangle inequality (Figure 7) may be implied by work showing that distance judgments made between places separated by barriers such as walls, hills, or regional boundaries (Cohen, Baldwin, & Sherman, 1978; Kosslyn, Pick, & Fariello, 1974; Shemyakin, 1961), and between cities separated by large geographic distances (Wilton, 1979) are often exaggerated, as are paper-and-pencil estimates of inter-city distances made in the vicinity of salient reference points (Holyoak & Mah, 1982). Cohen et al. (1978) asked children (9 and 10 year olds) and adults to report the distances between 45 pairs of familiar targets (15 to 110 m apart) after 1 month of exposure in a summer camp by estimating the number of steps (relative to a 100 step standard) and by placing tiles on a board with no scale/dimensions indicated. Children and adults did not differ in their distance estimates, and hills or barriers (trees, buildings) consistently led to overestimates of the distance between targets where they were present, and underestimates where they were absent. The authors claim that ease of travel or “functional distance” was the critical factor influencing distance estimates. To visualize how the results of these studies might imply violations of the triangle inequality, assume that locations A, B and C form an equilateral triangle but A and B are located in Room 1, and C is located in a second, adjacent Room 2. If the AC distance 20 Figure 7. The triangle inequality. Violations of the triangle inequality may be implied by work showing that distance judgments made between places separated by barriers such as walls, hills, or regional boundaries (Cohen, Baldwin, & Sherman, 1978; Kosslyn, Pick, & Fariello, 1974; Shemyakin, 1961). For example, if the distance between A and C is sufficiently overestimated due to the presence of a barrier, this would imply a violation of the triangle inequality. estimate were sufficiently exaggerated relative to the AB estimate because of the intervening wall, AC would be “longer” than the sum of AB and BC, violating the triangle inequality. Even if this condition is not met in the strictest sense, the fact that participants overestimate cross-boundary distances relative to within-boundary distances supports the notion that human spatial knowledge may not preserve accurate distance and angle relations, including those defining relatively simple geometric shapes such as triangles. Despite these violations of the metric postulates, many researchers still adhere to Siegel and White’s (1975) proposal that spatial knowledge proceeds from topological to metric Euclidean survey maps following increased experience and familiarity. Specifically, Siegel and White propose a progression from landmark knowledge, to non- metric (ordinal) route knowledge about landmark sequences, to comprehensive metric Euclidean survey knowledge. This is often referred to as the landmark, route, survey (LRS) model. Shemyakin (1961) observed that when 6-8 year old children draw maps of their neighborhood they turn the paper while tracing out familiar routes, rather than 21 drawing a full survey map from an arbitrary viewpoint. Shemyakin also found that older children do not need to turn the paper when drawing routes, which suggests that they have learned the layout in more allocentric terms. Anooshian & Wilson (1977) found that children (but not adults) overestimated the distances between objects in a table-top layout when objects were connected by circuitous tracks as opposed to direct tracks. In a study of larger scale space (a familiar neighborhood, approximately 4 km2), Anooshian & Young (1981) found that the absolute accuracy of children’s angular pointing estimates to prominent landmarks improved considerably (although errors were high overall for all groups) from 1st/2nd to 4th/5th grade, less from 4th/5th to 7th/8th grade, and pointing consistency (i.e. precision) increased with age. These results appear to be consistent with the suggestion that metric survey knowledge is acquired gradually over the course of development (Piaget & Inhelder, 1956; Siegel & White, 1975). Ishikawa and Montello (2006) tested the LRS model by having participants learn the locations of eight landmarks on two connected but independent routes. They found that participants acquired both ordinal and coarse metric knowledge about landmarks following a single exposure to a novel environment, contrary to Siegel & White’s (1975) hypothesis that landmark knowledge is initially unscaled and purely ordinal. Mean absolute errors for between-route direction estimates were as large as 45-50°, and distance correlations between estimated and actual distances were at chance levels, even after seven experimental sessions. Although direction estimates for 8 out of 24 participants improved by 17.5° on average, and distance correlations improved from 0.1 to 0.4 (Pearson’s r), this improvement does not appear to reflect a progression from 22 purely topological to fully metric spatial knowledge. In addition, 8% of participants failed to improve at all, and only 50% of those at intermediate levels of performance estimated directions as well as the most accurate participants had from the beginning (absolute errors less than 30º). This reveals that there are often large individual differences in the acquisition and geometry of spatial knowledge in humans, and casts doubt on Siegel and White’s (1975) proposal that spatial knowledge proceeds from a purely topological to an accurate metric cognitive map following increased experience and familiarity. Another prediction of Siegel and White’s (1975) LRS model is that survey knowledge should be acquired with sufficient experience regardless of environmental complexity. However, Moeser (1988) found that nurses with over two years of experience in a hospital with an irregular layout made poor angular (mean absolute error, 21.75°) and distance (mean error, 173 feet) estimates among frequently visited rooms, and did not integrate their route knowledge into a globally consistent cognitive map of the hospital. It is important to point out that the nurses had to rely primarily on coarse information from path integration and route knowledge because short corridors, fire doors, and a general lack of windows prevented visual access to external landmarks; so contrary to Gallistel (1990) and others, these results imply that global survey maps may not be built up through path integration.. Control subjects who were allowed to look at hospital floor plans made significantly better distance and angle estimates, suggesting that global survey maps may depend on experience with two dimensional view-from-above maps (Moeser, 1988). While these results do not permit inferences about violations of 23 particular metric postulates, it is clear that the nurses could navigate successfully in the hospital without relying on Euclidean metric information. Other work suggests violations of the metric additivity postulate, as route properties sometimes influence distance estimates (Byrne, 1979; Sadalla & Staplin, 1980; Sadalla, Burroughs & Staplin, 1980). Sadalla & Staplin (1980) had participants estimate distances in two conditions: (a) a laboratory environment with equidistant paths that differed in the number of junctions along their length, and (b) a field study in which participants estimated distances from a mall to two equidistant locations with differing numbers of intersections. Participants in the laboratory condition (a) overestimated the length of routes with more intersections despite traveling along all routes in the same amount of time; participants in condition (b) showed the same pattern of distance estimation errors despite familiarity with the routes and locations. These results suggest that spatial knowledge of real-world environments might tolerate violations of additivity postulate (Figure 5). Byrne (1979) used ratio scaled distance estimates of familiar routes and found that routes with several bends and routes lying within a town center are overestimated, and that short routes are generally overestimated relative to long routes. Furthermore, when participants traveled along paths joined at up to 120º they tended to orthogonalize angles to 90º. If these results generalize to more complex real world environments consisting of several interconnected routes they may be seen as evidence of violations of metric postulates (symmetry and additivity), suggesting that the number of junctions and turns along routes coupled with the warping of perceptual space by the presence of landmarks 24 (McNamara & Diwadkar, 1997) may lead to spatial representations that do not depend on Euclidean metric structure. Evidence for topological navigation strategies. While these studies provide evidence against purely Euclidean representations, several others (Byrne, 1979; Ishikawa & Montello, 2006; Weiner & Mallot 2003, 2004; Harrison & Warren, under review) suggest that successful navigation can be accounted for by reliance on weaker geometric structure resembling a topological graph. Weiner and Mallot (2003; 2004) found results consistent with Byrne’s (1979) suggestion that rather than building up a “vector-map” that preserves accurate distance and angle information, humans build up a “network- map” that preserves only the connectivity and ordinal relationships among locations and turns. They report evidence that human spatial knowledge includes explicit representations of the topological relations among environmental regions to be used in route planning, and that the connectivity of these regions might be preserved in a graph structure that does not perfectly preserve the metric relations among nodes. They had participants navigate through a virtual environment consisting of twelve distinct locations in a hexagonal arrangement, connected by a network of streets. Regions were defined by landmarks falling into three different semantic categories: cars, animals, and buildings. During testing, participants were given the option of traveling either of two routes, both of which were equivalent in terms of metric distance, but which differed in the number of region boundaries that would be crossed in the course of executing the route. Of the subjects that recognized that distinct regions were present in the environment, more than 25 half chose routes that crossed fewer region boundaries, indicating that knowledge of regional boundaries does indeed influence route planning. In a follow-up experiment (Weiner & Mallot, 2003) employing islands that defined regions, they found that participants will chose routes that permit faster access to the target region. This led them to propose that navigation involves "hierarchical planning,” that is, places are grouped into regions and navigational decisions are then made based on the topological relations among these regions; however, they acknowledge that the regions in their experiment were very clearly defined which is rarely the case in real-world environments where regions may even overlap (Weiner & Mallot, 2003). While this does not constitute definitive evidence for topological representations, these results do indicate that humans use connectivity relations in conjunction with coarse representations of distance to inform navigational decisions. Support for a topological strategy also comes from Harrison & Warren (under review) who found that participants rely on the ordinal relations among paths when the metric structure of the environment varies. Participants first learned the locations of several objects located at the ends of paths in a virtual hedge maze and were subsequently asked to walk through the maze to the remembered location of a target object. In experiment 1, the maze was stretched on probe trials, altering metric structure while preserving the ordinal relations among paths; they found that participants walked to the ordinally correct (rather than metrically correct) target locations on nearly all trials. In experiment 2, paths were added or deleted on probe trials, altering the ordinal relations among paths while leaving the metric structure roughly constant. On test trials 26 participants did not strictly rely on either a metric or ordinal strategy (ordinal on 43% of probe trials when paths were deleted, and 62% of probe trials when paths were added), indicating that participants do not necessarily depend upon a metric strategy even when metric structure is preserved. When local visual landmarks (e.g. paintings) were added near junctions and shifted by one junction during testing (Zhong et al., 2005a, 2005b), performance was primarily consistent with an ordinal (70-75% of trials) rather than a landmark (18-25% of trials) strategy. Applying an affine transformation to surrounding visual landmarks during a novel shortcut task (Zhong et al., 2007) led shortcuts to shift in the direction of the centroid of the surrounding landmarks rather than minimizing the difference between the configuration they learned and the configuration encountered during testing. The results of this series of experiments imply that participants acquire and rely upon the ordinal structure of paths and landmarks rather than metric distance and angle; if the ordinal structure becomes unreliable, participants tend to fall back on local distance and angle information. There is also some evidence from neurological research supporting the notion that topological structure is employed in navigation. Poucet (1993) has suggested that the parietal cortex might handle metric information while the hippocampal formation might handle topological information. Studies on taxi drivers (Maguire, Frackowiak & Frith, 1997; Maguire et al., 2000) implicate the hippocampus in memory for spatial layouts that have been engaged with over long time spans, and the medial and right inferior parietal cortex in ordinal or sequential information. Evidence from the animal navigation literature suggests that rats do not form global Euclidean cognitive maps, even when they 27 are given a great deal of practice in relatively small-scale environments. Benhamou (1996) found that rats search at random when they must find a hidden platform with a fixed location (with respect to the external environment) in a water maze when the entry to the maze is rotated between test trials. This result was seen despite the presence of extra-maze landmarks that should, in principle, provide enough information to build a survey map of the environment and perform the navigational task accurately. Navigation in non-Euclidean virtual environments. Asking participants to learn a non-Euclidean environment that violates the Euclidean metric postulates provides a strong test of the Euclidean survey map hypothesis (Gallistel, 1990; Piaget & Inhelder, 1956; Siegel & White, 1975), which predicts that if the navigation system attempts to construct a Euclidean cognitive map, spatial learning should be derailed by violations of Euclidean metric structure. That is, a non-Euclidean environment should be harder to learn and participants should make more navigational errors than in a matched Euclidean control environment. If participants are able to learn and navigate successfully in a non- Euclidean environment in which global Euclidean structure has been made unstable, this would imply that navigators ordinarily acquire weaker geometric structure (e.g. neighborhood, place, or landmark topology). Seyranian & D'Zmura (2001) used a desktop VR paradigm to investigate human navigation in a 4-dimensional (4D) spaceship. Participants used the mouse to control position, orientation, and 4D turns that moved participants through continuous sequences of 3D cross sections within the encompassing 4D space. Turns in the 4th spatial dimension were referred to as "nim" and "bor" instead of left and right. Orientation could 28 also be manipulated via "zaw," or rotation in the ZW plane (where the W-axis was orthogonal to the X, Y, and Z axes). Participants learned 3D and 4D environments with matched graph structures in desktop VR. On each trial, participants had to find an object and return to home as quickly as possible. They found that the time required to return to home approached optimality over the course of 28 trials in the 4D environments, whereas return times were nearly optimal from the beginning of testing in 3D environments. These results were later corroborated by Ambinder et al. (2009), who found that participants were able to make accurate judgments about the distance (R2 = .80, .66, .54, . 90) and angle (R2 = .83, .85, .54, .77) relations of 4D objects viewed as successive 3D cross sections in a cave automatic virtual environment (CAVE). Seyranian & D’Zmura (2001) interpret their results as evidence that the initial difficulty navigating the 4D spaceship environment is due to the extra spatial dimension, but that this difficulty can be overcome with experience. A potential shortcoming of the study is that artifacts like texture streaking may have provided orientation cues when making 4D turns, explicitly revealing violations of Euclidean structure and thereby providing cues to 4D structure that would not available if the participants were actually embedded in a 4D space. Also, participants couldn't turn left/right and nim/bor at the same time; they could move forward/backward and nim/bor, but the environment would appear to undergo a non-rigid deformation, providing another explicit cue to the 4D manipulation. The authors also acknowledge that the layout of the environment (a series of rooms connected by doors and hallways) may have forced participants to rely on local landmark strategies rather than more global cognitive maps of the environment. It is also 29 not clear whether the transition between successive 3D cross sections of the 4D space took the same amount of time to complete as corresponding movements through the matched 3D space. As Harrison et al. (under review) have noted, most of the evidence regarding the geometry of cognitive maps has been derived from pointing tasks, distance estimates, sketch maps, or desktop VR. These paradigms do not provide participants with many of the idiothetic (path integration; motor, inertial, and proprioceptive) cues that provide information about the environmental layout. This limits the ecological validity of these studies because participants are not able to employ the idiothetic and proprioceptive information that is normally available when actively walking, and must rely instead on visual information in the absence of cues for actively generated self-motion during both encoding and retrieval. Chrastil & Warren (2013) have shown that podokinetic (motor/ proprioceptive) information during walking significantly improves the accuracy of novel shortcuts, although mean absolute direction errors are still on the order of 70˚. The present series of dissertation experiments builds on several recent studies (Rothman & Warren, 2006; Schnapp & Warren, 2007; Warren et al., under revision) comparing the spatial knowledge acquired in paired Euclidean and non-Euclidean virtual environments with matched graph structures, using immersive VR. In these studies, the geometry of the Control Maze paralleled real world geometry while the Wormhole Maze violated Euclidean structure by making the distance and direction between any two objects in the maze dependent on the path taken. Unlike the Seyranian & D’Zmura (2001) study, any visual cues that might explicitly reveal violations of Euclidean structure 30 (e.g. sudden texture changes) were eliminated. The connectivity (place graph) among places was identical in both mazes; however, the Wormhole Maze violated the four metric postulates. First, positivity was violated because the distance between any point and itself was non-zero: several objects in the maze occupied multiple locations in the physical room depending upon the rotation of the maze at a given time. Second, symmetry and additivity were violated because the distance between two objects depended on the direction taken through the maze. Finally, the triangle inequality was violated because the distance from an object A to object B (which occupies both a “Euclidean” location, B, or “wormhole” location, B1) is simultaneously greater and less than the distance from A to another reference object, C. Rothman and Warren (2006) had two groups of participants walk between pairs of objects through the maze and found that they took significantly more “wormhole” routes in the non-Euclidean maze than in the Euclidean Maze. Contrary to the Euclidean cognitive map prediction, both mazes were equally easy to learn and participants in both mazes reported that some of the paths seemed confusing. In addition, participants who learned the wormhole maze did not report noticing the substantial violations of Euclidean structure created by the wormholes. To probe participants’ spatial knowledge, Schnapp and Warren (2007) employed a novel shortcut task in which participants walked from Home to a starting object (A) whereupon the walls of the maze disappeared and they were told to turn in place and walk directly, “as the crow flies,” to a target object (B). They found that the Euclidean Maze group walked towards the Euclidean target, on average, despite large variable errors, whereas the Wormhole Maze group walked in the 31 wormhole direction, on average. Comparisons of probe and control trials suggested that participants may walk in a similar direction to two targets that are not neighbors in the maze, indicating that spatial knowledge tolerates radical violations of Euclidean structure such as “rips and folds” in space. Perhaps if participants were provided with sufficient information to reveal violations of Euclidean structure in the Wormhole Maze, they would detect geometric inconsistencies and their shortcuts would either become more variable or would shift towards the Euclidean location of the target object on probe trials. To investigate this, several follow-up experiments added external landmarks (Ericson & Warren, 2010, 2012), and cast shadows (Ericson & Warren, 2012) to the Wormhole Maze. Despite adding cast shadows and external landmarks, participants were largely unaware of violations of Euclidean structure, and still used the wormholes when taking shortcuts. Collectively, these experiments suggest that the spatial knowledge used for navigation does not depend on Euclidean metric structure. But they raise the possibility that spatial knowledge reflects whatever geometric properties are stable during learning. For example, when metric structure is violated in the wormhole mazes, successful navigation can still be supported by acquiring knowledge of the place graph; this primarily topological knowledge might be supplemented with local information about distance and angle, and on-line perceptual information from landmarks and other salient local environmental features (Harrison & Warren, under review; Warren et al., under revision). These results are also be compatible with the proposal that spatial knowledge has a hierarchical organization (Hirtle & Jonides, 1985; Montello, 1992), perhaps 32 consisting of a “collage” of representations (Tversky, 1993), each characterized by different geometries and spatial scales (Anooshian, 1996; Montello, 1992). The present experiments This dissertation critically examines the geometric structure of “cognitive maps” by evaluating three hypotheses about the structure of spatial knowledge: the Euclidean Hypothesis, which claims that spatial knowledge has a primarily constant Euclidean metric structure; the Topological Hypothesis, which claims that spatial knowledge has a primarily topological structure, and the Stability Hypothesis, which claims that spatial knowledge reflects the specific geometric properties that remain stable during learning. To test these hypotheses, the present experiments selectively destabilized three kinds of geometric properties in a virtual hedge maze during learning: Euclidean metric structure, neighborhoods, and graph structure (Figure 7). The Control Maze preserved all three kinds of structure under investigation. Elastic Maze I preserved the place graph while making Euclidean and neighborhood structure unstable by stretching paths so that a target occupied two different metric locations in two different neighborhoods during learning. Elastic Maze I provided metric (via path integration) but not visual information about neighborhood boundary relations. Elastic Maze II modified Elastic Maze I to provide visual information about neighborhood boundary relations by connecting stretched paths to the major paths that they crossed. Finally, the Swap Maze preserved neighborhood structure while destabilizing metric structure and the place graph by swapping pairs of targets within a neighborhood during learning. 33 Figure 8. Full experimental design. A total of 10 conditions were included in the full experimental design. N=12 participants (6 male, 6 female) were included in final analyses for each condition (N = 120 total). Virtual environments were designed to selective destabilize metric structure, neighborhood structure, and the graph. Strike-through text denotes geometric structure that was unstable during learning. In Elastic Maze I, neighborhoods were preserved visually but not via path integration; in Elastic maze II, information about neighborhood boundary relations was provided by connecting stretched paths to the major paths they crossed. Xs indicate conditions omitted from the design. Tasks were designed to probe the acquisition of each kind of structure. Experiment 1 compared performance in the Control Maze and Elastic Maze I on metric shortcut, neighborhood shortcut, and Route Tasks. Experiment 2 compared performance in Elastic Maze I and Elastic Maze II environments on neighborhood shortcut and Route Tasks. Experiment 3 compared performance in the Euclidean and Swap mazes on neighborhood shortcut and Route Tasks. In the Shortcut Task, participants took as-the-crow flies shortcuts to targets. The Neighborhood Shortcut Task was the same with the addition of outlines of major paths on the ground plane during shortcuts. The Route Task asked participants to walk through the maze and down infinite hallways to the remembered location of targets. Highly schematized examples of predictions for shortcuts in each environment and task are shown as arrows in each maze diagram. 