The mathematics of mental rotations
Journal of Mathematical PsychologyPublished 1 December 1990
R. Patrick Goebel
Citations14
SJR quartileQ2
SJR score0.74
SNIP1.10
Generate an AI Snapshot to get a quick, structured summary of this paper.
Study Snapshot
ObjectiveStudy objective
MethodsResearch methodology
PopulationPopulation studied
Sample sizeSample sizes
OutcomesStudy outcomes here
ResultsStudy results comes here
LimitationsResearch study limitations comes here
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
Abstract
A mathematical analysis of the structure of rotations and the dynamics of linear recurrent connectionist networks suggests a simple strategy for implementing mental rotations on a parallel distributed processing (PDP) architecture. Rather than encode a separate map for each combination of rotation axis and degree of rotation, the net embodies generators for infinitesimal rotations about three linearly independent axes. Finite rotations about an arbitrary axis are then generated in real time by the natural dynamics of the net operating under the influence of the appropriate linear combination of the three generators. The resulting transformation proceeds at a constant rate.
Keywords
Computer ScienceEngineering
Springer series in information sciencesSelf-Organization and Associative Memory
8,770 Citations1989Teuvo Kohonen
ScienceMental Rotation of Three-Dimensional Objects
6,230 Citations1971Roger N. Shepard, Jacqueline Metzler
The time required to recognize that two perspective drawings portray objects of the same three-dimensional shape is found to be a linearly increasing function of the angular difference in the portrayed orientations of the two objects.
The American Journal of PsychologyImage and Mind
1,835 Citations1982Peter Podgorny, Stephen M. Kosslyn
Cognitive ScienceConnectionist Models and Their Properties
1,324 Citations1982Jerome A. Feldman, D.H. Ballard
A general connectionist model is introduced and how it might be used in cognitive science is considered, among the issues addressed are: stability and noise-sensitivity, distributed decision-making, time and sequence problems, and the representation of complex concepts.
Journal of Experimental Psychology GeneralDistributed memory and the representation of general and specific information.
876 Citations1985James L. McClelland, David E. Rumelhart
IEEE ExpertInduction: Processes of Inference, Learning, and Discovery
824 Citations1987John H. Holland, Keith J. Holyoak +3 more
Induction is the first major effort to bring the ideas of several disciplines to bear on a subject that has been a topic of investigation since the time of Socrates and is included in the Computational Models of Cognition and Perception Series.
MIT Press eBooksA general framework for parallel distributed processing
710 Citations1986D. E. Rumelhart, Geoffrey E. Hinton +1 more
This chapter contains sections titled: Classes of PDP Models, Specific Versions of the General Parallel Activation Model, Sigma-Pi Units, Conclusion, Acknowledgments.
Cambridge University Press eBooksGeometrical Methods of Mathematical Physics
662 Citations1980B. F. Schutz
Elsevier eBooksSchemata and Sequential Thought Processes in PDP Models
635 Citations1988D. E. Rumelhart, Paul Smolensky +2 more
It is, as previously noted, these maxima that the authors believe correspond roughly to instantiations of schemata for kitchens, offices, bathrooms , living rooms, and bedrooms.
Group theory in physics
534 Citations1984J. F. Cornwell
American Journal of Physics<i>Geometrical</i> <i>Methods</i> <i>of</i> <i>Mathematical</i> <i>Physics</i>
478 Citations1982B. F. Schutz, Mark Peterson
Medical Entomology and ZoologySpatial cognition : brain bases and development
399 Citations1988Joan Stiles-Davis, Mark Kritchevsky +1 more
This book discusses the development of Spatial Cognition in Adults with a focus on young children and the role of the Parietal Lobe in this process.
Journal of Experimental Psychology Human Perception & PerformanceMental transformation in the identification of left and right hands.
293 Citations1975Lynn A. Cooper, Roger N. Shepard
It is proposed that subjects determine whether a visually presented hand is left or right by moving a mental "phantom" of one of their own hands into the portrayed position and by then comparing its imagined appearance against the appearance of the externally presented hand.
Journal of Mathematical PsychologyThe Lie algebra of visual perception
282 Citations1966William C. Hoffman
Perception & PsychophysicsDemonstration of a mental analog of an external rotation
258 Citations1976Lynn A. Cooper
Transformational Studies of the Internal Representation of Three-Dimensional Objects 1
202 Citations2024Jacqueline Metzler, Roger N. Shepard
International Joint Conference on Artificial IntelligenceA parallel computation that assigns canonical object-based frames of reference
196 Citations1981Geoffrey F. Hinton
A viewpoint-independent description of the shape of an object can be generated by imposing a canonical frame of reference on the object and describing the spatial dispositions of the parts relative to this object-based frame.
Representation in Memory
187 Citations2013David E. Rumelhart, Donald A. Norman
Major controversies within psychology -- such as distinctions between declarative and procedural representation, propositional and analogical representation, and the nature of visual images -- are analyzed and found not to reflect fundamental disagreements.
Perception & PsychophysicsThe Lie transformation group model of visual perception
157 Citations1983P. C. Dodwell
This tutorial paper expound the mathematical basis for LTG/NP and to evaluate that model against a reasonable set of criteria for a neuropsychological theory, and shows that this approach to spatial vision is closer to the mainstream of current theoretical work than might be assumed.
Mathematical BiosciencesHigher visual perception as prolongation of the basic lie transformation group
77 Citations1970William C. Hoffman
The group is extended via prolongation of its Lie derivatives to the complicated orbits characteristic of the contours recognizable in higher form vision and it is argued that perception consists of an exact sequence of prolonged Lie derivatives acting on the corresponding differential form.
Journal of Mathematical PsychologyPsychologically simple motions as geodesic paths I. Asymmetric objects
59 Citations1990Eloise H. Carlton, Roger N. Shepard
Biological CyberneticsVisual apparent motion and some preferred paths in the rotation group SO(3)
50 Citations1975David Foster
The results obtained indicate that the visual system behaves more in accordance with the second scheme than with the first, which specifies the action-minimizing curves in the rotation group SO(3) for a particular Riemannian metric.
Quarterly of Applied MathematicsThe neuron as a Lie group germ and a Lie product
27 Citations1968William C. Hoffman
Murphy: A neurally-inspired connectionist approach to learning and performance in vision-based robot motion planning
21 Citations1990Bartlett W. Mel
A series of strong correspondences are outlined between the representations and algorithms used by MURPHY, and the psychology, physiology, and neural bases for the programming and control of directed, voluntary arm movements in humans and animals.
Biological CyberneticsConnection between internal representation of rigid transformation and cortical activity paths
7 Citations1988E. H. Carlton
The key hypothesis of the paper is that an action of the Euclidean group on the Gabor transform space brings the perceived paths into relation with corresponding successive neural microstructure activity patterns.