This web-site is about the Poggendorff and related illusions
First posted Feb 2003, last revised May 2009
The discussion here is a bit specialised
Are you looking for general sites about optical illusions?
Try these!
Michael Bach’s brilliant introduction to the Poggendorff illusion
My new visual illusion and special effects blog
My optical illusion comic-book story
Undergrad Students
Welcome to the site,
But if you have a visual perception course component,
beware.
This is not peer reviewed, professionally validated stuff.
Don't believe a word of it.
You might not get credit
for using it in coursework.
A Summary of the case being made here
It has often been suggested that cardinal axes in the observer's visual field contribute, but only in a minor way, to the Poggendorff illusion. Here it is suggested that they play a larger role, but in combination with other axes, which run between opposed angles or normal to parallels within the figures. These "figure axes" may therefore be at any orientation in the visual field, depending on figure rotation. A rule is proposed for the effect on alignment whenever a cardinal or figure axis intersects an alignment traverse across the gap. What is then new about the proposal is that cardinal and figure axes may act co-operatively to enhance illusion, or competitively to inhibit it. A model of the summed effect of both types of axis suggestively matches some new observations of variation in observed misalignment, as acute and obtuse angle figures rotate. The model also plausibly matches data in the literature from earlier studies of variation in apparent misalignment with figure rotation. A speculative physiological basis for the process is proposed, in a conflict between pre-attentive signalling of virtual eye movements, imagined as recruited both in making alignment judgments across gaps, and also in a novel role in sketching in dominant axial and orientational features of the scene.
Who’s the author?
I am David Phillips, but that’s a common name, so for identification my background was in the UK museum world in Nottingham and Manchester from 1970. I am an amateur as a vision researcher, but with two published peer reviewed papers:
"Constancy scaling and conflict when the Zollner illusion is seen in three dimensions", Perception, 28, 375-386, (1999). (for an abstract see: http://www.perceptionweb.com/abstract.cgi?id=p2711 and
“The Poggendorff illusion: premeditated or unpremeditated misbehaviour?”, Perception, 35, 1709-1712 (2006) (In the section “Last but not least”), online to academic libraries subscribing to the journal at:
http://www.perceptionweb.com/abstract.cgi?id=p5682
To contact me, please use the contact form on my site on illusions in pictures at http://www.OpticalOctopus.com
1. The Poggendorff Illusion and cardinal axes in the visual field
Researchers into the Poggendorff illusion have often attributed most of the illusory effect to depth processing. The brain tries to make sense of the figure as a three dimensional array, it is argued, and error arises because of size-constancy effects, or inconsistencies in the depth judgments involved (ref 11 - ref. nos. in brackets apply to sources listed at the end). These are at first glance very plausible explanations. But looked at in detail, they seem to me to pose major problems. Some new demonstrations of my reservations about Poggendorff depth processing accounts are on my separate visual illusions blog site.
Other researchers focus on two dimensional characteristics of the figures. In one account, illusion occurs because of our tendency to misjudge the size of angles (ref 24). In another, misalignments are observed because the figures trick the brain into getting the aim wrong in projecting alignment across the gap between the test lines, either because the position of the target points is misjudged (ref 30), or because the projected traverse orientation is misjudged (ref 22).
It is usually recognised that those processes do not account for all the illusion we see. They act in combination with other, secondary processes. The additional process that is most often invoked seems to stem from the cardinal axes of the observer's visual field. (If we imagine the figures projected onto the centre of a Cartesian co-ordinate system, the X and Y axes of the field are the cardinal axes). This is seen even when, as in figure 1, a stimulus consists simply of aligned line segments at 22.5° from vertical. Since the effect can be slight, the direction in which illusion is usually reported is indicated by arrows in this figure.
In this form the illusion is sometimes called the Poggendorff without parallels. A line at 67.5° shows a similar shift, but now as if due to the horizontal axis (ref 24, fig.4a). It is as if each segment had been marginally translated away from the axis, that is, nudged out of position but with no change in orientation. (Some researchers however prefer to think of the misalignment as a slight rotation of the lines, so as to align them more with the nearest cardinal axis).