34 By asking participants to learn and navigate these environments, the present experiments evaluate several specific predictions about the structure of spatial knowledge: (1) if spatial knowledge has a primarily constant Euclidean metric structure,then performance on all tasks should deteriorate when metric structure is unstable during learning, (2) if spatial knowledge is primarily topological, performance should reflect preserved topological structure even when metric structure is unstable, and (3) if spatial knowledge adapts to whatever geometric structure remains stable during learning, performance should always reflect stable structure. 35 Chapter 2: Experiment 1 Experiment 1 employed a 2 (environments) x 3 (tasks) design to investigate the relationship between metric structure, neighborhoods, and the place graph when metric and neighborhood structure were either stable (Control Maze environment) or unstable (Elastic Maze I environment) during learning (see Experiment 1 Methods below). Three tasks were used to assess the spatial knowledge acquired in the two virtual environments: (1) the Shortcut Task, which asks participants to take as-the-crow flies shortcuts between objects while only a ground plane is visible; (2) the Neighborhood Shortcut Task, which modifies the Shortcut Task to show neighborhood boundaries (the major paths of the maze) on the ground during shortcuts; and (3) the Route Task, which asks participants to walk through the maze to the correct target hallway, and down a visually infinite corridor to the remembered location of a target. Elastic Maze I was designed to destabilize Euclidean structure and neighborhood structure during learning while preserving the place graph. This leads to several predictions for each task. Shortcut Task. Prediction 1: If spatial knowledge is primarily Euclidean, then destabilizing Euclidean structure should make shortcuts less accurate or more variable in Elastic Maze I than in the Control Maze. Alternatively, shortcuts may be poor in both conditions, implying that spatial knowledge is either highly inaccurate or not Euclidean. Prediction 2: If neighborhoods are derived from Euclidean structure, then destabilizing the metric information for object location should make shortcuts more bimodal (in both the long and short neighborhoods) in Elastic Maze I than shortcuts in the Control Maze. 36 Neighborhood shortcut task. Based on previous research (Zhong et al., 2006a), providing information about neighborhood boundary relations by making major paths visible during testing should lead shortcuts to be less variable overall in both the Euclidean and Elastic I mazes. Prediction 3: If neighborhoods are based on topological boundaries, neighborhood shortcuts in Elastic Maze I should be less variable and more unimodal (i.e. in the short neighborhood) than the metric shortcuts in Elastic Maze I because participants did not cross a boundary on the stretched path to reach the object during learning. Route Task. Both the Control Maze and Elastic Maze I preserved graph structure. Elastic Maze I destabilized metric information and the resulting neighborhood structure. This leads to two main predictions for the Route Task. Prediction 4: If graph knowledge is derived from Euclidean or neighborhood structure, participants may walk down various paths in Elastic Maze I because they are uncertain about the object’s location. Prediction 5: If spatial knowledge is primarily graph-like (topological), then participants should walk down the correct path in Elastic Maze I despite unstable metric and neighborhood structure. Methods Participants Participants were recruited through advertisements and were compensated for their time at a rate of $10/hour. All participants signed forms indicating their informed consent to participate in accordance with the requirements of the Brown University IRB. A total of 91 people (38 female) participated in Experiment 1. Of these participants, 72 37 (36 female) completed the study and were included in the final analysis. Each condition consisted of 12 participants (6 female). Mean age of participants who completed the study was 20.7 (SD = 3.9). 8 other participants (2 female) withdrew due to symptoms of simulator sickness. The number of participants who dropped out due to simulator sickness in each condition was: Control/Metric, 2; Control/Neighborhood, 2; Control/ Route, 2; Elastic/Metric, 0; Elastic/Neighborhood, 0; Elastic/Route, 2. Two additional participants (Control/Metric, 1 male; Control/Route, 1 male) were excluded for failing to find all of the objects twice within the 25 minute verification phase time limit. 9 participants (4 female) were excluded due to technical problems in the first session. Equipment All experiments were conducted in Brown University's Virtual Environment Navigation Laboratory (VENLab). If participants went outside a 10.5m x 12.5m area Figure 9. The Virtual Environment Navigation Laboratory (VENLab). The useable area of the lab was 10.5 x 12.5m. An Intersense IS-900 tracking system collected position data (x, y , z, time) at a rate of 60 Hz. Displays were rendered using a Dell XPS graphics machine and presented at 1280x1024 pixels using a Rockwell-Collins SR80 head- mounted display (HMD) with a 63º (horizontal) x 53º (vertical) field of view. A “wrangler” followed the participant while they walked in the virtual environment so that participants would not have to carry the power supplies and wireless receivers for the HMD. during the experiment, virtual brick walls appeared approximately 1m in front of them, coincident with the physical walls of the lab (Figure 9). Stereo images of virtual 38 environments were presented using a Rockwell-Collins SR80 head-mounted display (HMD), with a resolution of 1280x1024 pixels, and a 63° (horizonal) x 53° (vertical) field of view for each eye (Figure 9). The distance between the HMD's lenses and the computed stereo disparity between the two images was calibrated for each participant based on their measured inter-ocular distance (IOD). An InterSense IS-900 tracking system (50ms latency, 1.5mm/ 0.10° spatial resolution) was used to track the participant’s head position in the lab. Virtual environments were generated using 3DS Max 2011-2012, and were displayed using Vizard 4.0 (WorldViz) software running on a Dell XPS 730X Desktop (Intel® CoreTM CPU 965 @ 3.2 GHz, 12GB RAM, 64 bit OS, Windows Vista Ultimate SP1), equipped with an NVIDIA GeForce GTX 280 graphics card. Participants carried a wireless USB mouse to provide responses between phases, and during test phases. Naturalistic background noise (e.g. crickets) was played throughout the experiment via wireless headphones to mask sounds that could potentially provide position or orientation information. Design Experiment 1 included 6 groups of participants in a 3 x 2 (task x environment) design, with three tasks (Shortcut, Neighborhood Shortcut, Route) and two environments (Control Maze, Elastic Maze). A given participant experienced only one virtual environment and one response task (Figure 10). Virtual Environments Each environment was a virtual hedge maze that contained 10 distinctive objects (Figures 10 and 11) that were not visible from the main corridors, but were only visible at 39 Figure 10. Design of Experiment 1. 3 tasks (Metric, Neighborhood Shortcut, Place Graph) were crossed with 2 virtual environments (Control Maze, Elastic Maze). the ends of hallways that branched off from the main paths. Each maze also contained 4 paintings (van Gogh, Monet, van Eyck, Dali) located at eye height, which served as local landmarks that could be seen from the main corridors of the maze; paintings were not placed randomly, but were deliberately placed so as to be visible in the main corridors. Object locations were held constant across environments except for probe objects, which alternated between normal and stretched locations in the Elastic Maze I (Figure 11). 40 Figure 11. Mazes and displays used in Experiment 1. Each maze contained 10 distinctive objects, and four paintings. Objects were designated control (Well, Sink, Cactus, Rabbit, Flamingo, Earth) objects if they remained in the same location in both environments, and probe (Bookcase, Clock, Moon, Gear) objects if their paths were alternately stretched in the Elastic Maze. Some probe objects were stretched across neighborhood boundaries (moon, gear), while other probe objects were stretched within neighborhood boundaries (clock, bookcase). Overhead views appear in the top left corner of each screen capture and are included for illustrative purposes only—they were not visible to participants during the experiment. p1-p4 indicate the locations of paintings which served as local landmarks. The geometry of the Control Maze was not manipulated experimentally. This environment was meant to mirror the geometry of the real world, and to establish baseline performance so that comparisons could be made among performance measures 41 in virtual environments and probe tasks. In the Elastic Maze I, the paths associated with the four probe objects (Bookcase, Clock, Gear, Moon) alternately appeared in normal (i.e. as in the Control Maze) or stretched (by 3m, 9.84ft) locations on alternate trials during the free exploration and verification phases. The alternation between normal and stretched paths was triggered by invisible gates that would track when participants had entered each probe path and would load the appropriate hallway before the object came into view. Participants never witnessed the alternation at the moment it occurred, and the transitions were visually seamless: all of the textural elements (e.g. on the wall and floors) in the maze were aligned, and the duration of the change was 1/60th of a second (1 frame). Thus, in order to notice that probe hallways were stretched on half of the probe trials, participants would either need to (1) integrate the distance traversed to the target via path integration, or (2) visually perceive the distance of the target on the probe path. It is important to note that visual distance estimates may suffer from depth compression of more than 50% in the saggital plane (Wagner, 1985); however, blindfolded walking does not appear to be subject to the same compression in natural environments (Loomis, Da Silva, Philbeck, and Fukusima, 1996), and is generally accurate out to 12m (Elliott, 1986; 1987), which roughly matches the dimensions of lab space (10.5m x 12.5m) used for the present experiments. Although blindfolded walking has been shown to suffer depth compression in an HMD, this effect is abolished after a few minutes of walking around in the virtual environment (Mohler et al., 2004; Richardson & Waller, 2007). Whether paths were initially seen as normal or stretched was randomized across participants and probe objects. For example, for some 42 participants, the moon would first be seen in a stretched location, overlapping the sink location; the next time a participant walked down the moon corridor, they would see the moon in its normal location, and so on. Based on the starting locations for shortcuts in the test phase, the actual angles between normal and stretched locations were: Moon, 40.34º; Gear, 23.01º; Clock, 15.17º, Bookcase: 37.21º. The mean of these 4 angles was 28.94º (AD = 10.27º). For objects whose paths were stretched within neighborhoods (clock, bookcase), the mean angle was 26.19º (AD = 11º); for those whose paths were stretched across neighborhoods (moon, gear), the mean angle was 31.68º (AD = 8.65º). Procedure Each participant completed two sessions total (Table 2). The first session generally lasted approximately 1 hour, and included Free Exploration (12 minutes), Verification (25 minute maximum), and Test (18 trials) phases. The second session generally lasted approximately 1.5 - 1.75 hours total, and included Verification (25 minute maximum) and Test (54 trials) phases, followed by 25-30 minutes of Debriefing tasks (see Debriefing). At the start of each session, participants were informed of the risks involved in the study (e.g. headache, motion sickness, nausea) and told that they could take a short break at any time or stop entirely and still be paid if they did not wish to continue (see Appendix). Approximately once every 10 minutes, the experimenters asked participants if they were feeling well enough to continue the experiment, or if they would like to take a short break. 43 Session 1 Session 2 (1 hour) (1.5-1.75 hours) Free Exploration (12 mins) Verification (25 mins max) Phases Verification (25 mins max) Test (54 trials) Sketch Maps, Test (18 trials) Questionnaire, SBSOD, RMT, PTSOT Table 1. Procedure for Experiment 1. Each participant attended two experimental sessions. In the first session, participants freely explored the maze, were trained to a criterion of finding each object 2 times from home (and were excluded if they failed to do so within 25 minutes), and underwent 18 (out of 72 total) randomized test trials. In the second session, participants were again trained to criterion, underwent 54 test trials, and then completed a series of questionnaires and spatial ability tasks (see Appendix). Free exploration phase. At the start of free exploration, participants were brought to the home location (Figure 11) and listened to the instructions (Appendix A). Participants were instructed that they would have 12 minutes to find all of the objects in the maze and learn their locations, and that they would be tested on their knowledge of the objects and their locations later on. Verification phase. The verification phase was designed to accomplish several objectives: (1) to ensure that all participants knew how to find each of the objects from the home location, thereby providing sufficient information for them to make novel shortcuts between objects via path integration, and (2) to provide explicit criteria for excluding participants who could not find the objects quickly enough to complete a reasonable number of test trials across the two sessions. Participants experienced the verification phase two times: (1) in the first session following free exploration, and (2) at the start of the second session, to refresh their knowledge of the maze prior to the second set of test trials. 44 Participants were required to successfully find each of the 10 objects two times from home, where success was defined as finding the target object from home in 30 seconds or less. If a participant found the target object between 30 and 45 seconds, they were asked to find the object again later in the verification phase. If the participant was not able to find the target object within 45 seconds, they were instructed to return to home, and the experimenter then guided them to the object along the most direct route; they were then asked to find the object again later in the verification phase. Participants who took longer than 25 minutes to complete the verification phase were excluded from all analyses. Test Phase. Test phases were designed to assess the spatial knowledge acquired during learning by probing participants’ acquisition of Euclidean metric structure (Shortcut Task), neighborhood structure (Neighborhood Shortcut Task), and the place graph (Route Task). To ensure that test phase data would reflect the influence of the experimental manipulations during learning, tasks were designed to be consistent across environments. The object pairs were subdivided into control (sink→earth, rabbit→well, earth→sink, well→flamingo) and probe (sink→bookcase, rabbit→gear, earth→moon, well→clock) trial pairs. On control trials, participants were asked to walk to targets that were not directly affected by the experimental manipulation (e.g. stretching in Elastic Maze I). On probe trials, they were asked to walk to targets that were directly affected by the manipulation. All trials in the Control Maze were control trials because the geometry of the environment was not manipulated during learning. By convention, the first object 45 Figure 12. First person view of shortcuts by task. Participants’ view and overhead views of the virtual environments during shortcuts between objects in the Shortcut Task, Neighborhood Shortcut Task, and Route Task. In the Shortcut Task, the maze disappeared upon arriving at object A, leaving only a gravel ground plane and sky visible during the shortcut to the remembered location of the target. In the Neighborhood Shortcut Task, the maze also disappeared upon arriving at object A, whereupon participants saw the outlines of the major paths of the maze overlaid on the ground plane during the shortcut. In the Route Task, participants walked through the maze to the remembered location of the target and saw an “infinite” (~300m long) hallway rather than the target (the appropriate infinite hallway was displayed even if participants chose the wrong hallway). 46 in any pair will be referred to as object A or the start object, and the second as object B or the target object. Object pairs were kept constant across all experimental conditions, and the order of trials was randomized for each participant. The Shortcut Task was designed to probe acquisition of Euclidean metric structure (Figure 12). On each trial, a participant walked from the Home location to object A, whereupon the maze walls, objects, paintings, and paths disappeared, leaving only a textured ground plane. They were then instructed “now turn and face the [target], then click the mouse”; they then turned in place and clicked the mouse to indicate that they had finished turning to align with the Target object. Next, they were instructed “now walk to the [target] and click the mouse again,”; they then walked in a straight line to the remembered location of the Target object and clicked the mouse to end their shortcut. Finally, to prevent feedback about their final position relative to the home location, the experimenter wheeled them back to the home location in a wheelchair along a circuitous route. The Neighborhood Shortcut Task (Figure 12) was designed to probe acquisition of neighborhood structure based on metric information from path integration alone (Elastic Maze I). The Neighborhood Shortcut Task was the same as the Shortcut Task, except that the outlines of the major paths remained visible on the ground plane during shortcuts after the maze walls disappeared, so that neighborhood boundaries were visible during the shortcuts. Thus, if participants learn which neighborhood contains the target object, shortcut accuracy should be improved. Moreover, if neighborhood locations are based on 47 Figure 13. Predictions for probe and control trials in Experiment 1. Sample predictions for shortcut performance are indicated by black arrows between objects for one of the four control trials (earth → sink) and one of the four probe (earth → moon) trials for Experiment 1. metric information, we would expect shortcuts to probe targets in Elastic Maze I to shift in the stretched direction. In the Route Task, participants walked from Home to Object A, whereupon all of the objects disappeared, but the maze walls remained visible (Figure 12). Participants then walked within the maze to the remembered location of the target object. If the participant entered a terminal path that had previously contained an object, the corresponding hallway was rendered as a visually infinite (~300m) corridor; this was done for all terminal paths so the correct path could not be identified.. They then walked down the corridor 48 to the remembered location of the object B, and clicked the mouse. Thus, if participants acquire a graph of object locations in the maze, routes should be highly accurate. Moreover, if they learn local metric information (such as edge weights), we might expect their endpoints to shift along the corridor in Elastic Maze I. Participants took a total of 72 shortcuts (9 trials x 8 object pairs) between pairs of objects during the test phase: 18 trials in the first session, and 54 trials in the second session. Although participants did not receive explicit feedback on their performance, three of the control targets (well, sink, earth) also served as start objects, so further learning about their locations occurred during the test phase. Questionnaires. Participants then completed a written questionnaire that included a series of free response questions about strategies used, location/orientation awareness with respect to the lab, whether they noticed anything unusual about the maze, their sense of immersion in virtual reality, whether they experienced any dizziness or nausea, and any relevant video game experience. Spatial Ability Tests. Next, participants completed three standard assessments of spatial abilities: (1) the Santa Barbara Sense of Direction Scale (SBSOD; Hegarty, Richardson, Montello, Lovelace & Subbiah, 2002), (2) a Road Map Test (RMT; Money & Alexander, 1966; Zacks, Mires, Tversky, & Hazeltine, 2000), modified to use a 20- second timer; and (3) the Perspective-Taking and Spatial Orientation Test (PTSOT; Kozhevnikov & Hegarty, 2001). The SBSOD asks for self-ratings of spatial abilities along a number of dimensions. The RMT assesses mental rotation ability by having participants name the sequence of left and right turns they would need to make to follow 49 a route marked on an overhead map of city streets. The PTSOT asks participants to imagine that they are located at one object in a spatial layout, and draw the angle between two other objects from that location. To ensure that performance on these tests reflected mental rotation and perspective taking abilities, participants were not allowed to turn the test booklets or their head while attempting to complete the RMT and PTSOT tests. Debriefing. After completing the questionnaires and spatial ability tests, all participants were asked if they noticed anything unusual about the maze regardless of experimental condition. Initial responses were recorded (“yes,” “no,” “maybe”), together any follow- up comments about whether they noticed the experimental manipulation. For the neighborhood shortcut test, participants were asked if they (a) noticed and (b) used the neighborhood paths that were superimposed on the ground plane during their shortcuts. For the route test, participants were asked if they used any explicit strategies to decide which path to take, and how far down the path to walk depending on the target. Analysis Analyses were performed using Python 3, R (version 2.15.2), MATLAB (MathWorks), Oriana (Kovach Computing Services) and SPSS (IBM). Positional data (x, y, z, time) from the Intersense IS-900 tracking system were the primary data used in the analyses, in addition to data collected from debriefing tests. Following Howell (2008), follow-up tests on repeated measures (mixed-model) ANOVAs were conducted using the more conservative Tukey’s HSD procedure which maintains a family-wise error rate at alpha = .05 for multiple comparisons. 50 Figure 14. Gate locations. 19 gates were placed throughout the maze to track when participants walked down a main corridor or when participants visited objects. H = hallway, C = control, P = probe. Free Exploration Phase. To track the patterns of exploration, 19 “gates” were placed throughout the maze to track when participants visited objects (i.e. when they were able to see an object down a path) or entered walked down a main corridor of the maze (Figure 14). For each participant, the mean and standard deviation of the number of gate visits to hallway, control target, and probe target gates were used to measure the extent and evenness of exploration during the free exploration phase. Verification Phase. Several measures were extracted from the verification phase to assess familiarity with the maze following the initial learning period (session 1) and upon returning to the lab for their second session (session 2). These included the total number of trials needed to reach criterion, and the number of times participants needed to be guided to the objects. Test Phase. The primary dependent measures for the shortcuts were extracted from the position data collected during travel from Object A to the target, Object B (Figure 15). It is important to note that angular and linear dependent measures were always computed 51 with respect to the canonical locations of target objects corresponding to those in the Control Maze. Measures of angular error included signed angular errors (-180º, +180º], initial angular error (IAE) and final angular error (FAE), and their unsigned counterparts (IAEabs, FAEabs) which were computed by taking the absolute value [0º, 180º] of the angular error. Collectively, these measures assessed the accuracy and precision of angle estimates among targets. IAE was not extracted in the Route Task due to the nature of the shortcuts: participants were confined to hallways, and would initially turn and walk down the Object A hallway, artificially inflating the IAE values. Final (signed) angular error is computed by measuring the angle between the vectors or rays defined by (1) the Shortcut Starting Point and Shortcut Endpoint, and (2) the Shortcut Starting Point and the Target Location. All probe trial angular errors were normalized so that 0º corresponded to walking toward the canonical target location and positive values corresponded to walking in the direction of the stretched target location. Because FAE and IAE revealed the same pattern of results and were highly correlated, (r = .90, n = 118, p < .001) only FAE results are reported in the text. The results for a selection of other dependent measures (FAEabs, DERabs, and FPE) are included in Appendix K. Absolute angular errors (IAEabs, FAEabs) are common measures in the navigation literature, but they confound constant and variable error present in the signed angular errors (IAE, FAE), and remove the circularity of the data. Absolute angular errors were analyzed using linear statistical techniques (means, standard deviations, and ANOVA), whereas circular variables (IAE, FAE) were analyzed using the one-way Watson- 52 Object B FPE Shortcut End Point Ideal Dist. Straight Dist. Home Integral Dist. FAE 10º IAE = Initial Angular Error º (degrees) FAE = Final Angular Error º (degrees) IAE IAEabs = Absolute (Unsigned) Initial Angular Error º (degrees) -30º FAEabs = Absolute (Unsigned) Initial Angular Error º (degrees) DEabs = Absolute Distance Error Ratio, Straight / Ideal Dist.