Weaker than EuclideanWords
I'm helping a grad student at UCSB draft a letter to "Nature" that will highlight exciting elements of 3 years of research he undertook in the Virtual Reality Lab while an undergrad at Brown University. His research seeks to understand the geometry underlying cognitive maps. In other words, he wants to know what kind of math humans use to navigate the world.
Previous studies say it's standard metric geometry, but he's not so sure...
Introduction only:
To understand how we as humans navigate our surroundings—to find detours, for example, or simply to walk next door—many researchers have first tried to understand how we represent the world mentally, in our heads. Researchers have sought to do this since 1948, when the term “cognitive map” was first introduced.
To date, every study assumes that humans impose a metric structure on mental representations of space. In other words, in order to navigate, we depend on knowledge of distances and directions between learned places along a one-dimensional route or in a two-dimensional environment. This outwardly logical assumption advocating metric structure is grounded in a series of experiments which shaped the constraints of cognitive maps, starting with visual tests performed by Shephard & Metzler in 1971.
Shephard and Metzler asked individuals to look at two shapes and decide as quickly as possible whether the two were the same. If the shapes were in the same view point, participants could identify matches within a second. When shapes were rotated at different angles, however, participants took noticeably longer to identify sameness.
A similar experiment by Bundensen, Larsen, & Farrel in 1981 using rectangles sized at different ratios gave concurring results; participants took longer to identify sameness the greater the scale adjustment—up or down—of the rectangle.
Together, these two studies seemed to imply that when we perceive an object in the world, we do it so accurately that we then have an almost physical copy, or map, in our brains. This was further affirmed by Kosslyn, Ball, & Reiser in 1978. They found that participants asked to memorize a map and then mentally scan from different target locations took longer to scan from objects the further apart they had been.
In 1976, Chedru localized where in the brain spatial maps exist by studying patients with spatial neglect—a condition in which an individual’s visual field is bisected by a vertical line. Chedru concluded that perception of space is dependent on location in the visual field, further laying the groundwork for our understanding of constraints on a cognitive map. His findings also indicated that processes related to spatial perception occur in the parietal lobe.
Studies by Taylor & Tversky in 1992 revealed that humans can learn a new environment equally well independent of their method of learning. The researchers taught participants the layout of a park using one of two methods—verbal description of a route navigation, or verbal description of a survey view. Participants then had to say whether statements about the layout of the park were true or false and they had uniform success despite different learning methods.
Learning our surroundings does seem to benefit from some physical input. Presson and Montello (1994) and Resier (1989) asked participants to imagine walking from home to point A, and then to point B. They then had to indicate the direction from point B back to home again, and they were consistently incorrect.
In a different experiment, when participants actually walked the route—from home to A, and A to B—the physical action of walking greatly improved their ability to indicate the direction from B to home.
The researchers doing this experiment concluded that we live in a physical world such that without physical inputs, we cannot construct an accurate mental representation of space (Klatzky, Loomis, Beall, Chance & Golledge).
Despite the number of studies pointing to metric structure as the basis of wayfinding ability, it is possible that cognitive maps are based on a less stringent geometry and in turn, have a less defined shape. The apparently metric behavior associated with navigation could in fact be accounted for by adaptive strategies based on topological geometry, which relies not on knowledge of distances and directions between learned places but on knowledge of environmental features—places, junctions, and landmarks.
This latter notion dissociates spatial knowledge from Euclidean geometry and opens the forum to progressively weaker geometries, which preserve fewer of the properties that remain invariant in metric structure. A cognitive map based on constraints of such geometries might only preserve ordinal structure, such as the adjacency relationships among neighborhoods, or relationships between places based on interlinking paths. Shortcuts and detours could be derived from knowledge of the connectivity or neighborhood structure together with visible landmarks.
The recent development of ambulatory virtual environments makes it possible to study the spatial knowledge used during actual navigation and in particular, to analyze the hypothesis that a geometry other than Euclidean facilitates human wayfinding...
.MGW.
Previous studies say it's standard metric geometry, but he's not so sure...
Introduction only:
To understand how we as humans navigate our surroundings—to find detours, for example, or simply to walk next door—many researchers have first tried to understand how we represent the world mentally, in our heads. Researchers have sought to do this since 1948, when the term “cognitive map” was first introduced.
To date, every study assumes that humans impose a metric structure on mental representations of space. In other words, in order to navigate, we depend on knowledge of distances and directions between learned places along a one-dimensional route or in a two-dimensional environment. This outwardly logical assumption advocating metric structure is grounded in a series of experiments which shaped the constraints of cognitive maps, starting with visual tests performed by Shephard & Metzler in 1971.
Shephard and Metzler asked individuals to look at two shapes and decide as quickly as possible whether the two were the same. If the shapes were in the same view point, participants could identify matches within a second. When shapes were rotated at different angles, however, participants took noticeably longer to identify sameness.
A similar experiment by Bundensen, Larsen, & Farrel in 1981 using rectangles sized at different ratios gave concurring results; participants took longer to identify sameness the greater the scale adjustment—up or down—of the rectangle.
Together, these two studies seemed to imply that when we perceive an object in the world, we do it so accurately that we then have an almost physical copy, or map, in our brains. This was further affirmed by Kosslyn, Ball, & Reiser in 1978. They found that participants asked to memorize a map and then mentally scan from different target locations took longer to scan from objects the further apart they had been.
In 1976, Chedru localized where in the brain spatial maps exist by studying patients with spatial neglect—a condition in which an individual’s visual field is bisected by a vertical line. Chedru concluded that perception of space is dependent on location in the visual field, further laying the groundwork for our understanding of constraints on a cognitive map. His findings also indicated that processes related to spatial perception occur in the parietal lobe.
Studies by Taylor & Tversky in 1992 revealed that humans can learn a new environment equally well independent of their method of learning. The researchers taught participants the layout of a park using one of two methods—verbal description of a route navigation, or verbal description of a survey view. Participants then had to say whether statements about the layout of the park were true or false and they had uniform success despite different learning methods.
Learning our surroundings does seem to benefit from some physical input. Presson and Montello (1994) and Resier (1989) asked participants to imagine walking from home to point A, and then to point B. They then had to indicate the direction from point B back to home again, and they were consistently incorrect.
In a different experiment, when participants actually walked the route—from home to A, and A to B—the physical action of walking greatly improved their ability to indicate the direction from B to home.
The researchers doing this experiment concluded that we live in a physical world such that without physical inputs, we cannot construct an accurate mental representation of space (Klatzky, Loomis, Beall, Chance & Golledge).
Despite the number of studies pointing to metric structure as the basis of wayfinding ability, it is possible that cognitive maps are based on a less stringent geometry and in turn, have a less defined shape. The apparently metric behavior associated with navigation could in fact be accounted for by adaptive strategies based on topological geometry, which relies not on knowledge of distances and directions between learned places but on knowledge of environmental features—places, junctions, and landmarks.
This latter notion dissociates spatial knowledge from Euclidean geometry and opens the forum to progressively weaker geometries, which preserve fewer of the properties that remain invariant in metric structure. A cognitive map based on constraints of such geometries might only preserve ordinal structure, such as the adjacency relationships among neighborhoods, or relationships between places based on interlinking paths. Shortcuts and detours could be derived from knowledge of the connectivity or neighborhood structure together with visible landmarks.
The recent development of ambulatory virtual environments makes it possible to study the spatial knowledge used during actual navigation and in particular, to analyze the hypothesis that a geometry other than Euclidean facilitates human wayfinding...
.MGW.
posted by Meagan G at
10:09 AM
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