Mueller-Lyer and Inconsistent Beliefs

Regular readers may know that we enjoy optical illusions, and I think Roy Sorensen, whose Vagueness and Contradiction I’ve been enjoying lately, may have put his finger on why.

Believing the Impossible

Here’s the famous Muller-Lyer illusion, courtesy of Wikimedia Commons:

As you probably know, the horizontal lines in this picture are all the same length, but they don’t look that way. The second looks shorter than the first, and what about the third? To me it looks the same length as both first and second, even though these two seem to be different lengths.

The lower half of the picture uses colour to make the equality of the lengths easier to see.How can I believe that A is shorter than B, while C is the same length as both? It seems I can’t, because to believe that would be to believe in a contradiction, like a square circle or something that’s simultaneously red all over and green all over. Can human beings believe in contradictions?

Sorensen thinks we can — in fact, he thinks much of what we believe is inconsistent, and there’s no way to fix it. His tentative explanation is that it was a survival advantage, as brains evolved, to respect large differences but ignore details, and to treat things that are similar as if they were the same. That’s highly speculative, but his point is that we believe the inconsistency, whatever the reason might be. Perhaps we enjoy optical illusions because they show us that we believe contradictions, something we don’t usually think is true at all.


The commonest explanation for the effectiveness of Mueller-Lyer is quite different; that we’ve learned to read flat pictures in terms of Renaissance perspective, and that’s what we’re doing here, although there are no obvious cues in the illusion to that effect.

A more obvious example is the Ponzo illusion:

The effect, of course, comes from the strong visual cue the “railway tracks” create. They’re not even realistic railway tracks, being little more than the construction lines used in one-point perspective (hat tip):

I think it’s the construction-line reading that leads us to see the upper bar as being further away, and therefore longer than the lower bar. We’ve learned to use lines like these to tell us how to read depth in a picture. It was hard to learn that, but essential if we were to understand many of the pictures that accompanied our childhood stories. So this is a theory that relies on nurture, not nature.

Here’s an extraordinary variation on the Ponzo illusion that’s in two-point perspective (hat tip):

Look at the lines created by the roads; those are the cues to see the cars as being different sizes, even though they’re not.

Here are the bare construction lines of a two-point perspective drawing; a framework for seeing depth that was invented more than half a millenium ago:

It’s perhaps this enculturation into perspective that leads us to see an “illusion” or “impossble object” in the Pericope of Henry II (hat tip):

Here the middle pillar, which should be in front, is tucked behind the Madonna and child, since otherwise it would get in the way. At least, that’s our modern way of reading it. In fact the picture was painted before the invention of perspective drawing, and I’m sure there was no intention to create an “illusion” here.

At the time it was made — the eleventh century — only a small handful of mathematically-minded painters were attempting what we would think of as perspective drawing (and none were really successful). Everybody else used size and position not to create a visually realistic effect but to combine symbolism — the importance of each figure, for instance — with decorative effect. I don’t think anybody seeing this image would have been struck by its “impossibility” in the slightest.

Do We Really Believe Something Impossible?

I think part of the pleasure of looking at illusions like these is that they remind us that our ways of looking at two-dimensional representations of the world are artificial and learned. We feel pulled in two opposite directions; on the one hand, one looking at the figure as a flat collection of lines and colours, on the other wanting to believe the illusion of perspective.

An evolutionary psychologist might retort that the image on the retina is two-dimensional, too, but of course we have two eyes, and they see different images; if you lose the use of one eye then you lose your “natural” depth perception (stereopsis) and have to re-learn it using perspectival cues. That can’t explain these two-dimensional perspectival illusions, but perhaps an indecision between two conflicting interpretative schemes does.