(m) Breach Point DEint = Integral Distance Error Ratio = Integral / Ideal Dist.(m) FPE = Final Position Error (m), End Point to B Dist. (m) 1m Object A Figure 15. Test phase dependent measures. The primary dependent measures for the test phase were extracted from position data collected during shortcuts between Object A and Thursday, July 11, 2013 the target (Object B). Measures of angular error included signed errors [-180º, +180º], initial angular error (IAE) and final angular error (FAE), and their unsigned [0º, 180º] counterparts (IAEabs, FAEabs). Distance error measures included distance error ratios (DEint, DEabs), and final position error (FPE). Williams test (Bateschelet, 1981). Two-way Watson-Williams tests are not available so it is not currently possible to compute interaction terms for circular variables. As a result, only main effects were examined for IAE and FAE using one-way Watson-Williams tests. Distance error measures included distance error ratios (DEint, DEabs), and final position error (FPE). DEabs is the straight line distance of the shortcut (from Object A to the Shortcut End Point), divided by the actual distance from Object A to Object B, while DEint is the integral or cumulative distance along the shortcut. For DEabs and DEint, undershooting the target would produce a negative value, and overshooting a positive value. Collectively, these errors assessed the accuracy and precision of distance estimates 53 among targets. Final Position Error (FPE) is the distance between the Shortcut End Point and Object B, and is a measure of distance error with respect to the target. DEint is the integral distance of the shortcut, divided by the ideal distance from Object A to Object B (calculated by summing all of the distances between consecutive points from Object A to the Shortcut End Point). Measures of mean error (denoted CE, or constant error for circular variables) and variable error (denoted VE, or variable error for circular variables; SD for linear variables) were computed for each participant on probe and control trials. Mean CE is a measure of accuracy, and mean SD or angular deviation (AD) is a measure of individual precision. Mean error was analyzed using one-way Watson-Williams tests for circular variables and ANOVAs for linear variables, while the variable error of both circular and linear variables was always analyzed using ANOVAs. If participants triggered the emergency walls (i.e. if they exited the 10.5 x 12.5m area) then the full response distance could not be recorded, leading to an undershoot bias in the data. Distance error measures (FPE, DEint, DEabs) were only analyzed for trials on which participants did not trigger the emergency walls. Finally, it is possible that targets whose locations are more difficult to remember might require longer response times. We thus recorded the time between the moment a participant arrived at Object A and the moment they clicked the mouse to indicate the remembered direction of the target. Just noticeable differences (JNDs). Just noticeable differences (JNDs) were estimated to determine whether targets were stretched far enough for a shift in direction 54 error to be detectable given the variability in each task. Estimated JNDs were obtained by multiplying the angular deviations in the Control Maze by √2 for each task. Target stretch angles were obtained by measuring the angle subtended by the short and long locations of the target object with the starting object location at the vertex. If JNDs are smaller than the stretch angle for a given target, then that target was stretched far enough for a shift in shortcuts to be detectable based on shortcut variability (AD) in the Control Maze. Bimodality of angular responses. To examine bimodality of angular responses in each of the two mazes, several approaches were taken. First, shortcuts to each of the four probe trial targets were submitted to two-component cluster analyses using the k-means algorithm (Forgy, 1965; Hartigan & Wong, 1979; Lloyd, 1982). Because there are currently no statistical tests that allow for comparisons of the degree of bimodality across samples, Hill & Lewicki (2005) recommend simply comparing the magnitudes of the F- ratios obtained for each environment (Control, Elastic). If the magnitude of the F-ratio (FELST) comparing the two clusters in Elastic Maze I (Elst1 vs. Elst2) is larger than the F- ratio (FCTRL) comparing the two clusters generated for the Control Maze (Eucl1 vs. Eucl2), then the clusters in the Elastic maze are more separated than clusters in the Control Maze, suggesting that shortcuts are more bimodal. Second, angular errors were examined by fitting kernel density estimates for mixtures of von Mises distributions using the maximum likelihood cross-validation (MLCV) method. A detailed comparison of several kernel density estimation methods appears in Appendix I. 55 Endpoints. Endpoint analysis provided measures of how often shortcut endpoints fell in different topological neighborhoods of the maze. For example, participants in both the Control and Elastic I mazes took shortcuts from the earth to the moon (Figure 16). Figure 16. Classification methods for endpoint analysis. Left: Elastic Maze classification scheme; percentages of shortcut endpoints falling in each of the four possible neighborhoods (long, short, paths, wrong) were computed by participants and object pairs before obtaining probe and control means. Right: Swap Maze classification scheme; percentages of shortcut endpoints falling in each of the three possible regions (A/B, paths, wrong) were computed for each participant using the same method. The same classification scheme was applied to shortcuts in the Control Mazes depending on the experiment (Experiments 1 & 2 used the Elastic Maze classification, while Experiment 3 used the Swap Maze classification). Because the moon was stretched into another neighborhood in Elastic Maze I, classifying the number of endpoints that fell in the short and long neighborhoods for both the Control and Elastic Maze I mazes allows for more fine-grained comparisons of the neighborhood knowledge acquired in the two mazes. To avoid over-fitting and interpreting effects that were not central to the predictions, only effects of interest with respect to the predictions will be presented (a more complete analysis can be found in Appendix C). Path choices. To determine whether participants learned the correct path to targets, paths chosen in the Route Task were classified as correct if they walked down the 56 Figure 17. Classification method for path choice analysis. Path choices in the Route Task were considered correct if participants walked down the target object’s path, and incorrect if they walked down any other path. In the example, the dotted shortcut is consistent with walking to the stretched location of the probe trial target (the moon) in Elastic Maze I, however this path would be marked incorrect because the moon could not be found in this hallway. In Experiment 3, the correct path was considered to be either of the A/B paths associated with the swapped target, and the incorrect path was considered to be any non-A/B path. target object’s path, and incorrect if they walked down any other path (Figure 17). Mean and SD of the number of correct and incorrect path choices were obtained for probe and control trials for each participant. Sketch Maps. Upon completing the second session, participants were given a piece of paper with a rectangle drawn on it and a list of object and painting names, and were asked to spend 5 minutes drawing a map of the maze, while doing their best to indicate the locations of all of the objects, paintings, and maze paths (see Appendix B). The dimensions of the rectangle were proportional to the dimensions of the room (10.5m x 12.5m). In order to determine whether the experimental manipulation associated with the virtual environments (Control, Elastic I) had any effect on the resulting sketch maps, 57 maps in which participants drew probe trial objects in both the normal and stretched locations were scored as though those objects were drawn in the stretched locations. Sketch maps were analyzed using Gardony’s (2012) Map Drawing Analyzer Software, which provided measures of both relative and absolute landmark placement. Relative measures included Canonical Organization (CO), and Canonical Accuracy (CA). Canonical Organization compares the relative position of each item (objects and paintings) to every other item in terms of cardinal directions (North, South, East, West) regardless of whether the participant included them on their sketch map (omissions negatively impact the score), and yields a score from 0 to 1 with scores closer to 1 indicating better map organization and item recall. Canonical Accuracy includes in the calculation only those items that the participant actually placed on the sketch map, rather than factoring in omitted items. Absolute measures include Distance Accuracy (DA) and Angle Accuracy (AA). Distance Accuracy ranges from 0 to 1 with higher scores indicating better inter-item distance estimation, and is computed based on normalized difference scores of the absolute value of (actual – observed) distance ratios for all possible inter-item comparisons. Similarly, Angle Accuracy (AA) ranges from 0 to 1 with higher scores indicating better inter-item angle estimation, and is computed in a similar manner using angular differences. Finally, bidimensional regression (Friedman & Kohler, 2003; Tobler, 1994) analyses compared drawn configurations to the actual configuration of objects in the Control Maze. 58 Results Free Exploration Traces of position data collected for all participants’ trajectories in each condition appear in Figure 18. These traces confirm that participants in the Elastic Maze I physically walked to the stretched locations of probe objects during learning. To assess extent and evenness of exploration, the mean and standard deviation of the number of control, hallway, and probe gate visits was computed for each participant, and submitted to separate 2 (environments) x 3 (tasks) x 3 (gate type) repeated-measures ANOVAs on CE and VE of mean number of gate visits, where gate type (control, hallway, probe) was the within-subjects factor, and environments (Control, Elastic I) and tasks (metric, neighborhood, place) were between-subjects factors. We predicted that if participants explored the mazes similarly across environments and tasks, we should not find any effects of environment (Control, Elastic I) or task (Metric, Neighborhood, Place). For mean number of gate visits, there was a main effect of gate type, F2,132 = 76.3, p < .001, ηp2 = .25. Post-hoc Tukey tests revealed that participants did not differ in the mean number of object visits associated with probe and hallway objects, but visited probe objects and hallways more often than control objects (p < .001). No other significant effects were found for mean or SD of gate visits. Consistent with our expectations, patterns of exploration did not differ between Control Maze and Elastic Maze conditions. This implies that performance differences between environments may be attributed to the spatial knowledge acquired rather than group differences in exploration patterns. 59 Figure 18. Experiment 1: free exploration path traces. Position Data (x,y) plotted for all participants in each of the six conditions. During exploration in Elastic Maze I, participants alternately encountered probe objects [bookcase (light blue), gear (gray), clock (green), moon (yellow)] in either normal or stretched paths. Two of the paths [bookcase (light blue), clock (green)] were stretched so that the objects remained in the same neighborhood; two of the paths [gear (gray), moon (yellow)] were stretched into another neighborhood so that participants had to walk across a major path to reach the object. Verification Phase A 2 (environments) x 3 (tasks) ANOVA on number of trials required to reach criterion did not reveal any significant differences among conditions in Experiment 1, nor 60 did the 2 (environments) x 3 (tasks) x 2 (trial type) ANOVA on the number of times participants were guided to objects (p > .05). Test Phase Shortcut Data. Traces of shortcuts in the Control Maze and Elastic Maze I are plotted for each of the three tasks in Figures 19-20. Individual shortcuts are shown as black paths radiating from the mean starting point (object A’s approximate location), and filled dots indicate the endpoints. For each object pair, a mean shortcut vector was computed based on constant (signed) final angular error (FAE), and mean distance from starting points to endpoints. Shortcuts that intersected with the emergency walls were not factored into the mean distance calculation; overall, this amounted to 43% of shortcuts in the Control Maze, and 53% of shortcuts in the Elastic I Maze. Percentages of by condition were: Control / Metric, 35%, Control / Neighborhood, 16%, Control / Route, 0%, Elastic I / Metric, 44%, Elastic I / Neighborhood, 18%, Elastic I / Route, 0%. The 95% confidence ellipses (shaded regions) were also computed based on the endpoints of shortcuts that did not intersect with the emergency walls. Final Angular Error (FAE). For the Shortcut Task, we predicted (Prediction 1) that if spatial knowledge is primarily Euclidean, shortcuts should be less accurate (higher CE) or more variable (higher angular deviation, AD; because Euclidean structure is unstable) in Elastic Maze I than in the Control Maze. Results are shown in Figure 22. JND analysis. To estimate whether the target shifts were large enough to be detectable, just noticeable differences (JNDs) were computed based on the shortcut variability in each task (Table 2). The JND analysis revealed that the Shortcut Task does 61 !),'(-:(*;&.$ 2,#(',1*6& 2,#(',1*;'"&1. =',4/*;'"&1. 2,#(',1*;'"&1. =',4/*;'"&1. *+,,$-&./ *+,,$-&./ %&'() %&'() !"#$ !"#$ !"#$ !"#$ 71&8"#9, 71&8"#9, 21,-$ 21,-$ 0/11 0/11 0/11 0/11 5/&' 5/&' 0/11 0/11 3&44"( 3&44"( 3&44"( 3&44"( 6,,# 6,,# !"#$ !"#$ %&'() %&'() %&'() %&'() Figure 19. Experiment 1: shortcut data, shortcut task. Position data for individual shortcuts in the Control Maze and Elastic Maze I. Individual shortcuts are shown as black paths radiating from the mean starting point (i.e. object A’s approximate location). Large circles: target objects. Medium circles: mean starting point of shortcuts (at the approximate location of of Object A). Large diamonds: stretched location of probe trial objects encountered on 50% of visits during learning in Elastic Maze I. Small circles: endpoints of shortcuts for “full trials” (i.e., trials on which participants clicked the mouse to indicate that they thought they had arrived at the target object location, and did not trigger the emergency walls). Dotted ellipses: 95% confidence ellipses for full trial shortcut endpoints. Arrows: mean vectors of shortcuts (length = mean distance from shortcut starting points to endpoints for full trials; angle = mean final angular error for all trials, including trials on which participants triggered the emergency walls). 62 :/"9)4,'),,;*!),'(-<(*=&.$ 2,#(',1*6&>/ %1&.("-*6&>/*@ 2,#(',1*='"&1. ?',4/*='"&1. 2,#(',1*='"&1. ?',4/*='"&1. *+,,$-&./ *+,,$-&./ %&'() %&'() !"#$ !"#$ !"#$ !"#$ 71&8"#9, 71&8"#9, 21,-$ 21,-$ 0/11 0/11 0/11 0/11 5/&' 5/&' 0/11 0/11 3&44"( 3&44"( 3&44"( 3&44"( 6,,# 6,,# !"#$ !"#$ %&'() %&'() %&'() %&'() Figure 20. Experiment 1: shortcut data, neighborhood shortcut task. Position data for individual shortcuts in the Control Maze and Elastic Maze I. Individual shortcuts are shown as black paths radiating from the mean starting point (i.e. object A’s approximate location). Large circles: target objects. Medium circles: mean starting point of shortcuts (at the approximate location of of Object A). Large diamonds: stretched location of probe trial objects encountered on 50% of visits during learning in Elastic Maze I. Small circles: endpoints of shortcuts for “full trials” (i.e., trials on which participants clicked the mouse to indicate that they thought they had arrived at the target object location, and did not trigger the emergency walls). Dotted ellipses: 95% confidence ellipses for full trial shortcut endpoints are also. Arrows: mean vectors of shortcuts (length = mean distance from shortcut starting points to endpoints for full trials; angle = mean final angular error for all trials, including trials on which participants triggered the emergency walls). 63 3,:(/*;&.$ 2,#(',1*6& 2,#(',1*;'"&1. =',4/*;'"&1. 2,#(',1*;'"&1. =',4/*;'"&1. *+,,$-&./ *+,,$-&./ %&'() %&'() !"#$ !"#$ !"#$ !"#$ 71&8"#9, 71&8"#9, 21,-$ 21,-$ 0/11 0/11 0/11 0/11 5/&' 5/&' 0/11 0/11 3&44"( 3&44"( 3&44"( 3&44"( 6,,# 6,,# !"#$ !"#$ %&'() %&'() %&'() %&'() Figure 21. Experiment 1: shortcut data, route task. Position data for individual shortcuts in the Control Maze and Elastic Maze I. Individual shortcuts are shown as black paths between mean starting point (i.e. object A’s approximate location) and the target location. Large circles: target objects. Medium circles: mean starting point of shortcuts (at the approximate location of of Object A). Large diamonds: stretched location of probe trial objects encountered on 50% of visits during learning in Elastic Maze I. Small circles: endpoints of shortcuts for “full trials” (i.e., trials on which participants clicked the mouse to indicate that they thought they had arrived at the target object location, and did not trigger the emergency walls). Dotted ellipses: 95% confidence ellipses for full trial shortcut endpoints are also. Arrows: mean vectors of shortcuts (length = mean distance from shortcut starting points to endpoints for full trials; angle = mean final angular error for all trials, including trials on which participants triggered the emergency walls). 64 Figure 22. Experiment 1: final angular error. Errors were normalized so that 0º corresponded to perfect accuracy to the control target on control trials, or the unstretched location of the probe trial target on probe trials. Thus, for probe trials, a positive shift in angular error indicates a shift towards the stretched location of the target. Error bars indicate +/- 1 SEM. Duncan flags indicate significant (p < .05) post-hoc Tukey tests. 65 Angular Just Noticeable Deviation (AD) Difference (JND) Stretch Angle Measurable Shift Shortcut Task Earth → Moon 39.4º 55.7º 40.3º Sink → Bookcase 45.7º 64.7º 23º Well → Clock 24.1º 34.1º 15.2º Rabbit → Gear 50º 70.8º 37.2º Neighborhood Shortcut Task Earth → Moon 16º 22.7º 40.3º * Sink → Bookcase 21º 29.9º 23º Well → Clock 10.3º 14.6º 15.2º * Rabbit → Gear 19.5º 27.5º 37.2º * Route Task Earth → Moon 7.4º 10.4º 40.3º * Sink → Bookcase 5.9º 8.4º 23º * Well → Clock 2.6º 3.6º 15.2º * Rabbit → Gear 4.5º 6.3º 37.2º * Table 2. Estimated just noticeable difference (JND) analysis. Stretch Angle = angle subtended by short and long locations of the target object with starting object location at the vertex. Asterisks denote that the target was stretched far enough for a shift in shortcuts to be detectable based on shortcut variability (AD) in the Control Maze. not have the precision required to reveal a significant shift in the mean direction of shortcuts. In the Neighborhood Shortcut Task, estimated JNDs are small enough that for three of the four targets, a shift in the mean direction of shortcuts should be detectable. Similarly, in the Route Task, JNDs were small enough for all four targets that a shift should be detectable. Shortcut task. Results for CE and VE are plotted in Figure 22; bars with the same flag are not statistically different, while different Duncan flags denote statistically significant differences (p < .05). Watson-Williams tests did not reveal any significant 66 effects of environment or task for either constant (CE) or variable (VE) error in the Shortcut Task (p > .05). JNDs for the Shortcut Task were too high for a shift in CE to be detectable; that is, targets were not shifted far enough for a shift in shortcut direction to be detectable given the variability in the Shortcut Task. This implies two possibilities: (1) targets were not stretched far enough to destabilize Euclidean structure, or (2) Euclidean structure is normally destabilized, preventing participants from accurately localizing objects in space. Neighborhood shortcut task. For the Neighborhood Shortcut Task, a Watson- Williams test on CE revealed a significant main effect of trial type, F1,46 = 8.2, p < .01, with control (M = -.88º, AD = 5.63º) errors lower than probe (M = -7.05º, AD = 8.66º) errors. No other significant effects were found for CE or VE. There was no main effect of environment for the Neighborhood Shortcut Task. JNDs for the Neighborhood Shortcut Task were lower than target stretch angles for three of the four probe targets; thus, a shift in shortcut direction was detectable in principle given the variability in the data, yet shortcuts did not shift towards the stretched location of targets in Elastic Maze I. Thus neighborhood shortcuts did not reveal sensitivity to the violations of Euclidean and neighborhood structure in Elastic Maze I. Results for the shortcut and Neighborhood Shortcut Tasks are consistent with (but do not distinguish between) two possibilities: (1) shortcuts in the metric and Neighborhood Shortcut Tasks are too variable to reveal whether neighborhood structure is derived from Euclidean structure, or (2) neighborhood structure is not derived from Euclidean structure. 67 Route task. For the Route Task, CE on probe trials in the Elastic Maze shifted in the positive direction, as expected (Figure 22c). The Watson-Williams test on CE revealed a main effect of environment, F1,46 = 21.6, p < .001, with errors in the Elastic Maze I (M = 2.23º, AD = 4.8º) shifted in the direction of the stretched targets more than in the Control Maze (M = -3.12º, AD = 2.73º). This can be observed in Figure 21, and results of post-hoc Watson-Williams are indicated by Duncan flags in Figure 22. The extent of the angular shift was approximately 33% (9.75º) of the mean angle (28.9º, AD = 10.27º) between normal and stretched location, which is not surprising given that the probe targets appeared in the stretched locations on 50% of the learning trials. JND analysis revealed that the angular shift was detectable given the variability in the data. This result is consistent with participants acquiring local metric information about object distance down the terminal path, but not global information about object locations in space. Such local metric distances would correspond to edge weights in a labeled graph. As expected, Route Task VE on probe trials was higher in the Elastic Maze (Figure 22f). The ANOVA on VE also revealed a significant main effect of environment, F1,22 = 4.47, p = .046, ηp2 = .16, with higher errors overall in the Elastic Maze than in the Control Maze. There was also a significant environment x trial type interaction, F1,22 = 7.32, p = .013, ηp2 = .017. Post-hoc Tukey tests revealed that VE of FAE was higher overall in Elastic Maze I (M = 9.74º, AD = 1.15º) than in the Control Maze (M = 4.01º, AD = 1.15º), p < .05. This result is consistent with participants detecting that the position of the target was unstable during learning (50% of the time in the short location, 50% in the long location). 