I am going to propose a scheme in which axes in the visual field play the starring role, instead of the bit part that is all they are usually allowed. For a start, in some earlier research, I included an experiment whose results are consistent with the idea that more of the illusion in Poggendorff figures may be due in some way to cardinal axes than we usually suppose. When the vertical axis between a pair of isolated lines at 22.5° was flagged just with blurry blobs, as in figure 2, the average misalignment reported by fifteen observers was about 2° out of true, strikingly enhancing the error reported without the blurry blobs of only about 0.5° (See the appendix for details of my experimental procedures).
2° is a level of misalignment comparable to that usually reported with obtuse angle Poggendorff figures (as for example in figure 13 below). In this figure there is no obvious scope for angle or spatial processing effects to be contributing, but an effect from the axis between the blobs is a candidate.
My first proposal is that one source of the contribution that inducing arms bring to Poggendorff figures may lie simply in the way that they enhance cardinal axes, as the blobs seem to do.
2. Other axes may also contribute - "figure axes" between opposed angles or across parallels.
However, cardinal axes effects alone certainly cannot account for the extent and variation of illusion seen in all the varieties of Poggendorff illusion. It has been noted, in a number of studies, that as Poggendorff and related misalignment figures are rotated, a contribution from cardinal axes can be measured separately from an effect due to some kind of more oblique figure characteristics (refs 24 & 31, p 443). I will try to demonstrate that these additional effects could come just from a second kind of axis in the observer's visual field. These would be axes that can be at any orientation, but which run between the "centres of gravity" of opposed angles, and also at right angles to parallels in the figures. In what follows I will call them figure axes to distinguish them from cardinal axes.
Let's leave till the end the question of the physiological nature of these supposed axes. We can do so because of the way that illusions whose strength varies as we rotate them, (which Poggendorff type illusions do), can often seem like little objective physical systems. As they change before our eyes with rotation, they seem to have behaviours that are quite independent of us. Let's therefore for now imagine that they are indeed independent systems, and merely try to characterise and measure their behaviour, leaving till the end the question of the physiological agency causing it.
How then might the two kinds of axis, cardinal and figure, interact in our rotating systems? The one feature common to every variety of Poggendorff figure is a gap, across which alignment must be projected. Of course, both cardinal and figure axes also run across the gap. Let's propose a simple rule, consistent with the observations we have already made of the cardinal axis effect. Wherever the projected alignment intersects with either a cardinal or a figure axis, across the gap, the effect is to push the test line segments out of alignment, as if pushing them apart them away from the axis, without changing their orientation. The key point is then that cardinal and figure axes may act either together to enhance misalignment, or in competition to reduce it. This is diagrammed in figure 3. Under the proposal (and in the models we will make based on it) there would also be a small effect from horizontal axes on test arms at the orientation shown, but this is small enough to omit from this diagram, for clarity.
The proposed axial rivalry successfully models an otherwise puzzling variation in illusion with acute angle figures. In the left hand angle pair in figure 3, to my eye, a small but robust misalignment appears, shifting the upper test arm to the right and the lower test arm to the left, so that extensions of the test arms would intersect the inducing arms of the opposing angles. In the figure to the right, the misalignment is eliminated or reversed. (The interfering broken demonstration lines in the figure seem to have little effect, if comparison is made with figures without them). This cancellation or reversal of illusion occurs with rotation of acute and right angle pairs (see data below), when, in figures with handedness as to the right above, the test lines rotate clockwise between vertical and about 45°. As test lines rotate through 45° and on to horizontal (always assuming handedness as to the right above) illusion reappears. Illusion then likewise reduces or reverses with test arm rotation between horizontal and about 135°, thereafter increasing as the test arm approaches near vertical.
