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A hypothesis is proposed that predicts regions where fixations are most likely to occur on a visual pattern viewed by a human test subject. A scheme is developed for evaluating a fixation probability density function for any given pattern, based on an assumed fixation probability density function for a pattern consisting of a single dot. Predictions of the hypothesis agree with experimental data available in the literature. When the hypothesis is used in conjunction with some assumptions, it also provides interpretations for some well-known geometrical illusions, such as the Muller-Lyer illusion. It is hoped that a study of this kind will aid in understanding some aspects of the mechanisms subserving human visual pattern recognition.