68 That the JNDs (Table 2) for the Shortcut Task were too high for a shift in CE to be detectable in the Shortcut Task implies two possibilities: (1) targets were not stretched far enough to destabilize Euclidean structure to a measurable degree, or (2) that Euclidean structure is normally destabilized so much that participants are not able to locate objects in space accurately. In contrast, JNDs in both Neighborhood and Route Tasks were low enough to detect shifts; however, no shifts were found in the Neighborhood Task, while a significant shift was found in the Route Task. Taken together, this implies that neighborhood and graph knowledge are not derived from Euclidean knowledge. The Topological Hypothesis predicts (Prediction 3) that if neighborhoods are derived from topological boundaries, neighborhood shortcuts in Elastic Maze I should be (3a) less variable than the metric shortcuts in Elastic Maze I. Watson-Williams tests for CE did not reveal any significant effects of task. However, the ANOVA on VE revealed a main effect of task, F1,22 = 7.27, p = .013, ηp2 = .23; angular deviations were twice as large in the Shortcut Task (M = 32º, SD = 18.8º) as the Neighborhood Shortcut Task (M = 15.9º, SD = 9.8º) within Elastic Maze I. Consistent with the Topological Hypothesis (3a), variable error was lower for the Neighborhood Shortcut Task, suggesting that neighborhood knowledge depends upon topological boundaries. Bimodality of final angular responses. The Euclidean Hypothesis predicts (Prediction 2) that if neighborhoods are derived from Euclidean structure, metric shortcuts should be more bimodal in Elastic Maze I than the metric shortcuts in the Control Maze. However, kernel density estimates using the MLCV method (Agnostinelli & Lund, 2011; Sharma & Tarboton, 1997) and plots of within-group sums of squares vs. 69 the number of clusters used in the k-means algorithm (Forgy, 1965; Hartigan & Wong, 1979; Lloyd, 1982) did not suggest the presence multiple modes for any of the samples examined in Experiment 1. Thus, two-component cluster analyses for Experiment 1 are not well-motivated and should be treated with caution. Shortcut task. F-ratios comparing cluster means were larger in the Elastic Maze for two of the four probe targets (Sink → Bookcase, FELST (1,70) = 134 > FCTRL (1,96) = 37.2; Rabbit → Gear, FELST (1,71) = 291 > FCTRL (1,74) = 73.8), and larger in the Control Maze for the other two of the four probe targets (Earth → Moon, FCTRL (1,87) = 201 > FELST (1,79) = 107; Well → Clock, FCTRL (1,90) = 160 > FELST (1,70) = 146). All F-ratios computed were significant at p < .001. Thus, no definitive conclusions can be drawn from these data regarding the relative degree of bimodality in these two environments for the Shortcut Task. Neighborhood shortcut task. A similar pattern of results was found for the Neighborhood Shortcut Task. F-ratios comparing cluster means were larger in the Elastic Maze for the same two probe targets as before (Sink → Bookcase, FELST (1,91) = 233 > FCTRL (1,90) = 231; Rabbit → Gear, FELST(1,94) = 104 > FCTRL (1,91) = 95.3), and larger for the other two (Earth → Moon, FCTRL (1,97) = 155 > FELST(1,93) = 102; Well → Clock, FCTRL (1,90) = 103 > FELST (1,83) = 51.5). All F-ratios computed were significant at p < .001. As in the Shortcut Task, no definitive conclusions can be drawn from these data regarding the relative degree of bimodality for the Neighborhood Shortcut Task. Route task. For the Route Task, F-ratios were higher for shortcuts in the Control Maze than the Elastic Maze for three of the four probe targets (Earth → Moon, FCTRL 70 (1,105) = 305 > FELST (1,104) = 133 ; Well → Clock, FCTRL (1,114) = 195 > FELST (1,95) = 98.5 ; Rabbit → Gear, FCTRL (1,114) = 103 > FELST (1,108) = 255) and higher in the Elastic Maze for the one remaining probe target (Sink → Bookcase, FELST (1,108) = 192 > FCTRL (1,99) = 255). While these results suggest that responses were more bimodal in the Control Maze than in the Elastic Maze, variability of angular error was lowest in the Route Task, and preliminary analyses did not suggest bimodality for either group; as a result, this analysis of the degree of bimodality should be treated with caution. Taken together, analyses of the bimodality of angular responses suggest that— contrary to the Euclidean Hypothesis—the degree of bimodality was not greater in the Elastic Maze I than in the Control maze for any of the response tasks. Endpoints. The percentages of endpoints falling in the four possible neighborhoods (wrong, path, short, long) appear in Figure 23. Shortcut task. For the Shortcut Task, there were no effects of environment or trial type on the percentage of endpoints falling into any of the four neighborhoods. This is consistent with the FAE results, and suggests that either targets were not stretched far enough to destabilize Euclidean structure, or that Euclidean structure is normally destabilized, preventing participants from accurately localizing objects in space. Neighborhood shortcut task. For the Neighborhood Shortcut Task, there were no effects of environment or trial type on the percentage of endpoints falling into any of the four neighborhoods. Thus, the percentage of endpoints did not shift towards the stretched location of probe targets in the Neighborhood Shortcut Task. As with the FAE results, this is consistent with two possibilities: (1) shortcuts in the metric and Neighborhood 71 Figure 23. Experiment 1: percentages of endpoints falling in neighborhoods. c = control trials, p = probe trials. Eucl. = Control Maze, Elst.I = Elastic Maze I. For the shortcut and Neighborhood Shortcut Tasks, no predicted effects of environment and no interactions were found for any of the four neighborhoods. For the Route Task, the mean percentage of endpoints falling in the long neighborhood was higher in Elastic Maze I (M = 5.3%, SD = 6.2%) than the Control Maze (M = 0%, SD = 0%), and higher on probe trials (M = 5.3%, SD = 9.4%) than on control trials (M = 0%, SD = 0%). Full results including F-ratios, p-values, and effect sizes, appear in Appendix C. Shortcut Tasks are too variable to reveal whether neighborhood structure is derived from Euclidean structure, or (2) neighborhood structure is not derived from Euclidean structure. Route task. For the Route Task, no significant effects were found on the percentages of endpoints falling in the wrong neighborhood, while all main effects and interactions were significant for short, long, and path neighborhoods (p < .05). The mean percentage of endpoints falling in the long neighborhood was higher in Elastic Maze I (M = 5.3%, SD = 6.2%) than the Control Maze (M = 0%, SD = 0%), and was higher on probe trials (M = 5.3%, SD = 9.4%) than on control trials (M= 0%, SD = 0%). The mean percentage of endpoints falling on major paths was higher in Elastic Maze I (M = 10.7%, SD = 5.9%) than the Control Maze (M = 2.3%, SD = 3.9%). The mean percentage of endpoints falling in the short neighborhood was lower in the Elastic Maze (M = 82.2%, SD = 9.4%) than in the Control Maze (M = 97%, SD = 3.8%), and was lower on probe trials (M = 81.1%, SD = 18.7%) than control trials (M = 98.4%, SD = 3.9%). A similar 72 pattern of results was found for SD of percentage of endpoints falling in the four neighborhoods (see Appendix C). For the Route Task, the increased percentage of endpoints falling in long neighborhoods on probe trials in Elastic Maze I relative to the Control Maze is consistent with spatial knowledge that takes the form of a labelled graph. However, these results could also be explained by the fact that shortcuts in the metric and Neighborhood Shortcut Tasks depend on knowledge acquired by integrating over 2 dimensions (x,y coordinates, or distance and angle), whereas the Route Task only requires knowledge along a single dimension (distance down a path). Comparisons across tasks. The Topological Hypothesis predicts (Prediction 3) that if neighborhoods are derived from topological boundaries, neighborhood shortcuts should be (3b) more unimodal (i.e. in the short neighborhood) than the metric shortcuts in Elastic maze I. In contrast, the Euclidean Hypothesis predicts that if neighborhoods are derived from metric path integration, there should be no difference between these two tests. Mean percentages of endpoints falling in short neighborhoods were submitted to a 2 (task) x 2 (trial type) mixed-model ANOVA, where task (metric shortcut, neighborhood shortcut) was the between-subjects factor and trial type (control, probe) was the within- subjects factor. The ANOVA revealed a significant main effect of task, F1,22 = 15.2, p < . 001, ηp2 = .38, and no other effects or interactions; the mean number of endpoints falling in short neighborhoods was significantly higher for the Neighborhood Shortcut Task (M = 83.8%, SD = 17.1%) than the Shortcut Task (M = 47%, SD = 27.9%) within Elastic Maze I. This can be seen in Figures 19 and 20. This result is consistent with the Topological Hypothesis (Prediction 3) that neighborhoods are derived from topological boundaries. 73 Path choices. Mean number of correct and incorrect path choices were submitted to separate independent samples t-tests. No statistically significant differences were found between the Control and Elastic Maze I environments for paths chosen on either control or probe trials in the Route Task. Percentages of correct path choices were high in both the Control (M = 99.8%, SD = 1%) and Elastic (M = 97.3%, SD = 4.4%) mazes; thus, participants learn the graph of both mazes. Response time. For the Shortcut Task, the ANOVAs on mean and SD of response time did not reveal any main effects or interactions. Thus, shortcut errors do not appear to be influenced by a speed accuracy trade-off. Sketch maps. Each of the sketch map measures were submitted to separate 2 (environment) x 3 (tasks) ANOVAs; the ANOVAs on Canonical Accuracy, Distance Accuracy, Bi-Dimensional Regression, and Angle Accuracy measures did not reach significance. Group Differences No significant differences in self-reported VR experience (location awareness, level of immersion, nausea) or video game experience were found. The Santa Barbara Sense of Direction (SBSOD) scale had high internal consistency when for Experiments 1-3 (Cronbach's α = .91). The ANOVA conducted on the aggregated SBSOD scale did not reveal any statistically significant differences between groups. The ANOVA conducted on Road Map Test (RMT) scores did not reveal any significant differences between groups. Thus, significant performances differences were not due to selection bias in the participant groups. Separate one-way Watston-Williams tests for differences 74 between angular means were also conducted on CE of angular error on the PTSOT across environments and across tasks; no significant effects were predicted or found. Debriefing Responses. Two participants (1 female) in the Control Maze / Metric condition reported thinking that some of the objects might be “overlapping” or in the same physical location despite being located down different paths. One of these participants (male) reported thinking that the bookcase and gear were in the same location, and that the moon and cactus might be in the same location as well. A (female) participant in the Control Maze / Neighborhood condition thinking that some objects were overlapping, and suspected that the clock and flamingo might occupy the same location. None of the participants in the Elastic Maze I conditions reported noticing that probe target paths stretched. Discussion The Euclidean Hypothesis predicts (Prediction 1) that if spatial knowledge is primarily Euclidean, metric shortcuts should be less accurate or more variable in Elastic Maze I (because Euclidean structure is unstable) than in the Control Maze. Contrary to this prediction, the analyses of angular errors did not reveal any significant differences between the two mazes for either constant or variable errors. In addition, percentages of endpoints falling in long/short neighborhoods did not differ between Euclidean and Elastic I mazes for either metric or Neighborhood Shortcut Tasks. These results for the metric shortcut and Neighborhood Shortcut Tasks are consistent with—but do not distinguish between—two possibilities: (1) we did not destabilize Euclidean structure enough to be revealed by noisy metric shortcuts (2) Euclidean knowledge is normally so 75 crude that neighborhoods and places are not derived from it. It is important to note that high variability in metric task does not imply that spatial knowledge is not Euclidean: on average metric shortcuts are accurate, but precision is poor. However, this pattern of results is also compatible with spatial knowledge that resembles a labeled graph. The Euclidean Hypothesis also predicts (Prediction 2) that if neighborhood structure is derived from Euclidean structure, Neighborhood Shortcut Task shortcuts should be more bimodal in Elastic Maze I than the metric task shortcuts in Elastic Maze I. However, contrary to the Euclidean Hypothesis (Prediction 2), our analyses failed to find greater bimodality in the Control Maze c. The Topological Hypothesis predicts (Prediction 3) that if neighborhoods are derived from topological boundaries, neighborhood shortcuts should be (3a) less variable and (3b) more unimodal (i.e. in the short neighborhood) than the metric shortcuts in Elastic Maze I. Consistent with the Topological Hypothesis, analyses of final angular error revealed that neighborhood shortcuts were less variable than metric shortcuts, and analyses of endpoints revealed that shortcuts were more unimodal (fell more frequently in the short neighborhood) than metric shortcuts within Elastic Maze I. Both the Control Maze and Elastic Maze I preserved graph structure, while Elastic Maze I destabilized metric structure and neighborhoods. This provided to two main predictions for the Route Task: (Prediction 4) if graph knowledge is derived from Euclidean structure, participants may walk down different paths to get to the stretched location in the Elastic Maze I than in the Control Maze, and (Prediction 5) if spatial knowledge is primarily graph-like (topological), then participants should walk down the 76 correct path despite unstable metric and neighborhood structure in Elastic Maze I. The analysis of path choices in Experiment 1 did not reveal any significant differences between environments, and the analysis of endpoints for the Route Task revealed that more probe trial endpoints fell in long neighborhoods in Elastic Maze I, and in the short neighborhood in the Control Maze. Thus, participants walked down the same paths in both the Euclidean and Elastic Mazes, and walked to the long neighborhood more in the Elastic Maze when tested on the Route Task but not when tested on the Shortcut Tasks. These results are inconsistent with Prediction 4 (graph knowledge is derived from Euclidean structure) because participants did not walk down incorrect paths more in one environment than another. They are also inconsistent with Prediction 1 (spatial knowledge is primarily Euclidean), because participants did not walk to long neighborhoods when taking shortcuts, yet did so when walking down the terminal hallway in the Route Task. In addition, the results are consistent with Prediction 5 (spatial knowledge is primarily graph-like), because participants walked down the correct path despite unstable metric and neighborhood structure in Elastic Maze I. Finally, no statistically significant differences between groups were found for free exploration and verification phases, sketch maps, VR experience, video game experience, or spatial ability tests (SBSOD, RMT, PTSOT). This indicates that performance differences in the test phase are not attributable to group differences in spatial ability or experience. Three participants in the Control Maze conditions reported thinking that some objects might occupy the same spatial locations, which suggests that participants 77 may be uncertain about the metric locations of objects even when Euclidean structure is preserved. Conclusions Experiment 1 reveals a pattern of results consistent with primarily topological spatial knowledge. Metric and neighborhood shortcuts were roughly accurate on average (across participants), but were highly imprecise, and did not shift toward the metric location of stretched targets. This indicates that neighborhoods were not derived from metric Euclidean knowledge. In contrast, endpoints in the Route Task were accurate, precise, and shifted towards the stretched locations of probe trial targets. These results are consistent with spatial knowledge resembling a labeled graph that is supplemented with local metric information about the distances to objects along paths. However, this pattern of results could also be explained by primarily Euclidean spatial knowledge that is very imprecise, yielding highly variable shortcuts in the metric and Neighborhood Shortcut Tasks. In Experiment 1, information about neighborhood boundary relations was only available through path integration; Experiment 2 was designed to determine whether neighborhood structure depends on visual information specifying topological neighborhood boundaries by providing visual information about neighborhood boundary relations during learning. 78 Chapter 3: Experiment 2 Introduction The results of Experiment 1 were consistent with the Topological Hypothesis, and inconsistent with the Euclidean Hypothesis that neighborhood structure is derived from Euclidean structure (based on metric information from path integration). The results of Experiment 1 suggest that the spatial knowledge acquired is consistent with a topological graph supplemented with local metric information, even when Euclidean structure and metric information for neighborhood structure are unstable during learning. Experiment 2 was designed to determine whether the acquisition of neighborhood structure instead depends on visual information specifying topological neighborhood boundaries. Elastic Maze II was designed to destabilize Euclidean structure and topological neighborhood structure (visible neighborhood boundaries) by making the stretched hallways visibly cross a major path during learning (Figure 12). In Elastic Maze II, the stretched paths for two objects (gear, moon) crossed neighborhood boundaries and were connected to the main hallways that they crossed, providing both path integration and visual information about neighborhood boundary relations. This is in contrast to Elastic Maze I, in which information about neighborhood boundary relations was only available through path integration. Data from two new groups of participants were collected in Elastic Maze II and compared to the Elastic Maze I data collected in Experiment 1. Predictions Neighborhood shortcut task. Prediction 6: If neighborhoods are based on topological boundaries, probe trial shortcuts in Elastic Maze II should (6a) shift towards 79 the stretched target location, (6b) be more variable, or (6c) be more bimodal than shortcuts in Elastic Maze I. Route Task. Prediction 7: If spatial knowledge is primarily graph-like, participants should walk down the correct path despite varying metric and neighborhood structure. Prediction 8: Given that graph knowledge includes neighborhood boundary relations (Kuipers, Tecuci, & Stankiewicz, 2003), endpoints should (8a) shift towards stretched target location, (8b) be more variable, or (8c) be more bimodal in Elastic Maze II than in Elastic Maze I. Methods Participants A total of 33 additional people participated in Experiment 2. Of these participants, 24 (12 female) successfully completed the study and were included in the final analysis. The mean age these participants was 20.9 (SD = 2.6). Three participants (1 female) withdrew due to symptoms of simulator sickness in the Elastic II / Place condition. Two participants (Elastic II / Neighborhood, 1 female; Elastic II / Place, 1 female) were excluded for failing to find all of the objects two times each within the 25 minute verification phase time limit. Four (3 female) participants were excluded due to technical problems. Equipment The equipment used was the same as in Experiment 1. 80 Figure 24. Experiment 2: virtual environments. Yellow circles on overhead map diagrams highlight the differences between stretched paths in Elastic Maze I vs. Elastic Maze II. In Elastic Maze II, additional visual information about boundary relations was provided by connecting stretched paths to the hallways they crossed for the two probe objects (moon, gear) that stretched into new neighborhoods. Virtual Environments Elastic Maze I was the same as in Experiment 1. Elastic Maze II was identical to Elastic Maze I, with one exception: during free exploration and verification, when a stretched path crossed a neighborhood boundary (a main path), a visible junction with the 81 Figure 25. Design of Experiment 2. Two tasks (Neighborhood Shortcut, Route) were crossed with two virtual environments (Elastic Maze I, Elastic Maze II) for a total of four conditions. main path was displayed. This provided visual information for topological neighborhood boundaries and the unstable locations of probe objects that was not available in Elastic Maze I. Displays for each environment are shown in Figure 24. Design Experiment 2 employed a 2 (environment) x 2 (task) x 2 (trial type) design, where environment (Elastic I, Elastic II) and task (Neighborhood, Route) were between-subjects factors, and trial type (control, probe) was the repeated measure (Figure 25). Data from 82 two new groups of participants in Elastic Maze II conditions were compared to the Elastic Maze I data from Experiment 1. Procedure The procedure in the free exploration, verification, and test phases was the same as Experiment 1. Results Free Exploration Phase Visual examination of path traces (Figure 26) indicate that participants visited both normal and stretched locations of probe trial objects during the free exploration phase. As in Experiment 1, the mean and SD of the number of gate visits were submitted to separate 2 (environments) x 2 (tasks) x 3 (gate type) repeated-measures (mixed-model) ANOVAs, where gate type (control, hallway, probe) was the within-subjects factor, and environments (Elastic I, Elastic II) and tasks (neighborhood, route) were between- subjects factors. For mean number of visits, there was a significant main effect of gate type, F2,88 = 39.1, p < .001, ηp2 = .21. As in Experiment 1, post-hoc Tukey tests revealed that participants visited control objects less than probe objects and hallway gates, p < . 001. No other significant effects were found. As in Experiment 1, these results imply that groups did not differ in their exploration patterns. Verification Phase A 2 (environments) x 3 (tasks) between-subjects ANOVA on number of trials required to reach criterion did not reveal any significant differences among groups, nor did a 2 (environments) x 3 (tasks) x 2 (trial type) between-subjects ANOVA on the mean 83 number of times participants were guided to objects. This implies that neither maze was more difficult to learn than the other. Figure 26. Experiment 2: free exploration path traces. Position Data (x,y) plotted for all participants in each of the four conditions. In both mazes, participants alternately encountered probe objects (bookcase, gear, clock, moon) in either normal or stretched paths. Two of the paths (bookcase, clock) were stretched so that the objects remained in the same neighborhood; two of the paths (gear, moon) were stretched into another neighborhood, such that participants would need to walk across a major path to get to object. In Elastic Maze II, the stretched paths for two objects (gear, moon) crossed neighborhood boundaries and were connected to the main hallways that they crossed, providing both path integration and visual information about neighborhood boundary relations. This is in contrast to Elastic Maze I, in which information about neighborhood boundary relations was only available through path integration. 84 Test Phase Figures 27 and 28 illustrate traces of shortcuts for the test phase. Final Angular Error. If neighborhoods are derived from topological boundaries (Prediction 6) or if graph knowledge includes neighborhood boundary relations (Prediction 8), Final Angular Errors should shift towards stretched target location, be more variable, or be more bimodal in Elastic Maze II than in Elastic maze I. Thus, ANOVAs for dependent measures should reveal significant main effects of environment, or environment x trial type interactions for constant errors (CE) or variable errors (VE). Neighborhood shortcut task. For the Neighborhood Shortcut Task, the Watson- Williams test on CE did not reveal any statistically significant effects of environment or trial type. For VE, a repeated-measures ANOVA on angular deviations revealed a main effect of trial type, F1,22 = 5.94, p = .023, ηp2 = .03; shortcuts were more variable on probe trials (M = 18.3º, SD = 2.3º) than on control trials (M = 14.3º, SD = .57º). No other significant effects were found; post-hoc Tukey tests did not reveal any statistically significant pairwise differences. Route task. For the Route Task, a one-way Watson-Williams test on CE revealed a main effect of trial type, F1,46 = 39.9, p < .001; participants were more accurate to control targets (M = -1.51º, SD = 1.56º) than probe targets (M = 6.79º, SD = 6.12º). No other significant effects or interactions were found. For VE, there was a significant main effect of trial type, F1,22 = 12.5, p < .01, ηp2 = .03, and post-hoc Tukey tests did not reveal any significant pairwise differences. 85 :/"9)4,'),,;*!),'(-<(*=&.$ %1&.("-*6&>/*? %1&.("-*6&>/*?? 2,#(',1*='"&1. @',4/*='"&1. 2,#(',1*='"&1. @',4/*='"&1. *+,,$-&./ *+,,$-&./ %&'() %&'() !"#$ !"#$ !"#$ !"#$ 71&8"#9, 71&8"#9, 21,-$ 21,-$ 0/11 0/11 0/11 0/11 5/&' 5/&' 0/11 0/11 3&44"( 3&44"( 3&44"( 3&44"( 6,,# 6,,# !"#$ !"