This variation within the first segment of each quadrant with rotation, in places straying beyond cancellation and into reversal of misalignment, is not readily explained by existing theories. However we can model it quite successfully by defining separate axial contributions and summing them.
3. An experiment with rotating acute and obtuse angle figures
To explore the effect demonstrated in figure 3, we will need new data. There is extensive data in the literature about misalignment observed in classic Poggendorff and obtuse angle Poggendorff figures with rotation. There is much less about acute angle figures, and the results are not consistent. Acute angle Poggendorff figures have even been shown to present illusion in different directions with different methods of experiment (ref 6). I therefore decided to make new observations of variation in apparent misalignment with rotation of four pairs of angles: two acute angle Poggendorff figures (with angles of 40° and 65°); a pair of opposed right angles; and an obtuse angle Poggendorff of 115°. I wanted to test effect at ten degree intervals, as each pair rotated through 180°, with three tests at each orientation in one handedness, and three in a mirror figure (full experimental details once again in appendix). That required 432 test adjustments, far more than I could inflict on a proper sample of observers. The data are therefore only from my own observations. That disqualifies them as proper experimental evidence - the results may be skewed either by experimental method, peculiarities of my own vision, or because, unconsciously, I allowed expectations to affect what I observed. But they can provide a basis for speculation based on comparison with other results.
Before looking at the data, a note about scoring. Firstly, I have scored rotation of test arms in degrees, with zero at twelve o'clock, positive angles clockwise and negative anti-clockwise. Positive and negative measurements of misalignment, however, follow a different convention, shown in figure 4.
Imagine in each case extending one test segment towards the other. If there is apparent misalignment, the extension will either intersect the inducing arm of the opposing angle, or pass the angle by without touching it. I shall score all misalignments that appear to intersect as positive, and all the ones that would appear to pass by without touching as negative. (Newcomers to the field, Beware! Other studies follow very different angular and sign conventions). The advantage of this scheme is that the directions of misalignment scored as positive and negative do not reverse with mirror-reversal of the figures, as happens in alternative, more mathematically consistent schemes, which tend to have been adopted in recent work.
The angles tested are shown in figure 5. They appear here in one handedness only, but a mirror reflected array was also tested.
With the scoring convention of figure 4 in mind, here in figure 6 is the data, (figures in appendix), for 40° angle pairs in red, 65° ones in blue, 90° in yellow, and 115° in orange. Each data point is from six trials, three in one handedness, three mirror reversed.
The salient features of these curves are:
* they go through two cycles of minima and maxima in a 180°
rotation of the angle pairs.
* they seem, as plotted here, to "run uphill". Relative to the scores
at -5°, a big dip is followed by a smaller rise, followed by a
smaller dip, followed by a big rise.
* they dip into negative scores, but are mostly in positive space -
that is, they are not symmetrical about the zero line.
* the obtuse angle (115°) curve is not only raised into positive
space clear of the other curves, but phase shifted, so that its
minima and maxima tend to appear to the left of the others.
4. Matching the data with a model
There would be more than one way of modelling the characteristics of these curves by summing contributions from more simple component curves. What I have to do therefore is to identify a set of component curves that do the job, but with each contributing curve matching the proposed variation in effect from just one of the axes as the figures rotate. To do that I have made a number of assumptions, which I have arrived at by trial and error, because they yield curves that match the data from observations. That may be acceptable if the assumptions are reasonable, and can be applied consistently across data from quite different sets of experiments.
