# Other Theories of the MÃœller-Lyer Illusion

 Contributors: Robert T. Arrigo: Programming Gordon Redding: Author
 Additional Credits: FundingThis module was supported by National Science Foundation Grants #9981217 and #0127561.

The Müller-Lyer illusion (Müller-Lyer, 1889/1981) is the observation that a line with angles less than 90° (i.e., arrow junctions) looks shorter than the same line with angles greater than 90°(i.e., fork junctions). There are many explanations of this illusion and it certainly does not have a single cause (e.g., Coren & Girgus, 1978). Moreover, testing among various explanations is difficult because they often make the same predictions. We will illustrate this problem with two kinds of theories: linear perspective and stimulus averaging theories.

## Linear Perspective Theory:

One explanation of this illusion is that the two-dimensional line junctions are interpreted as linear perspective drawings of three-dimensional corners (Redding, 2000; Redding & Hawley, 1993; cf. Gregory, 1963; Gregory & Harris, 1975). These drawings activate brain mechanisms that compute the inverse perspective to recover the size of the virtual corners that produced the drawings. Lines arrow junctions depict convex corners in front of the picture plane of the drawing, while lines with fork junctions depict concave corners behind the picture plane. To produce the same size line in the drawing the virtual corners must be different sizes; the convex corner must be smaller than then concave corner. Hence the illusion: The line with arrow junctions looks smaller than the line with fork junctions.

## Stimulus Averaging Theory:

Another explanation of the Müller-Lyer illusion is that perceived size reflects an average size of the two-dimensional stimulus (e.g., Pressey, 1967; Pressey & Pressey, 1992). According to this kind of explanation, the illusion occurs because the average size includes the distance between opposing junction lines and is different from the size of the line alone. For the line with arrow junctions the average size is less than the line, while for the line with fork junctions the average size is greater than the line. Perceptual judgment is biased toward the average size and the line with arrow junctions looks smaller than the line with fork junctions. Averaging stimulus attributes is a natural part of the perception of complete objects (Redding, Winson, & Temple, 1993).

## Predictions:

The linear-perspective hypothesis may be tested by rotating the virtual corners about their vertical picture plane axes to produce drawings with asymmetrical angles. To maintain the same size lines in the drawings the size of the virtual corners must be changed because rotation moves the corners closer to the picture plane. The rotated convex corner in front of the picture plane (depicted by asymmetrical arrow junctions) must be increased in size, while the size of the rotated concave corner behind the picture plane (depicted by asymmetrical fork junctions) must be decreased in size. Hence, the prediction is that the illusion will decrease for both the drawings of with arrow junctions and fork junctions.

However, averaging theories make the same directional prediction! Changing the size of rotated virtual corners also changes the distance between opposing junctions in the drawing. The average size of stimulus with asymmetrical arrow junctions is greater than the average size of the stimulus with symmetrical arrow junctions. And, the average size of the stimulus with asymmetrical fork junctions is less than the average size of the stimulus with symmetrical fork junctions.

## Conclusion:

Thus, the manipulation of virtual corner rotation and consequential asymmetrical junctions does not provide a definitive test between linear perspective and stimulus averaging theories. Changing size of rotated virtual corners covaries with average size of the drawing. Can you think of a way to break this natural covariation and provide a more definitive test between the theories? This is the creative part of research!