#$ %&'() %&'() %&'() %&'() Figure 27. Experiment 2: shortcut data, neighborhood shortcut task. Position data for individual shortcuts in Elastic Maze I and Elastic Maze II. Individual shortcuts are shown as black paths radiating from the mean starting point (i.e. object A’s approximate location). Large circles: target objects. Medium circles: mean starting point of shortcuts (at the approximate location of of Object A). Large diamonds: stretched location of probe trial objects encountered on 50% of visits during learning in Elastic Maze I. Small circles: endpoints of shortcuts for “full trials” (i.e., trials on which participants clicked the mouse to indicate that they thought they had arrived at the target object location, and did not trigger the emergency walls). Dotted ellipses: 95% confidence ellipses for full trial shortcut endpoints are also. Arrows: mean vectors of shortcuts (length = mean distance from shortcut starting points to endpoints for full trials; angle = mean final angular error for all trials, including trials on which participants triggered the emergency walls). 86 /#0.%&')+1 2*)+.(3&4)5%&6 2*)+.(3&4)5%&66 ,#-."#*&'"()*+ !"#$%&'"()*+ ,#-."#*&'"()*+ !"#$%&'"()*+ &9##13)+% &9##13)+% 2)".8 2)".8 7(-1 7(-1 7(-1 7(-1 <*)=(-># <*)=(-># ,*#31 ,*#31 :%** :%** :%** :%** ;%)" ;%)" :%** :%** /)$$(. /)$$(. /)$$(. /)$$(. 4##- 7(-1 4##- 7(-1 2)".8 2)".8 2)".8 2)".8 Figure 28. Experiment 2: shortcut data, route task. Position data for individual shortcuts in Elastic Maze I and Elastic Maze II. Individual shortcuts are shown as black paths between mean starting point (i.e. object A’s approximate location) and the target location. Large circles: target objects. Medium circles: mean starting point of shortcuts (at the approximate location of of Object A). Large diamonds: stretched location of probe trial objects encountered on 50% of visits during learning in Elastic Maze I. Small circles: endpoints of shortcuts for “full trials” (i.e., trials on which participants clicked the mouse to indicate that they thought they had arrived at the target object location, and did not trigger the emergency walls). Dotted ellipses: 95% confidence ellipses for full trial shortcut endpoints are also. Arrows: mean vectors of shortcuts (length = mean distance from shortcut starting points to endpoints for full trials; angle = mean final angular error for all trials, including trials on which participants triggered the emergency walls). 87 Figure 29. Experiment 2: final angular error. Errors were normalized so that 0º corresponded to perfect accuracy to the control target on control trials, or the unstretched location of the probe trial target on probe trials. Thus, for probe trials, a positive shift in angular error indicates a shift towards the stretched location of the target. Error bars indicate +/- 1 SEM. Duncan flags indicate significant (p < .05) post-hoc Tukey tests. Thus, mean accuracy and precision of Final Angular Error did not differ between Elastic Mazes I and II for Neighborhood Shortcut and Route Tasks. This appears to be inconsistent with the predictions; however, analyses presented in the next section suggest multimodal distributions of errors for probe trial shortcuts may have masked any mean group differences. Multimodality of angular errors. Because exploratory data analysis suggested multimodal shortcuts when targets were stretched across neighborhood boundaries, the 88 Final Angular Errors for both within- and across- neighborhood probe trials were submitted to two-component cluster analyses. Kernel density estimates are included in Figure 30. The maximum likelihood cross-validation (MLCV) model strongly suggests that the distributions of final angular errors were multimodal when participants were asked to walk to targets stretched across neighborhood boundaries in Elastic Maze II, but not in Elastic Maze I. In addition, F-ratios were larger in Elastic Maze II than Elastic Maze I for both within- and across-neighborhood stretches, suggesting a greater degree of overall bimodality in Final Angular Error on probe trials in Elastic Maze II than in Elastic Maze I. Cluster 2 means (light blue vectors in Figure 30) in Elastic Maze II shifted towards the stretched location (dotted lines within circular histograms in Figure 30) of probe trial targets when targets were stretched across neighborhood boundaries (Figure 30, top row) but not when targets were stretched within neighborhood boundaries (Figure 30, bottom row). This was not observed for Elastic Maze I. Thus, consistent with the Topological Hypothesis (Prediction 6), the cluster analysis revealed that probe trial neighborhood shortcuts shifted towards the stretched location when targets were stretched across neighborhood boundaries (6a) and were more bimodal when targets were stretched both within and across neighborhood boundaries (6c) in Elastic Maze II. Because visual information about neighborhood boundary relations was available in Elastic Maze II but not in Elastic Maze I, these results suggest that neighborhoods may be derived from topological boundaries. However, because the boundary was not visible during the test phase, this is not a very strong effect. 89 Figure 30. Experiment 2: cluster analysis of angular errors, neighborhood shortcut task. Dark blue arrow: cluster 1 mean. Light blue arrow: cluster 2 mean. Red lines: kernel density estimates for a mixture of von Mises distributions using the maximum-likelihood cross-validation (MLCV) method. (S): stretched location of the target; the ideal direction to the unstretched target was always set at 0º. Dotted lines inside circular plots: angle (relative to 0º) by which target was stretched. Comparison of this angle to the angles between cluster means (M2 - M1) and to the cluster 2 mean (M2, light blue arrow) provides an estimate of whether the degree of bimodality in the sample maps onto the angular separation between unstretched and stretched locations of the target. Comparisons of F-ratios (Hill & Lewicki, 2005) between environments provides an indication of whether the degree of bimodality was greater in one environment than in another. All F-ratios were larger in Elastic Maze II for both within- and across- 90 neighborhood stretches. Cluster 2 means were shifted towards the stretched location (S) of targets in Elastic Maze II when targets were stretched across neighborhood boundaries (top row, rabbit → gear and earth → moon), but not when targets were stretched within neighborhood boundaries (bottom row, well → clock, sink → bookcase). Figure 31. Experiment 2: cluster analysis of angular errors, route task. Dark blue arrow: cluster 1 mean. Light blue arrow: cluster 2 mean. Red lines: kernel density estimates for a mixture of von Mises distributions using the maximum-likelihood cross-validation (MLCV) method. (S): stretched location of the target; the ideal direction to the unstretched target was always set at 0º. Dotted lines inside circular plots: angle (relative to 0º) by which target was stretched. Comparison of this angle to the angles between cluster means (M2 - M1) and to the cluster 2 mean (M2, light blue arrow) provides an estimate of whether the degree of bimodality in the sample maps onto the angular separation between unstretched and stretched locations of the target. Comparisons of F- ratios (Hill & Lewicki, 2005) between environments provides an indication of whether 91 the degree of bimodality was greater in one environment than in another. F-ratios were larger in Elastic Maze II for both (top row, rabbit → gear, earth → moon) of the across- neighborhood probe trials, and for one (bottom row, well → clock) of the within- neighborhood probe trials. The cluster 2 mean was shifted towards the stretched location (S) of the target in Elastic Maze II for one (top row, rabbit → gear) of the across- neighborhood stretches, but not the other (top row, earth → moon). For the Route Task, MLCV kernel density estimates (Figure 31) did not suggest the presence of bimodal Final Angular Errors in either maze, except for a slight bump in the density estimate for one of the across-neighborhood probe targets (top row, rabbit → gear); therefore, the results of the cluster analysis for the Route Task should be treated with caution. F-ratios were larger in Elastic Maze II for both of the across-neighborhood probe trials (top row, rabbit → gear, earth → moon), and for one of the within- neighborhood probe trials (bottom row, well → clock). To the extent that these results are reliable despite the apparently unimodal distributions of errors, results for the Route Task are tentatively consistent with The Topological Hypothesis (8c) because shortcuts in the Route Task were more bimodal in Elastic Maze II than Elastic Maze I on across- neighborhood probe trials, when visual information about boundary relations was available. However, because the F-ratio was larger for one of the within-neighborhood stretches (bottom row, well → clock), these results should be treated with caution. For example, participants may have been looking for a boundary during test but did not see one and thus stopped short. Taken together, cluster analyses for the Neighborhood Shortcut Task suggest that neighborhoods can be derived from topological boundaries, and that graph knowledge includes knowledge of topological neighborhood boundaries. Endpoints. For the Neighborhood Shortcut Task, no significant effects were found for mean or SD of the percentage of endpoints falling in the wrong neighborhood. 92 Figure 32. Experiment 2: percentages of endpoints falling in neighborhoods. Percentages of endpoints falling neighborhoods appear in Figure 32. For mean percentage of endpoints falling in the short neighborhood, the ANOVA revealed a significant main effect of trial type, F1,22 = 5.1, p = .034, ηp2 = .02, and a significant environment x trial type interaction, F1,22 = 7.5, p = .012, ηp2 = .032, consistent with prediction 6a; the marginal mean was higher for Elastic Maze I (M = 83.8%, SD = 17%) than Elastic Maze II (M = 81.5%, SD = 14.9%), and higher on control trials (M = 85.1%, SD = 14.6%) than on probe trials (M = 80.2%, SD = 18.9%). Thus, more endpoints fell in the short region in Elastic Maze I. For mean percentage of endpoints falling in the long region, the ANOVA revealed a main effect of trial type, F1,22 = 9.77, p = .005, ηp2 = . 31; more endpoints fell in long regions on probe trials (M = 5.4%, SD = 9.1%) than control trials (M = 0%, SD = 0%). Thus, it was not the case that more endpoints fell in long regions in Elastic Maze II. For SD of the percentage of endpoints falling on major paths in the Neighborhood Shortcut Task, there was significant environment x trial type interaction, F1,22 = 7.03, p = .015, ηp2 = .033; shortcuts were more variable in Elastic Maze II (M = 6.1%, SD = 93 8.7%) than Elastic maze I (M = 5.4%, SD = 6.6%), and more variable on probe trials (M = 6.1%, SD = 9.1) than on control trials (M = 5.4%, SD = 7.1%). For SD of the percentage of endpoints falling in the long neighborhood, there was a main effect of environment, F1,22 = 6.49, p = .018, ηp2 = .23, a main effect of trial type, F(1,22) = 10.7, p = .003, ηp2 = 0.33, and a significant environment x trial type interaction, F1,22 = 6.49, p = . 018, ηp2 = .23; the percentage of endpoints falling in long neighborhoods was more variable in Elastic Maze II (M = 8.6%, SD = 9.6%) than Elastic Maze I (M = 1.1%, SD = 3.7%), and more variable on probe (M = 9.7%, SD = 16%) than control (M = 0%, SD = 0%) trials. Higher variability for endpoints in Elastic Maze II, especially on probe trials, is consistent with the Topological Hypothesis (Prediction 6b), suggesting that neighborhoods are based on topological boundaries. In sum, for the Neighborhood Shortcut Task, the mean percentage of shortcuts falling in the short neighborhood was higher in Elastic Maze II (Prediction 6a), as was the SD of percentage of shortcuts falling in the long neighborhood (Prediction 6b), consistent with the Topological Hypothesis, and suggesting that neighborhoods are derived from topological boundaries. For the Route Task, no main effects of environment and no environment x trial type interactions were found; results of the endpoint analysis cannot be meaningfully interpreted with respect to the topological hypothesis, Prediction 8 (a-c). Path choices. Mean number of correct and incorrect path choices were submitted to separate independent samples t-tests. No statistically significant differences were found between Elastic Maze I and Elastic Maze II for paths chosen on either control or 94 probe trials in the Route Task. The percentages of correct path choices in each environment on probe trials were: Elastic Maze 1, 97% (SD = 4%); Elastic Maze II, 98% (SD = 4%). These results are consistent with Prediction 7 because participants walked down the correct path despite varying metric and neighborhood structure in both mazes, consistent with primarily graph-like spatial knowledge. Response Time. The ANOVAs for mean response time in the Neighborhood Shortcut Task did not reveal any significant effects or interactions for mean or SD of response time. Thus, shortcut errors do not appear to be influenced by a speed-accuracy trade-off. Group Differences Separate 2 (environment) x 2 (tasks) ANOVAs were conducted on the same group difference measures as in Experiment 1. For sketch maps, no significant effects were found. For self-reported level of immersion in VR, a 2 (environment) x 3 (tasks) ANOVA revealed a significant main effect of environment, F1,44 = 5.02, p = .03, ηp2 = . 10. Post-hoc Tukey tests revealed that participants in the Elastic Maze II (M = 6.167, SD = .83) rated themselves as being more immersed than participants in Elastic Maze I (M = 5.2, SD = .96), p = .02. The ANOVAs conducted on location awareness, nausea, and video game experience did not reveal any significant effects or interactions. A 2 (environment) x 2 (tasks) between-subjects ANOVA conducted on aggregated Santa-Barbara Sense of Direction (SBSOD ) scores revealed a main effect of environment, F1,44 = 4.58, p = .038, ηp2 = .09. Post-hoc Tukey tests did not reveal any statistically significant pairwise differences between groups. For the Road Map Test 95 (RMT), the ANOVA on percent correct did not reveal any statistically significant differences between groups. For the Perspective-Taking and Spatial Orientation Test (PTSOT), a one-way Watson-Williams test did not reveal any significant differences across the 4 conditions. Debriefing Responses. In the Elastic Maze II / Neighborhood condition, All participants reported noticing the outlines of the major paths superimposed on the ground plane during shortcuts.  9 of 12 participants reported using the outlines to guide their shortcuts. 10 of 12 participants reported that the gear and moon paths stretched across major paths. 4 of 12 participants noticed the bookcase and clock paths stretching, and one participant reported that the cactus path may have stretched. In the Elastic Maze II / Route condition, 10 of 12 participants reported that the gear and moon paths stretched across major paths; 4 participants reported that the bookcase path stretching, 3 reported noticing the clock path stretching; and several participants reported thinking that some control object paths were stretching (Cactus, 2; Earth, 1; Sink, 1). Thus, in contrast to Elastic Maze I, most participants in Elastic Maze II noticed the stretching due to the topological boundary information. Discussion The results of Experiment 1 suggested that neighborhoods may depend on topological rather than metric structure, Experiment 2 was design to directly address whether neighborhoods are derived from visual information about neighborhood boundary relations. For the Neighborhood Shortcut Task, we predicted (6) that if neighborhoods are based on topological boundaries, probe trial shortcuts in Elastic Maze 96 II should (6a) shift towards stretched target locations, (6b) be more variable, or (6c) be more bimodal than shortcuts in Elastic Maze I. For the Route Task, we predicted that (7) if spatial knowledge is primarily graph-like, participants should walk down the correct path despite varying metric and neighborhood structure; and (8) given that graph knowledge includes neighborhood boundary relations (Kuipers, Tecuci, & Stankiewicz, 2003), endpoints should (8a) shift towards stretched target location, (8b) be more variable, or (8c) be more bimodal in Elastic Maze II than in Elastic Maze I. Contrary to our predictions for both neighborhood shortcut (6a, 6b) and Route Tasks (8a, 8b), final angular errors did not show a significant shift in the direction of normal or stretched target locations, and the variability of angular errors did not differ significantly between Elastic Mazes I and II. However, bimodality can mask differences in constant and variable error. Indeed, consistent with Prediction 6c, circular kernel density estimates strongly suggested more multimodal shortcuts in Elastic Maze II than Elastic maze I for across-neighborhood probe trials, but not for within-neighborhood probe trials. F-ratios on cluster means were also consistent with Prediction 6c, and revealed that responses were more bimodal for across-neighborhood probe trials in Elastic Maze II than in Elastic Maze I. For the Route Task, endpoints were more bimodal in Elastic Maze II for both of the across-neighborhood probe trials, and one of the two within-neighborhood probe trials; one of the cluster means shifted towards the stretched location (S) of the target in Elastic Maze II for one of the across-neighborhood targets, but not for the other. Results for the Route Task should be treated with caution as the MLCV method did not suggest the presence of bimodal angular responses for any groups. 97 To the extent that the cluster analysis for the Route Task is well motivated, results are generally consistent with Prediction 8c because responses were more bimodal on across- neighborhood probe trials in Elastic Maze II. Taken together, the patterns of angular errors for the Neighborhood Shortcut and Route Tasks in Experiment 2 support the Topological hypothesis. Because responses were more multimodal when visual information about neighborhood boundary relations was available during learning, results suggest that neighborhoods are based on topological boundaries rather than metric structure. However, it is important to note that responses on within-neighborhood stretch trials may have been affected by the absence of boundaries in the infinite hallways during testing. Endpoint analysis for the Neighborhood Shortcut Task revealed a pattern of responses consistent with Prediction 6a: the percentage of endpoints falling in short neighborhoods was significantly lower in Elastic Maze II than in Elastic Maze I. The endpoint analysis was also consistent with Prediction 6b: the percentage of endpoints falling on paths and in long neighborhoods was more variable in Elastic Maze II than in in Elastic Maze I. Contrary to prediction (8a-c), endpoint analysis did not reveal any significant differences between environments for the Route Task. That endpoint differences were found for the Neighborhood Shortcut Task but not the Route Task may be due to the fact that neighborhood boundaries were visible (major paths were superimposed on ground plane) during the Neighborhood Shortcut Task, but no visual information about neighborhood boundaries was available in the Route Task (major paths were not visible on the ground in the infinite hallways). If this is the case, these results 98 provide further support for Prediction 6, and suggest that neighborhoods are derived from topological boundaries. Finally, consistent with Prediction 7, analyses of path choices did not reveal any significant differences between Elastic Mazes I and II. Consistent with the Topological Hypothesis, participants walked down the correct paths despite varying metric and neighborhood structure in both mazes, suggesting that spatial knowledge that is primarily graph-like. Conclusions The results of Experiment 2 revealed that providing visual information about neighborhood boundary relations during learning and testing (Neighborhood Shortcut Task in Elastic Maze II) resulted in more bimodal responses and a shifted distribution of endpoints in short and long neighborhoods. This suggests that neighborhoods are derived from topological boundaries rather than metric structure. When information about neighborhood boundaries was available during learning but not during testing (i.e. in the Route Task) no differences were found between Elastic Mazes I and II. Participants in both mazes also walked down the correct paths in the Route Task despite varying metric and neighborhood structure, consistent with primarily graph-like spatial knowledge. The spatial knowledge acquired is therefore consistent with a labeled graph that includes local information about the distances to targets (e.g. down a path, as in Experiment 1); although neighborhoods appear to based primarily on visual information specifying topological neighborhood boundaries, the results of Experiment 2 are also tentatively 99 consistent with graph knowledge that includes some knowledge of topological neighborhood boundaries. 100 Chapter 4: Experiment 3 Introduction Experiments 1 and 2 destabilized neighborhoods while holding the place graph constant; the spatial knowledge acquired was consistent with a topological graph supplemented with local metric information about the distances to objects along paths. The results of Experiment 1 further suggested that neighborhood structure is not derived from Euclidean structure (via path integration), and the results of Experiment 2 suggested that neighborhoods are primarily derived from visual information about neighborhood boundaries Experiment 3 was designed to destabilize the graph while holding neighborhoods constant. The Swap Maze was designed to destabilize the place graph while keeping neighborhoods intact by swapping pairs of objects within neighborhoods during learning. If spatial knowledge preserves whatever structure remains stable during learning, participants should learn the neighborhoods. But if neighborhoods depend on the graph, then neighborhood performance should be worse in the Swap Maze than in the matched Control Maze conditions from Experiment 1. This leads to several more specific predictions for the two tasks. Predictions Neighborhood shortcut task. Prediction 9: If spatial knowledge always preserves stable structure, shortcut endpoints should fall in the preserved neighborhoods containing the correct (A/B) locations of swapped targets in both mazes. Prediction 10: If neighborhood structure depends on the place graph, shortcuts in the Swap Maze should 101 be (a) less accurate or more variable than in the Control Maze, and (b) should fall outside of the neighborhoods that are preserved despite object swapping. Route Task. Prediction 11: If spatial knowledge always preserves stable structure, participants should walk down paths in the correct (A/B) neighborhood in both mazes. Prediction 12: If spatial knowledge is primarily topological, participants should walk down the correct (A/B) paths in the Swap Maze despite varying metric and neighborhood structure. Methods Participants A total of 24 (12 female) additional participants were run the in the two Swap Maze conditions. Each condition consisted of 12 participants (6 female), and the mean age of participants who completed the study was 20.8 (SD = 3.2). One (female) participant was dropped from the Swap / Place condition due to simulator sickness. Pilot testing of the Swap Maze conditions revealed that some participants would not be able to complete the verification phase within the 25 minute time limit and would not continue to the test phase, limiting the amount of test phase data that could be collected. Thus, if participants had not completed the verification phase after 25 minutes, they went on to the test phase and were included in the final analysis. As a result, participants who may have otherwise been excluded had they participated in one of the Euclidean control conditions were nonetheless included in the final analysis. Virtual Environments The Control Mazes were the same as in Experiment 1. The Swap Maze was 102 identical to the Control Maze except that pairs of probe trial objects (bookcase/gear; clock/flamingo) exchanged locations repeatedly during the free exploration and verification phases. The initial position of each object was randomized between participants. For example, the flamingo may have been initialized to the clock’s canonical (i.e., as in the Control Maze) location, while the clock would be initialized to the flamingo’s canonical location. The first time a participant walked down the path containing the flamingo, they would see the flamingo; the next time they walked down the same path, they would see the clock; next, they would see the flamingo, and so on. If participants walked to the incorrect path when trying to find an object during the verification phase, they were guided to the alternative location for that object. This was Figure 33. Experiment 3: virtual environments. In the Swap Maze, pairs of objects alternately exchanged locations during the free exploration and verification phases every other time a participant walked down the associated paths. For example, the first time a participant walked down the clock path they might see the flamingo instead. The next time they walked down the same path, they would see the clock, and so on. This manipulation violated metric structure, but preserved neighborhoods. The positions of these probe trial objects were randomized at the start of free exploration for all participants. 103 done to ensure that participants would learn that a given object could appear in multiple locations, rather than learning that multiple objects appear in a particular location. Thus, in the Swap Maze a particular path could lead to two different "places," as defined by the objects that occupied them. Design The design of Experiment 3 appears in Figure 34. Two virtual environments (Control Maze, Swap Maze) were crossed with 2 tasks (Neighborhood Shortcut, Route). Figure 34. Design of Experiment 3. 