We begin by isolating the contribution that would be made by the vertical cardinal axis, so that we ignore for the moment the counter contribution from the horizontal cardinal axis. Consider the effect with 40° angles, diagrammed in figure 7:
With the test arms of the angles at a small minus angle (far left), the axis will push the test lines apart to contribute an effect in the direction we are calling positive (shifted test arm projection cutting through the inducing arm of the opposing angle). However once the test arm has rotated through vertical to a small plus angle, the effect of the axis on the test lines will have reversed to negative (test arm projections passing by the opposing angle without touching it). The vertical axis will continue to affect misalignment in that negative way until the test arms have rotated through 90°. We can imagine that the effect might rise rapidly at first to a maximum whilst test arms are still near the vertical axis, and then tail off as they approach horizontal. Thereafter, from just after 90° through to 180°, the vertical axis will affect misalignment in the direction we are scoring as positive, but this time rising slowly, reaching a maximum only when test arm orientation returns to our starting point, and then dropping sharply to zero as the test arms align with vertical. The sequence, through negative and positive in the course of a 180° rotation of the test arms, is represented by the red line in the graph in figure 8:
Now we need to add a similar sequence for the counter effect of the horizontal axis, represented in blue, though this time with maxima and minima where test arms approach 90°, near the horizontal axis.
Note though that I have added a feature: I've attributed a stronger effect to the vertical (red) axis than to the horizontal one. That is going to present a problem later on, but the reason for doing it now becomes apparent when we sum the red and blue curves, to see the combined effect of the two cardinal axes as yellow. Setting vertical stronger than horizontal yields the first two of the characteristics of the curves in the data from observations that we seek to replicate in this model: the yellow curve not only goes through two cycles of negative and positive in a 180° rotation of the test arms, but thanks to the extra contribution from the vertical axis, a large negative dip is followed by a small rise, and a small dip followed by a large rise.
However, the yellow curve from the summed cardinal axes is symmetrical about the zero axis, whereas the curves from observation are lifted somewhat into positive space. To match that we need to add a component that is consistently positive. This is where the figure axes come in, running in figure 9 between "centres of gravity" in each angle pair, and additionally across the parallels that are presented by the obtuse angle pairs. The figure axis contribution, whether from angles alone or from angles and parallels in combination, is always in the direction we are scoring as positive: if acting in isolation, it would always cause test arms, if extended, to appear to intersect opposing inducing arms, rather than pass opposing angles by. It therefore offers just the consistently positive component we require in the model.
(Readers with a background in the field may note a useful feature of the separation of figure axis effect into angle and parallel components. It potentially extends the explanation to the illusion that can arise in the absence of angles in Tolanski type figures. It is also distinguishes pattern characteristics that might be involved in the variation in effect reported in blurred Poggendorff type figures (ref 22), though it does not explain it.)
Though the contribution from figure axes is always positive, it is not at a level that remains constant with rotation. As with the cardinal axis effect, misalignment falls to a minimum when test arms are vertical or horizontal, probably because at those orientations our powers of alignment are enhanced (ref 24). The maxima of the figure axes also vary. By trial and error I obtained the best match with data by setting the figure axis contribution from acute and 90° angles at maximum when the bisectors of the angles are vertical or horizontal. This of course varies from angle to angle. In figure 5, the bisectors of 40° angles are nearest horizontal, for example, when test arms are at 70°, bisectors of 65° angles nearest horizontal when test arms are at 57.5°, and bisectors of 90° angles are horizontal when test arms are at 45°. (See below, section 6 and figure 15, for a possible physiological basis for this variation in effect).
In the graphs in figure 10, we first establish the model curve for each acute angle pair (left hand charts) by summing the vertical, horizontal and figure components as described above, and then compare the resulting model curve with the data from observation (right hand charts). In each case the left hand chart shows the contribution from cardinal axes (red) and from the figure axis (blue) summed to give a model variation in error (yellow). The right hand chart compares the model (now changed from yellow into blue) with the observed data (red).
With the 115° angles the effect of the figure axis can be expected to increase. This is because with obtuse angle pairs, segments of the parallel inducing arms (that is, the arms that are not test arms) are directly opposite one another, whereas in the acute and right angle pairs the parallel arms are wholly offset from one another (you can check that in figure 5). We get a fit with the levels of the data for these obtuse angle pairs if we assume that, where parallels as well as angles contribute to figure axial effects, the combined effect is twice that attributed to the figure axis from acute angles alone. We also seem to get the best fit with data by assuming that the figure axes with obtuse angle pairs make their maximum contribution when the parallel arms are vertical or horizontal, rather than when the angle bisectors are aligned with these cardinal axes, as is the case with the acute angles. (See below, section 6 and figure 15, for a possible physiological basis for this variation in strength of effect). As above, to the left cardinal axis contribution in red, figure axis blue, sum yellow. To the right, the modelled sum is blue, data is red.
