One of the most obstinately puzzling illusions is Poggendorff’s, in which a slanting line interrupted by a gap no longer looks aligned. For over a century specialists have been unable even to agree whether it arises from 2D properties of the image, or as a result of attempts by the brain to interpret the configuration as 3D. Papers written a hundred years ago treat the problem in very much the same terms as we do today. I’m betting on 2D (I argue for that on another, website devoted to the Poggendorff illusion). It’s not likely my speculations are spot on, and they may well not even be in the right direction. But read on here if you’d like to see demonstrations that show why I don’t think depth processing can be the answer.

First, what’s the case in favor of depth processing as the source of the misalignment?  It starts from the assumption that any oblique line in the visual field is interpreted early in visual processing as probably receding away from the viewer. However, the argument goes, the brain does not necessarily interpret the interrupted oblique lines in Poggendorff figures as continuous in space, for example as if they were inscribed on a receding surface, as to the left in the picture at the start of the post. More probably, the oblique lines might be discontinuous, like the centre and right hand lines in the picture.  If this were a real scene, those lines would be laterally and vertically displaced from one another in space. They could only appear aligned because of an accident of viewpoint. On that assumption the illusion arises, some believe, because our brains make an adjustment for the assumed spatial displacement.

Others suggest that the interrupting lines are seen as edges of a surface, blocking our view of a continuous line, but at a different orientation in space, shown below as if seen from the side:

As a result, the argument goes, the brain is not sure exactly how far away the junctions between the oblique line and the edges of the interrupting rectangle are.  Is the upper junction, for example, where it would be on the oblique line, or only as far away as the interrupting edge?  Apparent misalignment might arise from the depth confusion, for example because apparent position as seen in one eye might then be expected by the brain to be different from position as seen by the other.

On the face of it these accounts sound very plausible. My problem with them is that it seems to me that the illusion can arise vividly when the Poggendorff figure is explicitly inscribed within a surface, so that there’s no scope for confusion about position. However, just showing the test lines on a surface in a picture, as to the left in the picture with the poultry at the start of the post, isn’t enough. It takes the brain a large fraction of a second to decide that a picture like that is of an odd-shaped block (I think up to a fifth of a second or so). Before the decision about what is represented, layout of the lines alone probably evokes preliminary assumptions about possible spatial layouts – for example that oblique lines are probably receding into depth. Illusion might arise just from those initial assumptions.

But check out this picture:

What I’ve done is to embed some Poggendorff figures in another illusion, Shepard’s tables. This is an amazing illusion. The two tabletops are identical, but it can be hard to believe, unless you measure them just as parallelograms on the screen. One presents its short side head on to us, and the other its long side, that’s all. We see one as squat, the other as long and thin, because of the size-constancy effect.  The oblique edges of the tables also seem to diverge slightly with distance.

Here’s the key point for us. The way that the tables appear to have quite different proportions, and that their edges appear to diverge with distance, tells us that in this case the spatial scheme was established very early in visual processing. (See the earlier post about paradoxical size-constancy). Could the Poggendorff figures on the tabletops be evoking a quite different spatial scheme, also in early processing, indicating that the lines may not be confined to the surfaces?

In principle, maybe. It could be that very early in processing, every possible spatial configuration of an array of geometric lines is, as it were, put on stand-by, and that all but one possibility is then eliminated (very rapidly) in a sort of Darwinian process of selection of the most probable. If so, it could be argued that illusion could arise even from a configuration that survives as a possibility only for a moment. I suspect that a process of elimination of possibilities like that could indeed go on. I find it much less likely that in the process two quite separate illusions, the effects associated with Shepard’s tables on the one hand, and Poggendorff effects on the other, could both survive the selection process. The Poggendorff misalignments here seem unequivocally inscribed in the table-tops, along with size-constancy effects, very early on in visual processing. And yet the misalignments, to my eye, are just as strong as in the bare reference figures either side of the tables – the three D context is making no difference.

However there’s another theory attributing the Poggendorff illusion to 3D processing which IS more consistent with the tabletop versions of the figures. This theory simply attributes the shift in apparent position of the lines to the size-constancy effect, which makes distant objects, especially in pictures, look larger than they are.  It’s as if, perceptually, space and objects expand with apparent distance. The expansion, on this account, causes the apparent shift in the oblique lines.

But then what about this next picture?

This time I’ve embedded a Poggendorff figure in the Ponzo illusion. Once again, the context presents depth unequivocally, complete with the kinds of size-constancy changes that we know arise early in visual processing. So the oblique edges of the geometric figure seem (to my eye) to diverge with distance, and the distant woman, objectively the same size as her sisters, looks larger. However, in this picture the oblique edges don’t seem to stretch into depth in the way they do in Shepard’s tables. The four edges of the parallelogram are all the same length, and to my eye that’s how they look. So now, with the Poggendorff test arms presented laterally, it doesn’t seem to me that we can attribute the misalignment to the apparent stretch into depth that might account for it on Shepard’s tabletops. In fact in the yellow picture there seems little scope for distance to be involved in the Poggendorff misalignment at all, since the junctions of the test arms and the parallels are not far off equidistant from our viewpoint. Once again, some will argue that very early in processing other spatial configurations are briefly in play, and that Poggendorff effects might arise because of these rapidly discarded possibilities.  But would the brain really discard one hypothesis about spatial arrangement in favour of another so early in processing, and still retain illusions arising from the first hypothesis?

Another argument in support of various depth processing accounts of the Poggendorff illusion is that when we see a Poggendorff configuration in a stereo image, but arranged so that the interrupting rectangle appears as a panel in front of a continuous rod, illusion is reduced or vanishes. That might indeed be expected if the configuration we see in 3D suppressed the hypothetical spatial arrangements that are supposed to give rise to misalignment in early processing. But is effect reduced in the stereo pair below?  (You won’t be able to see this as a 3D image unless you have the knack of viewing stereo picture pairs without a special viewer, in the mode known as cross-eyed viewing. If you need some practice, try this tutorial). To my eye, seen in 3D the panel is now floating in front of the rod. The Poggendorff effect seems to be little reduced if at all.

And even when I rig up a real set-up at home, with some kind of panel blocking my view of an oblique broomstick a few inches behind it, I still see the broomstick as misaligned.  I think there is some reduction in illusion in the real scene, but I suspect that in that case, and with the stereo pictures, any reduction in illusion could be because the Poggendorff effect, caused by physiological processes early in the visual pathway, is inhibited to an extent by our knowledge that in these cases the rod is continuous.

I reckon Poggendorff must be a 2D thing. I do agree that 3D context can enhance or suppress the effect. As another example, in the picture with the poultry at the start of the post, the left hand, surface-bound line sometimes looks to me less misaligned than the centre and right hand lines. For me, that would once again be because we’ve decided that, in this particular case, the oblique test lines are the ends of a continuous line on a surface.  Our brains the reduce the illusion, updating the assumptions arising in early processing in the light of our knowledge that in these cases the test lines are continuous. However the illusion still wins out. Thanks to whatever 2D characteristics of the figures produce the illusion, some misalignment persists.

Still, the brain is fabulously subtle. I wouldn’t be too dogmatic about this 2D/3D thing, not yet.