2 tasks (Neighborhood Shortcut, Route Task) were crossed with 2 virtual environments (Control Maze, Swap Maze) for a total of 4 conditions. Experiment 1 data from the Control Maze were compared to data collected in two new conditions in the Swap Maze. 104 Procedure The procedure was the same as in Experiments 1 and 2 except for the verification phase. The verification phase in Experiment 3 was modified based on pilot testing showing that several participants were not able to meet the training criterion (finding all 10 objects two times each within 30 seconds for a given trial) in the 25 minute time limit. In order to improve the attrition rate and ensure that enough participants would move on to the test phase to collect data, participants in the Swap Mazes automatically continued to the test phase if they had not met the verification criterion within 25 minutes. Analysis The analysis was the same as Experiments 1 and 2, except that probe objects in the Swap Maze were the clock, gear, moon, and bookcase, and control objects were the flamingo, well, earth, cactus, rabbit, sink. Thus, the control/probe designations of objects in the Control Mazes were re-coded for statistical comparisons between the Control and Swap mazes. Results Free Exploration Phase Combined traces of exploration data for all participants in each condition appear in Figure 35. The ANOVAs on mean and SD of gate visits did not reveal any statistically significant differences between groups, suggesting that participants explored Control and Swap mazes to similar extents. 105 Figure 35. Experiment 3: free exploration path traces. Position Data (x,y) plotted for all participants in each of the four conditions. ANOVAs on mean and SD of gate visits did not reveal any statistically significant differences between groups, suggesting that participants explored Control and Swap mazes to similar extents. Verification Phase The results for number of trials per object required to reach criterion revealed that participants required more attempts in the Swap Maze (M = 5.2, SD = 1.1) than in the Control Maze (M = 4.2, SD = 0.4) to reach the criteria set for the the verification phase, F1,44, = 27.1, p < .001, ηp2 = .12. To investigate this in more detail, a 2 (environment) x 2 (tasks) ANOVA was conducted on the total number of trials required to reach criterion, where environment (Control, Swap) and task (Neighborhood, Place) were between subject factors. The ANOVA revealed a significant main effect of environment, F1,44 = 106 27.1, p < .001, ηp2 = .38; number of trials to reach criterion was higher in the Swap Mazes (M = 25.8, SD = 2.89) than the Control Mazes (M = 20.8, SD = .97). This implies that the Swap Maze was harder to learn than the Control Maze. Participants needed to be guided to both probe and control trial targets more often in the Swap Maze (1.6 times on average) than in the Control Maze (.27 times on average). This suggests that the Swap Maze was more difficult to learn than the Control Maze. A 2 (environment) x 2 (tasks) x 2 (trial type) mixed-model ANOVA was conducted on the mean number of times participants were guided to targets, where environment (Control, Swap) and trial type were between subjects factors, and trial type (control, probe) was the within subjects factor. The ANOVA revealed a main effect of trial type, F1,38 = 16.8, p < .001, ηp2 = .31; participants were guided a greater number of times on probe trials relative to control trials. There was a large main effect of environment, F1,38 = 102, p < .001, ηp2 = .73; participants were guided more in the Swap Maze than in the Control Maze. There was also a comparatively smaller yet significant main effect of task, F1,38 = 5.45, p = .025, ηp2 = .13; participants in the Control/Route condition required more guidance than in the Control/Neighborhood condition. There was also a significant task x environment interaction, F1,38 = 2.29, p = .017, ηp2 = .14; participants in the Swap/Neighborhood condition were guided less (M = 1.28; SD = .41) compared to those in the Swap/Route condition (M = 1.91; SD = .48), whereas participants in the Control conditions required less guidance to targets in the both Neighborhood Shortcut Task (M = .27; SD = .45) and Route Task (M = .25; SD = .48). This may be due to the second verification block which occurs after the first set of 107 learning trials. There was also a significant environment x trial type interaction, F1,38 = 9.4, p < .01, ηp2 = .20. For participants in the Swap Maze, the mean number of times participants were guided was higher on probe (M = 2.12; SD = .58) than control (M = 1.08; SD = .58) trials, whereas participants in the Control Maze required relatively equal guidance to control (M = .19; SD = .38) and probe (M = .34; SD = .55) targets. Collectively, these results imply that participants had difficulty learning the locations of swapped targets. Test Phase Traces of shortcuts for Experiment 3 are plotted in Figures 36 and 37. It appears that the shortcuts in the Swap Maze diverge more than those in the Control maze, particularly so for Probe trials. Final Angular Error (FAE). Neighborhood shortcut task. Mean constant errors in the Neighborhood Shortcut Task appear in Figure 38a. Watson-Williams tests on CE did not reveal any effects of environment or trial type. Because there is no practical test for interactions in circular data, results of pairwise post-hoc Watson-Williams tests are shown as Duncan groupings in Figure 38. In the Swap Maze, constant errors were significantly more negative on control trials than probe trials; however, there is no obvious reason why this should be the case. Variable Error appears in Figure 38b, which indicates greater mean angular deviations on Probe trials in the Swap maze. The ANOVA on angular deviations revealed a main effect of environment, F1,22 = 7.81, p = . 011, ηp2 = .25, a main effect of trial type, F1,22 = 8.22, p < .01, ηp2 = .03, and an 108 !"#$%&'(%'')*+%'(,-.,*/012 3'4,('5*607" +@0A*607" 3'4,('5*/(#051 8('&"*/(#051 3'4,('5*/(#051 8('&"*/(#051 *:''2-01" *:''2-01" 90(,% 90(,% +#42 +#42 +#42 +#42 ="0( ="0( ;"55 ;"55 <0&&#, <0&&#, <0&&#, <0&&#, 6''4 6''4 35'-2 35'-2 90(,% 90(,% ;"55 ;"55 >50?#4$' >50?#4$' +#42 +#42 90(,% 90(,% ;"55 ;"55 Figure 36. Experiment 3: shortcut data, neighborhood shortcut task. Position data for individual shortcuts in the Control Maze and the Swap Maze. Control Maze data is reproduced from Experiment 1. Individual shortcuts are shown as black paths radiating from the mean starting point (i.e. object A’s approximate location). For the Swap Maze, swapped objects are plotted as circles connected by checkered black and white lines. Large circles: target objects. Medium circles: mean starting point of shortcuts (at the approximate location of of Object A). Small circles: endpoints of shortcuts for “full trials” (i.e., trials on which participants clicked the mouse to indicate that they thought they had arrived at the target object location, and did not trigger the emergency walls). Dotted ellipses: 95% confidence ellipses for full trial shortcut endpoints are also. Arrows: mean vectors of shortcuts (length = mean distance from shortcut starting points to endpoints for full trials; angle = mean final angular error for all trials, including trials on which participants triggered the emergency walls). 109 !"#$%&'()* +",$-".&/(0% 12(3&/(0% +",$-".&'-4(.) ?-":%&'-4(.) +",$-".&'-4(.) ?-":%&'-4(.) &7""*8()% &7""*8()% 5(-$6 5(-$6 14,* 14,* 14,* 14,* ;%(- ;%(- 9%.. 9%.. !(::4$ !(::4$ !(::4$ !(::4$ /"", +."8* +."8* /"", 5(-$6 9%.. 9%.. 5(-$6 <.(=4,>" <.(=4,>" 14,* 14,* 5(-$6 5(-$6 9%.. 9%.. Figure 37. Experiment 3: shortcut data, route task. Position data for individual shortcuts in the Control Maze and Swap Maze. Control Maze data is reproduced from Experiment 1. Shortcuts are shown as black paths between the mean starting point (i.e. object A’s approximate location) and target location. For the Swap Maze, swapped objects are plotted as circles connected by checkered black and white lines. Mean vector color corresponds to the target participants were asked to walk (e.g. pink = flamingo). Large circles: target objects. Medium circles: mean starting point of shortcuts (at the approximate location of of Object A). Small circles: endpoints of shortcuts for “full trials” (i.e., trials on which participants clicked the mouse to indicate that they thought they had arrived at the target object location, and did not trigger the emergency walls). Dotted ellipses: 95% confidence ellipses for full trial shortcut endpoints are also. Arrows: mean vectors of shortcuts (length = mean distance from shortcut starting points to endpoints for full trials; angle = mean final angular error for all trials, including trials on which participants triggered the emergency walls). 110 Figure 38. Experiment 3: final angular error. Errors were normalized so that 0º corresponded to perfect accuracy to the control target on control trials, or the A location of the swapped target on probe trials. Thus, for probe trials, a positive shift in angular error indicates a shift towards the B location of swapped targets. Error bars indicate +/- 1 SEM. Duncan flags indicate significant (p < .05) post-hoc Tukey tests. environment x trial type interaction, F1,22 = 4.45, p = .047, ηp2 = .016. Post-hoc Tukey tests revealed that VE was approximately 15º higher overall in the Swap Maze (M = 30.9º, SD = 5.7º) than in the Control Maze (M = 16º, SD = 1.15º). Results of tests for the interaction appear as Duncan flags in Figure 38. Route task. For the Route Task, the Watson-Williams tests on CE (Figure 38c) revealed a significant main effect of environment, F1,46 = 6.37, p = .015; mean angular errors in the Swap Maze (M = .31º, AD = 5.9º) were slightly but significantly different 111 from angular errors in the Control (M = -3.12º, AD = 2.73º) maze , and no other significant effects were found. Results of post-hoc Watson-Williams tests appear in Figure 38. Thus, CE was slightly greater on probe trials in the Swap Maze. For VE (Figure 38d), there was a main effect of environment F1,22 = 39.7, p < .001, ηp2 = .46, and a main effect of trial type, F1,22 = 13.5, p < .01, ηp2 = .25. There was also a significant environment x trial type interaction, F1,22 = 18.3, p < .001, ηp2 = 0.31. Once again, VE was significantly higher on probe trials in the Swap maze than in the other conditions. Results of post-hoc Tukey tests for the interaction are shown as Duncan flags in Figure 38. The analysis of angular errors for both the neighborhood shortcut and place graph are consistent with the Topological Hypothesis, Prediction 10a: for both tasks, shortcuts were more variable in the Swap Maze than the Control Maze, suggesting that neighborhood structure depends on the place graph. Endpoints. Endpoints were classified as falling within the correct (containing the A/B location of the swapped target), wrong, and path neighborhoods for Experiment 3. Percentages obtained from the endpoint classification analysis appear in Figure 39. Neighborhood task. The percentage of endpoints in the correct (A/B) neighborhood (Figure 39a) dropped from 85.5% (SD = 14.1%) in the Control Maze to 66.9% (SD = 32.9%) in the Swap Maze. Specifically, in the Swap Maze there were fewer correct endpoints on probe trials than control trials, whereas the opposite was the case in the Control Maze. The ANOVA revealed a significant environment x trial type interaction, F1,22 = 6.9, p = .016, ηp2 = .01. For the mean percentage of endpoints falling 112 in the wrong neighborhood, the ANOVA revealed a significant environment x trial type interaction, F1,22 = 7.4, p = .013, ηp2 = .012. Figure 39. Experiment 3: percentages of endpoints falling in neighborhoods. More endpoints fell in the wrong neighborhood in the Swap Maze (M = 24.1%, SD = 24%) than the Control Maze (M = 9.1%, SD = 10.5%). In the Swap Maze more endpoints fell into the wrong neighborhood on probe trials (M = 25.9%, SD = 25.5%) than control trials (M = 22.3%, SD = 23.1%), whereas in the Control Maze more did so on control trials (M = 11.5%, SD = 13.2%) than probe trials (M = 6.9%, SD = 8.2%). Route task. For the Route Task, over 90% of the endpoints fell in the correct (A/ B) neighborhood in all conditions (Figure 39b). Nevertheless, more endpoints fell in the wrong neighborhood on probe trials in the Swap Maze (M=8.7%, SD=10.1%) than on control trials in the Swap Maze (M-2.5%, SD=4.4%) or on probe trials (M=0%, SD=0%) or control trials (M=0.4%, SD=1.1%) in the Control Maze. The corresponding ANOVA confirmed significant main effects of environment, F1,22 = 8.53, p = .008, ηp2 = .21, and trial type, F1,22 = 6.98, p = .015 ηp2 = .10, and a significant environment x trial type 113 interaction, F1,22 = 5.12, p = .034, ηp2 = .07. The SD of percentage of endpoints falling in the wrong neighborhood was higher in the Swap Maze (M = 7.3%, SD = .05%) than the Control Maze (M = .4%, SD = .01%); the corresponding ANOVA revealed a main effect of environment, F1,22 = 14.3, p < .001, ηp2 = .25. Consistent with the Topological Hypothesis (Prediction 9) but not the Stability Hypothesis (Prediction 10), more endpoints fell in correct (A/B) neighborhoods in the Control Maze; in addition, more endpoints fell in wrong neighborhoods in the Swap Maze on both probe and control trials for both neighborhood and Route Tasks. This suggests that participants did not acquire stable neighborhoods even though neighborhoods remained stable despite object swapping. Path choices Participants took the correct path (either A or B) on over 95% of trials in both the Control Maze and the Swap Maze. Nevertheless, the chose incorrect paths more often in the Swap Maze (M=2.36%, SD=2.49%) than in the Control Maze (M=0.04%, SD=0.14%). A mixed-model ANOVA on the percentage of incorrect paths revealed a significant main effect of environment, F1,21 = 13.7, p < .01, ηp2 = .37, and no other significant effects. This suggests that instability in a portion the graph made it slightly more difficult to acquire the graph in the Swap Maze. Path choices within the swap maze. However, more detailed analyses of path choices within the Swap Maze suggest that participants were able to learn which portions of the graph remained stable. Percentages of path choices are shown in Figure 40. The canonical location of the target, A, was chosen more on control trials (M = 96.3%, SD = 114 6.6%) than on probe trials (M = 36.7%, SD = 21.7%); the corresponding ANOVA revealed a main effect of trial type, F1,11 = 74.8, p < .001, ηp2 = .79. This confirms that Figure 40. Experiment 3: path choices in the swap maze. Percentages of paths chosen on control and probe trials in the Swap Maze, broken down by A/B paths, A/B neighborhood, and wrong (non-A/B) neighborhood. participants did not always walk to the canonical location, A, of probe trial targets as they did in the Control Maze. Path B (swapped location of the target) was chosen more on probe trials (M = 54.1%, SD = 20.1%) than on control trials (M = 0%, SD = 0%); the corresponding ANOVA for path B revealed a main effect of trial type, F1,11 = 81.4, p < . 001, ηp2 = .88. Paths in the same neighborhood (A/B target pair neighborhood on probe trials) as the target were chosen more on control trials (M = 2.18%, SD = 3.75%) than probe trials (M = 0%, SD = 0%); the corresponding ANOVA revealed a main effect of trial type, F1,11 = 4.04, p = .07, ηp2 = .27. Paths in the wrong neighborhood were chosen more on probe 115 trials (M = 8.25%, SD = 10.7%) than on control trials (M = 1.56%, SD = 3.88%); the corresponding ANOVA revealed a main effect of trial type, F1,11 = 6.77, p = .025, ηp2 = . 16. Participants chose paths in the wrong neighborhood more for swapped targets than control targets, even though the swapping preserved neighborhoods. These results are generally consistent with the Topological hypothesis (participants learned that swapped targets occupied either of two locations), and inconsistent with the Stability Hypothesis (participants did not learn that neighborhoods were stable despite object swapping). It is possible that participants may have used strategies of minimizing metric or topological distance when walking to targets. Visual examination of Figure 40 suggests that participants may have preferred Path B to Path A. However, participants did not tend to favor either the canonical (path A) or the swapped (path B) path on probe trials; the ANOVA on the percentage of times path A vs. path B was chosen on probe trials did reach significance (p = .19) Visual examination of path traces (Figures 36 and 37) suggests that path preference may have varied depending upon the starting location and the target: participants seemed to prefer the bookcase location when asked to walk to the bookcase or gear, whereas participants seemed to prefer the clock and flamingo locations approximately equally when asked to walk to the clock or flamingo. This was confirmed by one-way ANOVAs conducted on A/B path choices for each of the target pairs. No significant differences in A/B path choices were found for the clock/flamingo target pair. For the bookcase/gear pair, participants preferred the bookcase location on both rabbit → gear trials (F1,11 = 15.1, p < .01, ηp2 = .46; M = 71.3%, SD = 24.2%), and on sink → bookcase trials, (F1,11 = 18.9, p < .01, ηp2 = .48; M = 63.9%, SD = 31.3%). 116 Both the metric distance (in meters) and topological distance (in terms of number of junctions encountered en route to the target path) from the rabbit to the B (bookcase) location of the gear were shorter (~10.5m, 4 junctions) than the metric and topological distance (~22.5m, 6 junctions) to the A (gear) location of the gear; thus, that participants walked to the B (bookcase) location than the A (gear) location of the gear more often on rabbit → gear trials does not distinguish metric and topological strategies for path selection. However, for sink to bookcase trials, the metric distances from the sink to the A and B locations of the bookcase were approximately equal (22m, 22.5m), while the topological distance from the sink to the canonical A (bookcase) location was less (3 junctions) than the topological distance to the B (gear) location (5 junctions). In addition to the ANOVA, a Pearson product-moment correlation coefficient was computed to assess the relationship between distance (topological/metric) and percentage of shorter paths chosen. Correlations both topological (r = .77, n = .77, p = .07) and metric (r = .61, n = 6, p = .19) distance both failed to reach significance, although the correlation for topological distance was both higher and marginally significant. Although this latter result shows a trend towards an implicit topological strategy, non-significant correlations between topological/metric distance and percentages of paths do not permit any definitive conclusions. On approximately 90% of probe trials in the Swap Maze, participants walked down either an A or B target path, suggesting that participants had learned that objects would appear in specific locations in the graph. The percentage of correct (A) paths chosen on control trials was also very high, at 96.2%, suggesting that participants had 117 learned that the locations of control targets remained stable despite the fact that the locations of probe targets were unstable. Although participants had some difficulty acquiring the Swap Maze graph, and did not acquire stable neighborhoods, graph performance was highly robust: results of the path choice analysis suggest that participants were able to learn which portions of the graph remained stable, and that probe targets with unstable metric locations occupied two specific places in the graph. Response time. For the Neighborhood Shortcut Task, the ANOVA on mean response time revealed a significant environment x trial type interaction, F1,22 = 6.01, p = .023, ηp2 = .008. Post-hoc Tukey tests for the main effect of environment revealed that response time was higher overall in the Swap Maze (M = 9.32s, SD = .28s) than in the Control Maze (M = 7.9s, SD = .74s), p = .037. Pairwise Tukey tests for the interaction did not reach significance. For SD of response time, no main effects or interactions were found. Higher mean response time in the Swap maze suggests that participants may have spent more time considering where to respond when turning to targets on both control and probe trials. Thus, uncertainty about probe target locations may have increased uncertainty about control target locations in the Swap maze. Group differences. No significant differences between groups were found for sketch maps, location awareness, immersion, nausea, and video game experience. Debriefing Responses. All participants reported noticing that some of the objects were swapping with one another, variously describing the pattern as the objects "changing," "alternating," "switching," "swapping." Two participants (1 female) thought it was necessary to visit a nearby object (e.g. the rabbit) in order to make another object 118 (e.g. the clock) alternate with its paired object. All but one of the participants correctly reported that the pairs of probe objects were swapping with one another in a predictable pattern; one male reported thinking that probe objects were randomly changing places with one another, and did not detect any discernible pattern. One male participant noticed that the sink might have been swapping with the clock and flamingo as well [interesting because the sink is in the same neighborhood]. Six participants (4 female) did not notice that the pairs of probe objects would swap positions in an alternating pattern in the first session, and therefore walked back and forth between the two possible probe trial target locations during the first verification phase session (if they did not find the target on the first try). All but two of these participants (1 female) discovered the alternating pattern by the end of the 2nd verification phase and took advantage of it when finding the objects during the verification phase. All participants reported noticing the outlines of the major paths superimposed on the ground plane during shortcuts. Two participants (both female) reported that they did not explicitly use these paths to help guide their shortcuts. Two other participants (1 female) reported explicitly using the paths to guide their shortcuts in the 2nd session, but not in the first. In the Swap/Route condition, all participants correctly reported noticing that pairs of objects were swapping with one another. One female participant reported noticing that the clock and flamingo were swapping with one another, but was not aware that the gear and bookcase were swapping. 119 Discussion For the Neighborhood Shortcut Task, we predicted that (9) if spatial knowledge always preserves stable structure, shortcut endpoints should fall in the preserved neighborhoods containing the correct (A/B) locations of swapped targets in both mazes; conversely (10) if neighborhood structure depends on the place graph, shortcuts in the Swap Maze should be (10a) less accurate or more variable than in the Control Maze, and (10b) should fall outside of the neighborhoods that are preserved despite object swapping. For the Route Task, we predicted that (11) if spatial knowledge always preserves stable structure, participants should walk down paths in the correct (A/B) neighborhood in both mazes, and (12) if spatial knowledge is primarily topological, participants should walk down the correct (A/B) paths in the Swap Maze despite varying metric and neighborhood structure. The Swap Maze was more difficult to learn than the Control Maze: more trials were required to reach the verification phase criterion in the Swap Maze, and participants also needed to be guided more to both probe and control targets in the Swap Maze. Consistent with the Topological Hypothesis (10a), the analysis of final angular error for both the neighborhood shortcut and Route Tasks revealed that shortcuts were more variable overall in the Swap Maze, especially when participants were asked to walk to swapped targets. Contrary to the Stability Hypothesis (9), for the Neighborhood Shortcut Task, more endpoints fell in the correct (A/B) neighborhood in the Control Maze, and more endpoints fell in the wrong neighborhood in the Swap Maze on both control and probe 120 trials. For the Route Task, more endpoints fell in the wrong neighborhood in the Swap Maze on both probe and control trials, and the percentage of shortcuts falling in wrong neighborhoods was also more variable in the Swap Maze. Results of endpoint analyses were consistent with the Topological Hypothesis (10a, 10b), suggesting that neighborhoods depend on the place graph. Contrary to the Stability Hypothesis (9), more endpoints fell in wrong neighborhoods on control trials in the Swap Maze than in the Control Maze on both the neighborhood shortcut and Route Tasks. A possible explanation for this result is that uncertainty about the neighborhood membership of probe trial targets increased uncertainty about the neighborhood membership of control trial targets, even though both control and probe targets always remained in the same neighborhood (i.e. neighborhoods were always preserved despite swapping). Higher overall response times when turning to face targets on probe and control trials in the Swap maze provide further support for this result. Consistent with the Topological Hypothesis (12), and contrary to the Stability Hypothesis (11), examination of path choices within Swap Maze revealed that participants chose paths in the A/B neighborhood less on probe trials than on control trials, and chose paths in the wrong neighborhood significantly more on probe trials than on control trials. These results suggest that participants did not learn that neighborhoods were stable despite swapping, and further suggest that neighborhood structure depends on the place graph: if the place graph is unstable, but neighborhoods are preserved, participants have a difficult time acquiring neighborhood structure. In addition, 121 participants walk predominantly to the A/B neighborhood of swapped targets when they are able to walk through the maze to targets in the Route Task, also consistent with neighborhoods depending on the place graph. In sum, although the Swap Maze was more difficult to learn than the Control Maze, participants still acquired relatively accurate graph knowledge: participants generally took shortcuts to—and selected paths within—the A/B neighborhood. Participants learned that the objects were swapping, but measures of neighborhood performance suggest that they had difficulty acquiring the neighborhoods preserved by swapping. Conclusions The results of Experiments 1 and 2 suggested that neighborhood structure is not derived from Euclidean structure (Experiment 1), but may be derived from visual information about neighborhood boundaries or from topological (graph) structure (Experiment 2). Taken together, the results of Experiment 3 provide support for the Topological Hypothesis but not the Stability Hypothesis. Graph structure appears to be the primary form of spatial knowledge acquired because participants have more difficultly learning an environment in which the graph is unstable during learning, and this instability in the graph interferes with the acquisition of neighborhoods. Participants were able to learn that objects were swapping between two locations, indicating that participants are able to learn the graph despite its instability, including which specific portions of the graph are stable, and which portions are unstable. As in Experiments 1 and 2, the spatial knowledge acquired is consistent with a labeled graph. Consistent with the Topological Hypothesis, Experiment 3 further suggests that graph structure is 122 primary, that neighborhood structure may depend on the graph, and that route choices through the graph may be based on an implicit strategy of minimizing topological distance to the target, even when the target’s location is unstable. 