Given those assumptions, there does seem to be a possible correlation between the model and the data. Against the proposal, the model consistently exaggerates negative effect when test arms are at 5°, and has not mimicked the phase shift of the maxima in the observed data for the 115° angle pair. However the model assumes that variation in axial effect is smooth and symmetrical with rotation, which is in practice not necessarily the case.
5. Matching the model with data from earlier studies
But how well would my scores or model match a proper experimental average, from a number of observers?
Firstly, judging by comparison between my new data above and averages from earlier experiments that I made, and also the match with the nearest comparable data in the literature, my own scores seem very high - about double the misalignment generally reported. For example, in an earlier series of experiments, which I did with with fifteen observers, (see appendix for details), a pair of 90° angles with test arms at 155°, as shown in figure 12, produced an average error of only some 2.75°, whilst at 25° they produced virtually no misalignment at all, instead of my own much more pronounced scores for comparable angles in figure 6. However, my high scores may not be too problematic, since it is well known that there are large variations in individual response to geometric illusions. My scores do also seem consistent. In adjusting my model to compare with results from other research, I therefore simply halved cardinal and figure axis effects.
A second idiosyncrasy in my data is more problematic. With vertical axis contribution set stronger than horizontal, to match the way that the curves from my observations "run uphill", a glance back at figure 8 shows that we might expect a cardinal axial effect on a Poggendorff without parallels even with the test arm orientation exactly half way between vertical and horizontal, at 45° and 135°. In fact a small misalignment has indeed consistently been measured at those orientations in the Poggendorff without parallels (refs 5 and 24). Unfortunately, it is in the wrong direction, as if the horizontal rather than vertical axis were dominant. The discrepancy does not seem to be a peculiarity of my vision. When I informally tested myself with parallel-less figures at 45° I obtained (rather to my surprise) results in line with those in the literature.
The small but significant discrepancy could of course mean that my scheme is simply wrong, but there could easily be artefacts in my experimental procedures. (See appendix. The trial alignments were made within a “portrait” format window on a computer screen, and in retrospect the vertical format of the window may have constrained judgments). The discrepancy also arises, of course, only with isolated line figures. It is possible that the addition of angles to the stimulus changes the strength of the cardinal axis contribution to an extent that would account for the mismatch. In adjusting the model for comparison with data from other experiments, I therefore set the contributions from the two axes so as to match the effect usually reported in the Poggendorff without parallels, with a small extra horizontal contribution with test arms at 45° and 135°. The compromise cardinal axis model is top left in figure 13, vertical axis in red, horizontal in blue and sum in yellow.
Given those adjustments, the nearest comparable data to mine that I know of are from experiments with rotation of a pair of 135° angles (ref 24 figures 6a & 6b). I took scores as accurately as possible from these charts, averaged the results from figures of different handedness (as I did with my own observations), and then translated them into the sign and angle and conventions I've described above. Figure 13 compares my model and the experimental data. (Upper left, cardinal axes sum to give a cardinal contribution in yellow; upper right, cardinal axis, now red, sums with blue figure axis to give model in yellow. Lower left, model, now blue, and data, red). Once again the match seems to admit the possibility that the scheme is on the right lines.
