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Topological constancy in the perception of Lissajous figures
by
Paul C. Kainen
Dept. of Math., Georgetown University
A phenomenon involving mathematical psychophysics is described and an interpretation is proposed which involves the KAM-theory of dynamical systems. Psychophysics studies various quantitative relationships of visual inputs to perceptual output. Mathematical, in this context, means that the input has a simple and explicit mathematical aspect and/or that perception is interpreted using an explicit mathematical model.
The display of a dot of light on a screen with a regular type of linear or rotational oscillation produced by two or more independently controllable mirrors or other deflection devices will cause the percept of a "trace figure" when the ratio between the periods of distinct oscillations is a low-order fraction. Such figures were first studied by Jules Antoine Lissajous and Nathaniel Bowditch and they can have a topological aspect, like a flexible wire moving in 3-dimensional space. It is the apparent constancy of the figure, under distorted frequency ratios and irregularity of oscillation, which is the phenomenon to be studied. In a reverse Doppler fashion, such distortion gives rise to writhing, twisting, and spinning of the figure whose fundamental topology nevertheless remains invariant up to a certain limit of distortion.
A brief description of portions of the Kolomogorov-Arnold-Moser (KAM) theory is given and a possible connection with topological constancy of Lissajous figure perception is considered. It is argued that the veridical perception of a topology as well as of changes in geometry (i.e., shape) show that object constancy does not suffice as an explanation.
Date received: October 24, 2004
Copyright © 2004 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # capf-10.