123 Chapter 5: Comparison of Experiments by Task Experiments 1-3 examined specific hypotheses about the relationship between metric structure, neighborhood structure, and graph structure by making focused comparisons between particular conditions included in overall experimental design. To integrate findings across all three experiments, this section compares results across all four environments (Control Maze, Elastic Maze I, Elastic Maze II, Swap Maze) by task (Metric Shortcut, Neighborhood Shortcut, Route), including new statistical analyses that compare all four environments. Results are discussed with respect to their general implications for the three main hypotheses under investigation: the Euclidean Hypothesis, the Topological Hypothesis, and the Stability Hypothesis. Summaries of statistical comparisons by task appear in Figure 41. Shortcut Task. The Shortcut Task was primarily designed to investigate The Euclidean Hypothesis by probing acquisition of metric structure when that structure was stable (Control Maze), or unstable (Elastic Maze I), and. The Euclidean Hypothesis predicts that if spatial knowledge is primarily Euclidean, shortcuts should be less accurate or more variable in Elastic Maze I (because Euclidean structure is unstable) than in the Control Maze. Contrary to this expectation, we found no significant differences in Final Angular Error between the Control Maze and Elastic Maze I. This result is summarized in Figure 41 (panels a and b) by the Absolute Errors of FAE. Watson-Williams tests conducted on final angular errors did not reveal any significant differences between Control and Elastic I mazes on probe or control trials, for either constant (control, p = . 41; probe, p = .38) or variable errors (control, p = .86; probe, p = .76). However, VE was 124 Figure 41. Comparison of experiments by task. Left: performance on the Shortcut Task as measured by mean absolute final angular error. Center: performance on the Neighborhood Shortcut Task measured by mean percentage of endpoints falling in incorrect neighborhoods; note that incorrect responses were considered to be any non- short/long neighborhood in the Elastic Mazes, and any non-A/B neighborhood in the Swap Maze. Bars represent percentage of shortcut endpoints falling outside the target neighborhood (but not on paths, and not in the long neighborhood for the Elastic Mazes) neighborhood. Right: performance on the Route Task. Bars indicate % of trials on which participants walked down the incorrect path. Eucl. = Control Maze, Elst.I = Elastic Maze I, Elst.II = Elastic Maze II, Swap = Swap Maze. n.s. denotes non-significant one-way ANOVA. Duncan flags denote significant Tukey tests. very high in both mazes (Figure 41, panels a and b). These results are consistent with two possibilities: (1) spatial knowledge may be primarily Euclidean but is highly imprecise, or (2) spatial knowledge is not primarily Euclidean. The ability to take imprecise shortcuts is also consistent with topological spatial knowledge resembling a labelled graph in which edges are labeled with local information about the distances 125 between neighboring nodes, and the nodes are labeled with information about the angles between adjacent edges. Neighborhood shortcut task. The Neighborhood Shortcut Task was primarily designed to investigate the Stability Hypothesis and the Topological Hypothesis by examing whether neighborhoods are acquired when all 3 kinds of geometric structure were stable during learning (Control Maze), when only graph structure was stable (Elastic Mazes I and II), and when only neighborhood structure was stable (Swap Maze). In Elastic Maze I, information about neighborhood boundary relations on stretched paths was only available through path integration; in Elastic Maze II, information about neighborhood boundary relations was available both visually and through path integration. The Stability Hypothesis predicts that performance on the Neighborhood Shortcut Task should be equivalent when neighborhood structure is preserved (i.e. in the Control Maze and the Swap Maze), and should decline when neighborhood structure is unstable (Elastic Mazes). The Topological Hypothesis predicts that if neighborhoods are derived from topological graph structure, performance on the Neighborhood Shortcut Task should be significantly worse in the Swap Maze (because the place graph is unstable) than in the other environments. Comparing across experiments, there were significantly more neighborhood errors in the Swap maze than in the other mazes. For control trials, a new one-way ANOVA on percent incorrect neighborhoods chosen was significant, F3,44 = 3.01, p = . 039, ηp2 = .10; however, none of the post-hoc Tukey tests reached significance, though there was a non-significant trend towards higher percent incorrect in the Swap Maze. For 126 probe trials, the one-way ANOVA was significant F3,44 = 4.36, p < .01, ηp2 = .23; post-hoc Tukey tests revealed that incorrect neighborhoods were chosen more often in the Swap Maze (M = 21.4%, SD = 4.5%) than in the other three mazes. Results for the Neighborhood Shortcut Task are contrary to the Stability Hypothesis, but consistent with the Topological Hypothesis. Performance on the Neighborhood Shortcut Task was poorest in the Swap Maze, despite the fact that neighborhoods were preserved (Figure 41d). Importantly, when the graph structure was destabilized in the Swap Maze, this undermined acquisition of the neighborhood structure. These results imply that graph structure is primary, and that neighborhoods depend on topological graph structure. The difference between Elastic Mazes I and II lay in the available information for neighborhood boundaries on the stretched paths. In Elastic Maze I, neighborhood boundaries were specified by metric information derived from path integration, whereas in Elastic Maze II boundaries were also specified by qualitative visual information for path intersections -- that is, for topological boundary relations. The results of Experiment 2 revealed that angular responses were more bimodal on probe trials in Elastic Maze II than in Elastic Maze I. Taken together, results for the Neighborhood Shortcut Task suggest that neighborhoods depend on topological boundary relations rather than metric structure. Given that these topological boundaries comprise the edges in a labeled graph, this implies that neighborhoods are derived from graph knowledge, rather than being derived from Euclidean knowledge or acquired independently. These results are thus 127 consistent with the Topological hypothesis, and are inconsistent with the Stability hypothesis. Route Task. The Route Task was designed to assess whether the place graph is acquired when all 3 kinds of geometric structure are stable (Control Maze), when only graph structure is stable (Elastic Mazes I and II), and when only neighborhood structure is stable (Swap Maze). The Euclidean Hypothesis predicts that if graph knowledge is derived from Euclidean structure or from neighborhoods, participants may walk down the wrong path more in the Elastic Mazes than in the other environments. The Topological Hypothesis predicts that if spatial knowledge is primarily graph-like (topological), then participants should walk down the correct in the Elastic Mazes, but may make errors when the graph is destabilized in the Swap Maze. The Stability Hypothesis hypothesis predicts that if stable structure is always acquired, participants should walk down the correct paths in the Elastic Mazes despite varying metric structure, but may make errors in the Swap Maze if the graph structure is destabilized. Performance on the Route Task was overwhelmingly accurate in all four mazes, with correct paths chosen on nearly all trials in every condition (Figure 41e,f). However, consistent with the Topological and Stability hypotheses, the Swap Maze was harder to learn and there were slightly more errors during the test phase. For control trials, an ANOVA on the percentage of incorrect paths was significant, F3,43 = 3.03, p = .039, ηp2 = .17; post-hoc Tukey tests revealed that incorrect paths were chosen more frequently in the Swap Maze (M = 1.55, SD = 2.16) than in the Control Maze (M = 0, SD = 0), and no other significant differences were found. For probe trials, the ANOVA was also 128 significant, F3,43 = 5.3, p = .003, ηp2 = 0.27; post-hoc Tukey tests revealed that incorrect paths were chosen more often in the Swap Maze (M = 3.18, SD = 3.54) than in the other three environments. Nonetheless, participants chose correct paths (to the A/B locations) on over 96% of Swap Maze trials, indicating that they were able to learn the paths to the two locations in which a target object appeared. That is, they successfully learned the graph of the Swap Maze despite our efforts to destabilize it. This implies that graph knowledge may be primary, consistent with the Topological hypothesis. Contrary to the Stability Hypothesis, participants chose more paths to incorrect neighborhoods on probe trials in the Swap Maze than the other three mazes. This indicates that object swapping undermined neighborhoods even though neighborhood structure was preserved, implying that neighborhood knowledge depends on graph knowledge. Results are also inconsistent with the Euclidean Hypothesis because participants walked down incorrect paths with roughly the same frequency in the Elastic I, Elastic II, and Control Mazes. The overall pattern of results for the Route Task demonstrates that participants successfully acquired the graph in all four mazes (Figure 41e,f), despite our attempt to destabilize graph structure in the Swap Maze. The findings are consistent with the Topological hypothesis that graph knowledge is primary. Conclusions In sum, the results across Experiments 1-3 do not support the Euclidean Hypothesis or the Stability Hypothesis, but are consistent with the Topological Hypothesis. In general, performance on a particular task declines when the associated geometric structure is unstable during learning: (1) when metric structure is destabilized, 129 performance on the Shortcut task remains poor; (2) when neighborhood structure is destabilized, performance on the Neighborhood Shortcut Task declines, but—contrary to the Stability Hypothesis—it also declines when the place graph is unstable but neighborhoods are preserved (i.e. in the Swap Maze); (3) when the place graph is destabilized (Swap Maze), performance on the Route Task declines slightly, but is still highly successful over all mazes when compared with to performance on the Shortcut and Neighborhood Shortcut Tasks. Taken together, these results suggest that human spatial knowledge is primarily topological, and neighborhood knowledge depends on the place graph (including visual information for topological boundaries) rather than metric structure. The spatial knowledge acquired is not consistent with Euclidean survey knowledge, but is consistent with a labeled graph, which incorporates approximate distances and directions among neighboring nodes. 130 Chapter 6: General Discussion This dissertation critically examined the geometric structure of “cognitive maps” by evaluating three main hypotheses: the Euclidean Hypothesis, which posits that spatial knowledge has a primarily Euclidean metric structure; the Topological Hypothesis, which posits that spatial knowledge has a primarily graph-like structure, and the Stability Hypothesis, which posits that spatial knowledge reflects the specific geometric properties that remain stable during learning. To test these hypotheses, the present studies selectively destabilized three kinds of geometric properties during learning: metric structure, neighborhoods, and graph structure. The Control Maze preserved all three kinds of structure under investigation; the Elastic Mazes destabilized Euclidean and neighborhood structure while preserving the place graph; and the Swap Maze destabilized metric and graph structure while preserving neighborhoods. Further, Elastic Maze I provided metric information about neighborhood boundaries via path integration, whereas Elastic Maze II added visual information for topological boundaries. We asked participants to learn these environments, and then perform three navigation tasks that assessed their metric, neighborhood, and graph knowledge. The overarching predictions for each of the three main hypotheses were as follows: (1) The Euclidean hypothesis predicts that navigation performance on all tasks should be poor when metric structure is destabilized, because these weaker forms of knowledge are derived from metric spatial knowledge. (2) The Topological hypothesis predicts that performance on the Route Task should be high in all three environments, even in the 131 Swap maze if swapped paths can be learned. (3) The Stability hypothesis predicts that participants will acquire whatever geometric properties are stable during learning: specifically, the metric structure of the Control Maze, the graph of the Elastic Mazes, and the neighborhoods in the Swap Maze. Experiment 1 compared the Control Maze and Elastic Maze I, and tested whether neighborhoods are derived from metric spatial knowledge. The results of were generally consistent with the Topological Hypothesis. First, even though we attempted to destabilize metric structure in Elastic Maze I, metric shortcuts were highly imprecise in both mazes, and did not shift measurably in the direction of stretched targets in the Elastic Maze. This suggests two possibilities: (1) spatial knowledge is primarily Euclidean, and metric shortcuts are too variable to reveal that metric structure was destabilized, or (2) spatial knowledge is consistent with a labeled graph. In the Route Task, participants successfully acquired the graph and the neighborhoods, for they walked down the correct path to bimodal locations despite unstable Euclidean structure in Elastic Maze I. This provides converging evidence for primarily graph-like spatial knowledge. Specifically, participants appear to have learned the local metric distances to targets down a path, consistent with edge weights in a labeled graph. Participants clearly learned neighborhoods, for neighborhood shortcuts were less variable than metric shortcuts in both the Euclidean and Elastic I Mazes. But it appears that neighborhoods were derived from topological information about visible boundaries rather than metric information via path integration. First, if neighborhoods were based on metric information during learning, neighborhood shortcuts should have been more 132 bimodal or shifted for stretched targets in Elastic Maze I, yet they were not. This was the case despite the fact that participants were able to detect the target shifts, as shown by the JND analysis. Second, if neighborhoods were based on visible topological boundaries, endpoints should have fallen more frequently into the short neighborhood in Elastic Maze I, and indeed this was observed. Specifically, participants did not cross visible boundaries on stretched paths during learning, and thus tended to walk to the same (short) neighborhood during test. This pattern of results suggests that neighborhoods are derived from topological boundaries, which correspond to edges in a labeled graph, rather than from metric knowledge of object and boundary locations, consistent with the Topological hypothesis. In Elastic Maze I, metric information specified that stretched paths crossed a neighborhood boundary during learning, whereas visual information for topological boundaries specified that they remained within the short neighborhood. In Elastic Maze II, both metric and visual information specified that stretched paths crossed neighborhood boundaries, because intersections with maze corridors were visible during learning. For the Neighborhood Shortcut Task, visible boundaries resulted in multimodal angular errors on shortcuts, a lower percentage of endpoints falling in short neighborhoods, and more variability in endpoint locations in Elastic Maze II than in Elastic Maze I. In contrast, for the Route Task these effects were not observed, because the "infinite corridor" on test trials did not cross visible boundaries and thus specified that participants remained within the short neighborhood. This pattern of results is consistent with spatial knowledge resembling a labeled graph, which includes local information about the distances along a 133 path to a target (edge weights) as well as information about topological boundaries (the edges themselves). Moreover, correct paths were chosen on over 97% of test trials in both Elastic Mazes, indicating that the stable graph structure was successfully learned.. The results of Experiments 1 and 2 suggested that neighborhoods are not derived from Euclidean structure, but depend on visual information specifying topological boundaries. Experiment 3 was designed to provide a stronger test of the Topological and Stability hypotheses, by destabilizing graph structure while preserving neighborhood structure in the Swap Maze. Even though neighborhoods were stable, participants had difficulty learning them when the graph was destabilized: neighborhood shortcuts were more variable, more shortcut endpoints fell in the wrong neighborhood and reaction times were longer in the Swap Maze than in the Control Maze. Although the Swap Maze took longer to learn and participants took a few more incorrect paths, they still chose the correct path to the (A/B) target on over 96% of probe trials. Thus, they learned the graph of the Swap Maze, including the two paths that led to a given target. These results underscore the Topological hypothesis that the primary form of spatial knowledge is graph structure. In contrast, they are inconsistent with the Stability hypothesis, for participants were less successful at acquiring the neighborhood structure of the Swap Maze, even though it was completely stable during learning. Taken together, the results of Experiments 1, 2 and 3 are overwhelmingly inconsistent with the Euclidean and Stability hypotheses, but support the Topological hypothesis. Graph knowledge was successfully acquired in all four Maze environments, even when we attempted to destabilize it in the Swap Maze. Thus, the spatial knowledge 134 acquired was primarily topological and is consistent with a labeled graph (Figure 1) in which edges are labeled with local information about the distances between neighboring nodes, and the nodes are labeled with information about the angles between adjacent edges (e.g. the routes between familiar places). The preliminary evidence discussed in the introduction suggests that human spatial knowledge does not preserve Euclidean metric structure, but may instead preserve topological structure. However, previous research has not systematically investigated the relationship between these kinds of geometric structure, nor has it investigated the possibility that spatial knowledge always preserves stable structure. The results of the present experiments are inconsistent with the hypothesis that human spatial knowledge preserves Euclidean structure. Rather, labeled graph structure appears to be the primary structure acquired, and this labeled graph structure is acquired not matter what--even when other structure (metric structure, neighborhoods) is unstable during learning. Furthermore, neighborhoods appear to be derived from graph knowledge, rather than from metric knowledge obtained via path integration. Neighborhood boundaries are paths in the environment, which correspond to edges in a labeled graph. If visual information about boundary relations is available, this information could be incorporated into the graph, and distance and angle information used to update node and edge labels. For example, noisy path integration estimates could be refined in a manner consistent with Thrun’s (2008) simultaneous localization and mapping (SLAM) approach to robot navigation, in which positional uncertainty is refined by propagating the certainty of early positional estimates throughout the graph. Because visual information about 135 neighborhood boundaries is presumably less subject to noise and error, certainty about the adjacency and inclusion relations of topological neighborhoods could be used to resolve discrepant edge and node labels that result in locational estimates (e.g. of landmarks or familiar places) implying violations of neighborhood boundaries. In addition to being consistent with a labeled graph structure, the results of the present experiments are also compatible with the proposal that spatial knowledge has a hierarchical organization (Hirtle & Jonides, 1985; Montello, 1992), perhaps consisting of a “collage” of representations (Tversky, 1993), each characterized by different geometries and spatial scales (Anooshian, 1996; Montello, 1992). One objection to a labeled graph is that the spatial knowledge acquired in the present experiments is also consistent with distorted Euclidean knowledge, or with spatial knowledge that may be globally non- Euclidean but locally Euclidean. While the results of the present experiments are not able to distinguish between these three possibilities, a candidate form of spatial knowledge that can potentially accommodate the results of the present experiments—as well as the results of many of the studies reviewed in Chapter 1—in a more continuous and surface- like manner is a manifold of 2 or more dimensions. Manifolds are arbitrarily shaped, locally Euclidean topological spaces that can be globally non-Euclidean (Weeks, 2002). That is, in the neighborhood of any given point, a manifold can be unambiguously mapped (i.e. is “homeomorphic”) to a Euclidean plane. The terminology used to describe this feature of manifolds closely parallels the familiar language of geography: a manifold can be fully specified by an atlas of charts, such that any given chart is homeomorphic to Euclidean space, and neighborhoods appearing in 136 multiple charts can be transformed into one another via “transition maps.” For example, in a map of Europe, Germany may appear in more than one chart; because of the distortion introduced by map projection to a Euclidean plane the geometry (e.g. the area) of Germany may differ across charts. This feature of manifolds agrees well with Kuipers’ (2000) “spatial semantic hierarchy,” which proposes that the accumulation of error via noisy path integration could be overcome through a topologically-linked “patchwork” of local geometric maps within which path integration is quite accurate. The surface of a sphere is globally elliptic, but locally Euclidean, and it is precisely this fact that allows us to use straight lines (in the Euclidean sense) over short distances on the Earth’s surface. However, in order to plan the shortest routes between distant cities, we must rely on elliptic geometry: on the surface of a sphere, the geodesics (intrinsically straight lines) lie on great circles. This feature of manifolds agrees well with experiments demonstrating that spatial knowledge can be locally but not globally Euclidean (Ericson & Warren, 2010, 2012; Rothman & Warren, 2006; Schnapp & Warren, 2007; Warren et al., unpublished manuscript). There is another sense in which the geodesics (straightest lines between two points) on the surface of the Earth are not straight lines: we cannot walk through walls, and environmental features may impede our locomotion to differing degrees (e.g. hills of different slopes), which can lead to exaggerated distance estimates (Cohen, Baldwin, & Sherman, 1978; Kosslyn, Pick, & Fariello, 1974; Shemyakin, 1961). In addition, distance estimates may vary depending direction (e.g., McNamara & Diwadkar, 1997), or may be compressed (Wagner, 1985), consistent with hyperbolic or elliptic geometry (Watanabe, 1999). Manifolds can 137 potentially accommodate distance asymmetries through isotropy, and the effects of barriers or other perceptual distortions through homogeneity. In an isotropic manifold, distances are equivalent in all directions; in an anisotropic manifold, distances vary depending on direction. Globally, a manifold can be homogeneous having constant (Euclidean, 0; elliptic, +; or hyperbolic, -) curvature everywhere, or inhomogeneous (e.g. globally elliptic, +, but locally Euclidean, 0), having varying curvature. Thus, in the language of manifolds, spatial knowledge may also be describable as an inhomogenous (because the local geometry varies so as to impede locomotion) and anisotropic (because remembered distance may vary depending on direction) manifold. A potential criticism of manifolds as a model of spatial knowledge is that manifolds can account for arbitrary distortions in spatial knowledge, and arbitrary patterns of behavior; positing that spatial knowledge is consistent with an anisotropic and inhomogeneous manifold does not make any specific predictions about the kinds of navigational errors people should make, whereas positing that spatial knowledge is Euclidean or graph-like does make specific predictions. This is a fair criticism; because the results of the present experiments were overwhelmingly consistent with a labeled graph, and it appears that any form of spatial knowledge can be accommodated by manifolds, the labeled graph is to be favored based on the principle of parsimony. The discussion of manifolds presented here is simply offered to encourage further applications of mathematics to the study of spatial knowledge. To better understand the nature of the labeled graph knowledge acquired, future experiments should probe graph knowledge using paradigms inspired by the literature on 138 graph search algorithms (Even, 2011), robotics (e.g., Leiser & Zilbershatz, 1989; Gopal & Smith, 1990; Thrun, 2008; Kuipers & Byun, 1987; Kuipers & Levitt, 1988; Elfes, 1989; Moravec, 1988; Thrun & Bücken, 1996), space syntax (Hillier, 1996, 1999; Hillier & Hanson, 1984; Klarqvist, 1993), and models of human navigation ability that aim to incorporate empirically derived constraints on human spatial knowledge and navigation (e.g. Kuipers, 2000; Chown, Kaplan, & Kortenkamp, 1995; Tovar, Guilamo, & LaValle, 2004). For example, systematically investigating the robustness of human navigation to novel detours or unstable graph structure (e.g. adding or removing new paths during learning while objects locations remain fixed) in maze-like environments may yield insights about the relationship between human path planning strategies, formal models of graph search, and existing models of path planning currently implemented in mobile robots. Concluding Remarks This dissertation critically examined the structure of human spatial knowledge by testing three general hypotheses about the structure of human spatial knowledge: the Euclidean hypothesis, the Topological hypothesis, and the Stability Hypothesis. 