Another set of comparable data come from a study of alignment between the arms of single angles and target dots (ref 31). Models and experimental data in this case appear in figure 14. I again first translated the data (from ref 31, table 1) into the conventions I used for my self-administered experiments. The model in this case, shown top left in figure 14, is the same for every angle. I next set the cardinal axis contribution, as in figure 13, once again at half the level in the model that matched my own observations, to allow for the unusually high level of my personal scores. For the same reason, I halved the figure axis contribution that I matched to my own data. I then halved again the figure axis contribution for the obtuse angles. That is because the single obtuse angle figures in this study lacked the opposed parallels that were assumed to enhance figure axis effect in paired obtuse angles. Figure axis contribution in this set of matches is therefore the same for acute and obtuse angles, and is assumed to be constant except where test arms align with horizontal and vertical. (In fig 14, upper left panel, cardinal axis contribution in red and figure axial contribution in blue sum to give model in yellow; in all the other panels, the model, now blue, is compared with data in red).
Here again there is a match between data and model, if not quite as good as for the more symmetrical figures. Model peaks and troughs tend to be too extreme, especially whenever test arms are at 112.5°, and there is a consistent extra negative contribution in the model very early in the cycle. However, these less symmetrical figures might evoke both smaller cardinal axis responses and generally less symmetrical axial contributions than the paired angle figures. Across all the examples looked at there remains (arguably) enough correlation to make a prima facie case for the proposal.
Extending the proposal to full Poggendorff figures is not straightforward. For a start a different scoring system would be required, since the one adopted above is only suited to opposed angles. More problematically, full figures should show slightly less misalignment than opposed obtuse angles, but in fact show more. They should show less because they present both an obtuse and an acute angle pair with each test line, and the effects of their axes must be in opposition to one another. This was the reason why the authors of the study from which the last set of comparison figures are taken ruled out the summing of cardinal axis effects and some kind of angular contribution as a basis for modelling illusion in full Poggendorff figures (ref 31, p444). According to the account proposed above, the problematic contribution from the acute angle axis, in full figures with parallels horizontal or vertical, would of course be much smaller than that from the obtuse angles and parallels, because of the fall-off in strength of their effect with increasing oblique axial orientation (see paragraphs following figure 9 and preceding figure 11). Even so, for full figures to show more illusion than obtuse angle figures, the contribution from the more oblique acute angle axis, however reduced, would still have to be more than compensated for by some other characteristics of the full figures.
A strong candidate would be an extra contribution from the much more salient parallels that full figures offer. Recall the apparent enhancement of cardinal axis effect by the blurry blobs in figure 2. This effect has not so far been quantitatively incorporated into the models above, in which cardinal axis contribution varies with test-line orientation, but not with the way in which cardinal axis emphasis is otherwise affected by figure characteristics. With the full parallels of classic Poggendorff figures, enhanced cardinal axis salience might become more significant. Full figures also present some positional imprecision at the junction of test and inducing lines as mediated in low frequency components of the signal, which obtuse angle figures do not offer. 22% of the effect in Poggendorff figures was convincingly attributed to this agency in one study (ref 4, pp 196,7). These characteristics should provide enough compensation, but without confirmation the anomaly must remain a potential problem for my account.
6. A speculative physiological basis for Poggendorff type illusions.
Moving on to a speculative physiological basis for a scheme like the one described above, we must first re-admit to the discussion of these figures an agency that has hovered on the margins of debate, only to be robustly rejected from the mainstream - the pre-attentive signalling of eye movements. I will now present some informal evidence to add to the case that at the very least they cannot be excluded from any full account of these figures.
Poggendorff illusions are especially puzzling in relation to eye movements. It seems established that the misalignments observed cannot depend on errors in realised saccades, since they appear even when the figures are fixed as after-images on the retina, moving when the eye moves, so that scanning is impossible (ref 9). Nonetheless they do seem to be affected by whereabouts in a figure we fixate and how we scan (refs 25, 26, 28). It has therefore been suggested that they might involve neural activation associated with the pre-attentive planning of saccades.
