This is what I was able to find out about this visual illusion by doing a 15 minute search on the world wide web. The comments in square brackets are mine.
Several effects are thought to contribute to the Mueller-Lyer illusion. The shafts of the two arrows above are equal, although the bottom looks longer. If the figures are interpreted in three dimensions, the upper image is seen as the outside edge of a box, while the lower image is seen as the inside edge of a box. Since it is assumed that the two "solid objects" rest on the same surface, the outside edge appears closer to the viewer than the inside edge. The more "distant" line is then interpreted as larger, to correct for the difference in apparent distance.
This figure [arrows replaced by arcs] shows a variant of the illusion
which does not contain depth cues, but still shows an apparent
difference in the length of lines. It is difficult to completely
ignore the suprouneifgs of the test stimulus; research has shown that
the illusion is stronger for children because it is more difficult for
them to ignore the extraneous parts of the stimulus. Other possible
explanations are confounding the length of the entire figure with the
length of the test line, or relying somewhat on the surround white
space to define the figure (a larger figure would contain more white space)."
Although many theories exist for this illusion, there is no certain explanation. One theory is based on eye movement. When the arrows point inwards, our gaze rests inside the angles formed by the arrows. When they point outwards, our eyes demarcate the entire perspective and our gaze rests outside the angles. The outward pointing arrows make the figure more open and so the horizontal line appears longer.
The illusion takes its name from Franz Carl Mueller-Lyer (1857-1916), who studied medicine in Strasbourg and served as assistant director of the city's psychiatric clinic. Mueller-Lyer's main works were in the field of sociology. He himself attempted to explain the illusion he had discovered as follows: `the judgment not only takes the lines themselves, but also, unintentionally, some part of the space on either side.' He published two articles on the illusion bearing his name. (`Optical Illusions' 1889, and `Concerning the Theory of Optical Illusions: on Contrast and Confluxion' 1896)
Favreau (1977) studied a number of textbooks in which Mueller-Lyer presented
and measured the figures. He noticed that in many cases, the figure had been
drawn the wrong way round so that the illusion appeared more forceful!"
"...[R.L.] Gregory uses The Inappropriate Constancy Theory to explain the
size distortion in the Mueller-Lyer illusion, though it does not work as
effectively. He uses the theory by comparing the two figures in the illusion
with corners of a building in the real world. Gregory was not trying to
say that everone sees the figure as buildings, but he claimed that the
visual system would use similar strategies in order to understand other
drawings, so it followed that they might be misused in this case. The line that
is enclosed by the normal arrowheads is seen as a being part of a convex
corner, and the lines enclosed by the reversed arrowheads are seen as
being part of a concave corner. If two corners like these were seen in the
real world, the line in the convex corner would be in front of the surrounding
arrowheads, and the line in the concave corner would be behind the arrowheads.
This would means that the first line is perceived as being closer to the
viewer, while the second line is perceived as being farther away. Since
lines are seen as being the same size, the visual system makes the assumption
that is made with the Ponzo illusion: lines that are the same size, but
at different depths must really be of different sizes. Therefore, the line
that is seen as part of the concave corner is perceived as being longer than
the line which is seen as part of the convex corner."
"A similar phenomenon is observed in the Mueller/Lyer illusion where the
vertical lines which are of the same length appear to be of different
lengths. Again this illusion makes sense in a spatial context, because
the line which is made to appear farther in depth becomes perpetually
larger, while the one that is made to appear closer in depth becomes
smaller. Again, this depth effect is pre-attentive and unconscious, thus
ruling out a cognitive explanation.
Apparent length in preattentive vision: Evidence for low lever grouping, by James T. Enns and Ronald A. Rensink, University of British Columbia, in Investigative Ophtalmology and Visual Science #32:1039, 1991, ARVO 1991; Sarasota, FL, [ARVO is the main vision conference]
"Theories of human vision have generally assumed that the features underlying
visual search and texture segmentation corresponding to simple measurements
made at the first stages of visual processing. In this paper, we describe
a series of visual search experiments that refute this assumption. Using
several variants of the Mueller-Lyer figure, we show that an illusion of
length exists in preattentive vision--search is easy when items contain
line segments of equal length, but becomes difficult when these segments
are adjusted to have the same apparent length. This illusion cannot be
reduced by selective inhibition of features, such as that used to facilitate
the rapid detection of feature conjunctions. For example, subjects are unable
to ignore the wings when making judgements of the test line, even when it
is advantageous to do so... This rules out the explanation based on interactions
among the features themselves. We shall also show that spatial filtering cannot
account for this illusion, since these effects are indifferent to the sign
of contrast of the line segments and can occur for textured lines having
the same first-order statistics. The illusion, however, can be explained
by a model in which line length is determined via grouping operations acting
at a level prior to the formation of preattentive features."
"Cool, eh? If you picked number 4 or 5, you were deceived by the Mueller-Lyer illusion! The correct answer was number 6!
The tails give depth cues to the brain that cause you to see the illusion."