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Journal of Vision, 9, 760a. 155 Appendices Appendix A: Instructions given to participants Risks Involved (verbal) "A small percentage of participants experience some dizziness, headache, or nausea while in virtual reality. You may not experience any of these things, but if you do, please let the experimenters know if you would like to take a break at any time. If you do not think you will be able to make it through 1 or 2 hours comfortably, please let us know as soon as possible; we will stop the experiment and you will still be paid for your time. There is a chance that we will get all of the data we need in the first session. If that happens, we will let you know at the end of this session, and there will be no need for a 2nd session. Do you have any questions?" Free Exploration Phase Instructions (recorded) "Feel free to walk around at your normal walking speed while in virtual reality, but please do not walk through any of the virtual walls. If you get too close to walls of the lab, virtual brick walls will appear and an audio file will tell you to stop where you are. If this happens, please stop immediately for your safety. In this experiment, you will be exploring a virtual hedge maze. You will begin the experiment at the home location which is marked by a baseball style home plate at your feet. First, you will explore the maze for 12 minutes, while trying to find all of the objects that are located in the maze. I cannot tell you how many objects there are, but you should try to find all of the objects and learn their locations. Later, you will be tested on your knowledge of the objects and their locations in the maze. When you get close to an object, you should walk up to it, and an audio file will play telling you the name of the object. In addition to objects, you will notice several paintings on the walls. You do not need to learn the names of the paintings or their locations—they are only meant to give you a sense of where you are in the maze. Please let the experimenters know if you have any questions. If you are ready to start the experiment, please click the mouse."& && Verification Phase Instructions (recorded) "During this phase of the experiment, you will be asked to find each of the objects from the home location. This is not a race, but you should try to find each object as quickly and directly as possible while walking at a comfortable speed. If you do not find the object soon, you will be asked to return to home. The experimenter will then guide you to the object along the most direct route. Try to remember the route that the experimenter has shown you, because you will be asked to find that object again until you can walk directly to it. Let the experimenters know if you have any questions. Otherwise, please click the mouse to begin." 156 Verification Phase Guiding Instructions (recorded) "Return to home and the experimenter will guide you to the object along the most direct route. Please try to remember this route because you will be asked to find the object again until you're able to walk directly to it." Test Phase Instructions, Shortcut and Neighborhood Shortcut Tasks (recorded) "During this phase of the experiment, you will be tested on your knowledge of the object locations by taking shortcuts between them. As in the previous phase, you will start by walking to an object. When you get there, the entire maze will automatically disappear and you will find yourself standing on an empty plain. You will then be asked to turn to face a target object. Try to remember where the target was located in the maze, then turn to face it, and click the mouse. You will then be asked to take a shortcut to the remembered location of the target. Because you are already facing the target, you should walk in a straight line and stop when you think you have arrived at the target location. You will then click the mouse again. Finally, the experimenter will wheel you back to home in a wheelchair. When you arrive at home, click the mouse to start the next trial. Test Phase Instructions, Route Task (recorded) "During this phase of the experiment, you will be tested on your knowledge of the object locations by taking shortcuts between them. As in the previous phase, you will start by walking to an object. When you get there, you will be asked to walk through the corridors of the maze to a target object. When you arrive at the target object location, stop and click the mouse. The maze will then disappear, and the experimenter will wheel you back to home in a wheelchair. When you arrive at home, click the mouse to start the next trial." 157 Appendix B: Questionnaire and Spatial Ability Tests Participant # __________________ Condition __________________ Please take a few minutes to answer the following questions Please do your best to draw a map of the virtual environment including the objects, paintings, and paths of the maze. Try to be as accurate as possible. List of Objects List of paintings Bookcase Cactus Dali (weird one with body parts) Clock Moon Monet (the reddish seascape) Flamingo Well Magritte (the one with the cat) Gear Sink Van Eyck (the married couple) Earth Rabbit PLEASE COMPLETE THIS PAGE BEFORE CONTINUING 158 Participant # __________________ Condition __________________ Did you use any conscious strategies to help you remember the locations of the objects while learning your way around the maze? Please describe. Did you use any conscious strategies during the test phase, when you started at one object and were asked to walk to a target object's location? Please describe. Did you learn anything new about the maze when taking shortcuts during the test phase? Did you notice anything unusual about the maze? Were some objects harder to find than others? If so, which ones? How aware were you of your location and orientation in the real environment (the lab) while you were walking around in the virtual world? 159 Participant # __________________ Condition __________________ How aware were you of your location and orientation in the real environment (the lab) while you were walking around in the virtual world? not at all aware very aware 1 2 3 4 5 6 7 How immersed did you feel in the virtual environment? not at all very immersed 1 2 3 4 5 6 7 Did you experience any dizziness or nausea while wearing the head-mounted display? none very nauseous 1 2 3 4 5 6 7 In the past year, how many hours per week (on average) did you spend playing video games (either on a computer or a console such as X-Box, PS3, etc.) that involve learning the layout of the game environment? (Examples: World of Warcraft, Halo, Call of Duty, Battlefield, Gears of War, Fallout, etc.) _____ hours per week In the past year, how many hours per week (on average) did you spend playing other kinds of video games that DO NOT involve learning the layout of the game environment (Examples: Tetris, Angry Birds, etc.) _____ hours per week The following statements ask you about your spatial and navigational abilities, preferences, and experiences. After each statement, you should circle a number to indicate your level of agreement with the statement. Circle 1 if you strongly agree that the statement applies to you, 7 if you strongly disagree, or some number in between if your agreement is intermediate. Circle “4” if you neither agree nor disagree. 1. I am very good at giving directions. strongly agree strongly disagree 1 2 3 4 5 6 7 2. I have a poor memory for where I left things. strongly agree strongly disagree 1 2 3 4 5 6 7 3. I am very good at judging distances. strongly agree strongly disagree 1 2 3 4 5 6 7 4. My “sense of direction” is very good. strongly agree strongly disagree 1 2 3 4 5 6 7 160 Participant # __________________ Condition __________________ 5. I tend to think of my environment in terms of cardinal directions (N, S, E, W). strongly agree strongly disagree 1 2 3 4 5 6 7 6. I very easily get lost in a new city. strongly agree strongly disagree 1 2 3 4 5 6 7 7. I enjoy reading maps. strongly agree strongly disagree 1 2 3 4 5 6 7 8. I have trouble understanding directions. strongly agree strongly disagree 1 2 3 4 5 6 7 9. I am very good at reading maps. strongly agree strongly disagree 1 2 3 4 5 6 7 10. I don’t remember routes very well while riding as a passenger in a car. strongly agree strongly disagree 1 2 3 4 5 6 7 11. I don’t enjoy giving directions. strongly agree strongly disagree 1 2 3 4 5 6 7 12. It’s not important to me to know where I am. strongly agree strongly disagree 1 2 3 4 5 6 7 13. I usually let someone else do the navigational planning for long trips. strongly agree strongly disagree 1 2 3 4 5 6 7 14. I can usually remember a new route after I have traveled it only once. strongly agree strongly disagree 1 2 3 4 5 6 7 15. I don’t have a very good “mental map” of my environment. strongly agree strongly disagree 1 2 3 4 5 6 7 161 Perspective-Taking and Spatial Orientation Test (PTSOT): Participants were given 5 minutes to complete 12 perspective-taking problems. 162 Road Map Test (RMT): Participants were given 20 seconds imagine themselves walking a path while trying to accurately name the required turning directions (left or right) at each bend in the path. 163 Appendix C: Experiment 1, full endpoint analysis F-tests and Effect Sizes for ANOVAs comparing Percentage of Shortcuts Falling in Neighborhoods (Wrong, Path, Short, Long) for Experiment 1. --------------------------------------------------------------------------------------------------------------------- Shortcut Task --------------------------------------------------------------------------------------------------------------------- Wrong Path Short Long ----------------- ----------------- ----------------- ----------------- Source F η p2 F η p2 F η p2 F η p2 Mean % endpoints ------ ------ ------ ------ ------ ------ ------ ------ E 0.22 0.00 1.07 0.04 0.027 0.00 2.14 0.09 TT 21.3*** 0.06 5.59* 0.05 0.279 0.01 14.8*** 0.40 E x TT 0.59 0.00 0.60 0.00 0.588 0.00 2.13 0.08 Source F η p2 F η p2 F η p2 F η p2 SD % endpoints ------ ------ ------ ------ ------ ------ ------ ------ E 1.54 0.05 0.005 0.00 0.59 0.01 2.11 0.08 TT 0.348 0.00 3.93 0.06 10.0* 0.12 16.2* 0.42 E x TT 0.723 0.00 0.013 0.00 1.02 0.10 2.11 0.08 --------------------------------------------------------------------------------------------------------------------- Neighborhood Shortcut Task --------------------------------------------------------------------------------------------------------------------- Wrong Path Short Long ----------------- ----------------- ----------------- ----------------- Source F η p2 F η p2 F η p2 F η p2 Mean % endpoints ------ ------ ------ ------ ------ ------ ------ ------ E 0.51 0.017 0.05 0.001 0.094 0.004 0.07 0.003 TT 3.39 0.031 1.27 0.011 3.19 0.009 3.50 0.137 E x TT 2.44 0.022 0.45 0.005 1.37 0.004 0.07 0.003 Source F η p2 F η p2 F η p2 F η p2 SD % endpoints ------ ------ ------ ------ ------ ------ ------ ------ E 1.09 0.034 0.05 0.002 0.19 0.006 0.07 0.028 TT 0.37 0.004 1.73 0.029 0.08 0.001 4.12 0.015 E x TT 1.13 0.014 0.03 0.000 0.11 0.002 0.07 0.028 --------------------------------------------------------------------------------------------------------------------- Route Task --------------------------------------------------------------------------------------------------------------------- Wrong Path Short Long ------------------ ----------------- ----------------- ----------------- Source F η p2 F η p2 F η p2 F η p2 Mean % endpoints ------ ------ ------ ------ ------ ------ ------ ------ E 2.38 0.084 16.8*** 0.272 27.4*** 0.428 9.263** 0.296 TT 1.19 0.008 30.9*** 0.417 53.0*** 0.491 9.263** 0.296 E x TT 0.00 0.00 11.2** 0.206 26.3*** 0.323 9.263** 0.296 Source F η p2 F η p2 F η p2 F η p2 SD % endpoints ------ ------ ------ ------ ------ ------ ------ ------ E 2.46 0.08 12.7* 0.21 22.3*** 0.40 9.49** 0.31 TT 0.97 0.01 29.5*** 0.41 57.2*** 0.52 9.49** 0.31 E x TT 0.13 0.00 7.08* 0.14 17.6*** 0.25 9.49** 0.31 --------------------------------------------------------------------------------------------------------------------- NOTE: F-tests and effect sizes for 2 (environment) x 2 (trial type) repeated-measures (mixed-model) ANOVAs where environment (Control Maze, Elastic Maze I) was the between-subjects factor, and trial type was the within-subjects factor. Separate ANOVAs were run for each of the four neighborhoods (Wrong, Path, Short, Long). E x TT = environment x trial type interaction. Degrees of freedom were (1, 22). Significance level of F’s: * = p < .05, ** = p < .01, *** p < .001. All other F’s shown were not significant, p > .05. 164 Appendix D: Experiment 1, correlations between selected dependent measures and trial number Final Position Error vs. Trial Number Absolute Final Angular Error vs. Trial Number 165 Response Time vs. Trial Number 166 Appendix E: Full Endpoint Analysis for Experiment 2 F-tests and Effect Sizes for ANOVAs comparing Percentage of Shortcuts Falling in Neighborhoods (Wrong, Path, Short, Long) for Experiment 2. --------------------------------------------------------------------------------------------------------- Neighborhood Shortcut Task --------------------------------------------------------------------------------------------------------- Wrong Path Short Long ---------------------------------------------------------------- Source F η p2 F η p2 F η p2 F η p2 Constant Error (CE) ------ ------ ------ ------ ------ ------ ------ ------ E 0.088 0.003 0.00 0.000 0.131 0.005 4.23 0.16 TT 0.299 0.003 0.168 0.002 5.1* 0.022 9.77** 0.30 E x TT 0.082 0.000 3.57 0.037 7.49* 0.032 4.23 0.16 Source F η p2 F η p2 F η p2 F η p2 Variable Error (VE) ------ ------ ------ ------ ------ ------ ------ ------ E 0.199 0.005 0.03 0.001 0.67 0.022 6.49* 0.22 TT 0.078 0.001 0.51 0.002 4.29 0.04 10.7** 0.32 E x TT 0.004 0.000 7.02* 0.033 7.07* 0.077 6.49* 0.23 --------------------------------------------------------------------------------------------------------- Route Task --------------------------------------------------------------------------------------------------------- Wrong Path Short Long ---------------------------------------------------------------- Source F η p2 F η p2 F η p2 F η p2 Constant Error (CE) ------ ------ ------ ------ ------ ------ ------ ------ E 0.549 0.019 0.17 0.003 0.07 0.001 0.06 0.003 TT 0.549 0.004 69.7*** 0.626 92.0*** 0.636 18.0*** 0.450 E x TT 0.033 0.000 0.25 0.006 0.02 0.000 0.06 0.003 Source F η p2 F η p2 F η p2 F η p2 Variable Error (VE) ------ ------ ------ ------ ------ ------ ------ ------ E 0.36 0.013 0.15 0.002 0.27 0.007 0.09 0.004 TT 0.75 0.006 68.6*** 0.641 176*** 0.756 22.4*** 0.504 E x TT 0.11 0.000 0.28 0.007 0.05 0.000 0.09 0.004 --------------------------------------------------------------------------------------------------------- NOTE: F-tests and effect sizes for 2 (environment) x 2 (trial type) repeated-measures (mixed-model) ANOVAs where environment (Control Maze, Swap Maze) was the between-subjects factor, trial type was the within-subjects factor. Separate ANOVAs were run for each of the four neighborhoods (Wrong, Path, Short, Long). E x TT = environment x trial type interaction. Degrees of freedom were (1, 22). Significance level of F’s: * = p < .05, ** = p < .01, *** p < .001. All other F’s shown were p > .05. 167 Appendix F: Experiment 2, correlations between selected dependent measures and trial number Final Position Error vs. Trial Number Absolute Final Angular Error vs. Trial Number 168 Response Time vs. Trial Number 169 Appendix G: Full Endpoint Analysis for Experiment 3 Experiment 3. F-tests and Effect Sizes for ANOVAs comparing Percentage of Shortcuts Falling in Neighborhoods (Wrong, Path, Correct A/B) for Experiment 3. --------------------------------------------------------------------- Neighborhood Shortcut Task --------------------------------------------------------------------- Wrong Path Correct (A/B) -------------- -------------- -------------- Source F η p2 F η p2 F η p2 Mean % endpoints ------ ------ ------ ------ ------ ------ E 3.86 0.144 1.43 0.051 3.32 0.127 TT 0.102 0.000 0.036 0.000 0.01 0.000 E x TT 7.385* 0.012 0.249 0.001 6.88** 0.009 Source F η p2 F η p2 F η p2 SD % endpoints ------ ------ ------ ------ ------ ------ E 1.48 0.053 0.56 0.020 0.173 0.005 TT 1.81 0.013 0.016 0.000 0.667 0.008 E x TT 2.31 0.017 2.71 0.023 0.581 0.007 --------------------------------------------------------------------- Route Task --------------------------------------------------------------------- Wrong Path Correct (A/B) -------------- -------------- -------------- Source F η p2 F η p2 F η p2 Mean % endpoints ------ ------ ------ ------ ------ ------ E 8.54* 0.206 0.72 0.017 3.91 0.098 TT 6.98* 0.094 2.43 0.048 9.44** 0.142 E x TT 5.12* 0.071 5.29* 0.098 0.01 0.000 Source F η p2 F η p2 F η p2 SD % endpoints ------ ------ ------ ------ ------ ------ E 14.3** 0.253 4.93 0.015 1.95 0.047 TT 4.16 0.082 2.19 0.043 8.25** 0.143 E x TT 1.94 0.040 4.92* 0.093 1.05 0.021 --------------------------------------------------------------------- NOTE: F-tests and effect sizes for 2 (environment) x 2 (trial type) repeated-measures (mixed-model) ANOVAs where environment (Control Maze, Swap Maze) was the between-subjects factor, and trial type (control, probe) was the within-subjects factor. Separate ANOVAs were run for each of the three neighborhoods (wrong, path, correct A/B). E x TT = environment x trial type interaction. Degrees of freedom were (1, 22). Significance level of F’s: * = p < .05, ** = p < .01, *** p < .001. All other F’s shown were p > .05. 170 Appendix H: Experiment 3, correlations between selected dependent measures and trial number Final Position Error vs. Trial Number Absolute Final Angular Error vs. Trial Number Response Time vs. Trial Number 171 Appendix I: Comparison of circular kernel density estimation methods ! Reproduced from an example in Agnostinelli & Lund’s (2011) documentation for the R package entitled “circular.” Agnostinelli & Lund (2011) note that Taylor’s (2008) method assumes an underlying von Mises distribution, but is not as sensitive to multimodality as the least-squares cross-validation (LSCV) and maximum likelihood cross-validation (MLCV; Duin, 1976; Habemma, 1974) methods. The MLCV method minimizes the integrated square error without the need for performing bias correction (Agnostinelli & Lund, 2011). In addition, Sharma & Tarboton (1997) have found that the MLCV method (followed by LSCV) provides a number of advantages for obtaining density estimates including: (1) better representation of nonlinearities such as asymmetry and bimodality, (2) avoiding the need to select a probability distribution, (3) lower variability and higher accuracy than other methods (e.g. Biased Cross Validation) for strongly bimodal data. 172 Appendix J: Comparison of performance across environments and tasks, chi-square analysis of endpoints All trials (control and probe combined) Metric Shortcut Task Neighborhood Shortcut Task Place Graph Task Place Graph Task 100 100 100 100 % Incorrect Neighborhood % Incorrect Neighborhood % Incorrect Neighborhood % Incorrect Path Choices 80 80 80 80 n.s. 60 60 60 60 51.1 % 52.5 % * 40 40 40 40 31.5 % 18.3 % * 18.4 % 16.6 % * n.s. 14.3 % 16.3 % 20 20 20 20 7% 8.4 % X X 2.6 % 0.1 % 2.1 % 1.4 % 0 0 0 0 Eucl. Elst.I Elst.II Swap Eucl. Elst.I Elst.II Swap Eucl. Elst.I Elst.II Swap Eucl. Elst.I Elst.II Swap Environment Environment Environment Environment Performance by task, control and probe trials combined. Shortcut Task, χ2 (1, N=417) = .11, p = .74. Neighborhood Shortcut Task, χ2 (3, N=554) = 10.2, p = .017. Route Task (% incorrect neighborhood), χ2 (3, N=668) = 14.8, p = .001. Route Task (% incorrect path choices), χ2 (3, N=672) = 1.32, p = .72. Probe trials Metric Shortcut Task Neighborhood Shortcut Task Place Graph Task Place Graph Task 100 100 100 100 % Incorrect Neighborhood % Incorrect Neighborhood % Incorrect Neighborhood % Incorrect Path Choices 80 80 80 80 n.s. 60 60 60 60 51.6 % 50.4 % * * * 40 40 40 40 33.7 % 33.7 % 31 % 23.5 % n.s. 16 % 20 20 20 20 11.7 % 10.1 % 11.6 % 5.3 % X X 0.2 % 2.8 % 2% 0 0 0 0 Eucl. Elst.I Elst.II Swap Eucl. Elst.I Elst.II Swap Eucl. Elst.I Elst.II Swap Eucl. Elst.I Elst.II Swap Environment Environment Environment Environment Performance by task, probe trials only. Shortcut Task, χ2 (1, N=397) = .001, p = .97. Neighborhood Shortcut Task, χ2 (3, N=406) = 14.93, p = .0018; post-hoc pairwise comparisons of proportions revealed that % incorrect neighborhoods was marginally higher in the Swap maze than the other conditions (p = .07). Route Task (% incorrect neighborhood), χ2 (3, N=365) = 30.91, p < .001. Route Task (% incorrect path choices), χ2 (3, N=338) = .67, p = .87. Control trials Metric Shortcut Task Neighborhood Shortcut Task Place Graph Task Place Graph Task 100 100 100 100 % Incorrect Neighborhood % Incorrect Neighborhood % Incorrect Neighborhood % Incorrect Path Choices 80 80 80 80 n.s. 60 60 60 60 54.7 % 50.7 % * 40 40 40 40 29.3 % 16.9 % 16.5 % n.s. n.s. 20 20 20 20 13 % 3.2 % 2.6 % 3.8 % 5.4 % X X 0% 0% 1.5 % 0.7 % 0 0 0 0 Eucl. Elst.I Elst.II Swap Eucl. Elst.I Elst.II Swap Eucl. Elst.I Elst.II Swap Eucl. Elst.I Elst.II Swap Environment Environment Environment Environment Performance by task, control trials only. Shortcut Task, χ2 (1, N=397) = .41, p = .51. Neighborhood Shortcut Task, χ2 (3, N=403) = 9.2, p = .02; post-hoc pairwise comparisons of proportions revealed that % incorrect neighborhoods was marginally higher in the Swap maze than in the other environments (p = .07). Route Task (% incorrect neighborhood), χ2 (3, N=367) = 3.93, p = .26, n.s. Route Task (% incorrect path choices), χ2 (3, N=334) = 2.76, p = .42, n.s. 173 Appendix K: Graphs of combined results for dependent measures (FAEabs, DERabs, and FPE) not reported in results sections for Experiments 1-3. Absolute Final Angular Error (FAEabs) Absolute Distance Error Ratio (DERabs) Final Position Error (FPE) 174