My observers drew my attention to just how much apparent misalignment can vary with scanning strategies. In doing the experiment that showed increased illusion with blurry blobs (figure 2), one observer noted that the effect depended where in the figure he fixated. That chimed with an observation made earlier by another observer in a different study, which I had paid little attention to at the time. I therefore went back to ten other of the observers of the blurry blob effect and asked them to note the extent of misalignment first when fixating one of the blobs, and then when fixating a line end, and to report any variation in illusion. Eight reported increased effect with fixation on a blob (the other two saw no difference). The enhancement seems to me quite strong if I fixate one or other blob, but strongest if I scan between the two blobs, switching from one to the other every two seconds or so. It is weakest if I scan between the line ends, or fixate the centre of the figure. For me there is a stronger effect still with full Poggendorff figures. In figure 15, try scanning between the angles, noting that you can still sense the extent of misalignment of the test segments as you do so. To my eye illusion is dramatically reduced when I scan obliquely between the acute angles.
These are informal observations, but they tie up with a subjective report in a study from about a century ago. Two researchers, after training themselves (with 3,200 alignments) to defeat Poggendorff misalignments by deliberate visual tracking up the line of the oblique, both independently observed "that even after the illusion had been overcome, it was possible to take a kind of general view of the figure, and to see the illusion again." (Ref 2, p90).
The idea that some kind of axial effect is at work seems consistent to my eye with appearances in figure 16. Here the strength of misalignment seems affected, in finely balanced versions of the Poggendorff figure, just by adjusting which axis is visually dominant. In these examples no deliberate scanning is involved. In the figure at 16(a), with the obtuse angles weighted, the usual Poggendorff shift appears. Weighting the axis between the acute angles at (b) seems to leave the test lines pretty much aligned. Yet this is not because the figure has in effect been reduced just to opposed acute angles. As seen in figure 5, at this orientation these always give a small reversed misalignment, in the opposite direction to the standard Poggendorff effect, as at figure 16(c). Then at 16(d), to my eye the test lines are pretty much aligned, whereas when the figure is given a salient vertical axis at (e), once again a small reversed misalignment appears, (upper bar shifted left, lower bar right), of just about the width of the test bar.
Reverting to classic Poggendorff figures, the important point to note now is that in informal scanning, of figures with parallels horizontal or vertical, we see strong misalignment, in the way we do when we scan deliberately between obtuse angles (figure 15a), or when the obtuse angles are weighted (figure 16a), rather than the negligible effect observed when we deliberately scan between the acute angle pair. Doesn’t that imply that we must therefore have a default tendency to privilege the obtuse angles, most probably because their axis lies nearest to vertical and horizontal? The default is only over-ridden by deliberately oblique scanning, or weighting. That would be consistent with the way in which, as a Poggendorff figure is rotated so that the parallels are oblique and the acute angle axis approaches horizontal and vertical, it is the acute angle pair that is privileged, and misalignment is cancelled or reversed. It also potentially accounts rather neatly for the increase in contribution attributed by trial and error to figure axes in the models above as they approached cardinal orientations, (see paragraphs following figure 9 and preceding figure 11), in order to achieve the best match with observations.
But just how might neural signalling of these default scanning axes be involved in the illusions? I speculate that it could be in a conflict between pre-attentive eye movement signals evoked on the one hand in order to establish dominant axes in a scene, and on the other in assessing alignment across a gap.
Imagine first that some saccade signalling may be evoked, not as a preparation for real eye movements at all, but instead, where no axis is deliberately scanned, in pre-attentive mapping of dominant axes in a scene. As if the brain was, as it were, using planned but unrealised eye scanning sequences to sketch in a skeleton of the orientations of dominant features in the field of view.
The visual system has a prodigious task to perform in keeping track of the ever shifting relationship in the visual field between gravitational vertical and head inclination. The link between the vestibular system and the eyes is of course the primary mechanism involved. It has been proposed that this might account for a number of illusions (ref 25B). However I envisage virtual saccades being recruited as well, where Poggendorff misalignments are involved, not so much in relation to the viewer’s own balance, but rather in helping to assess the postures and inclinations of whatever is in view - to help sketch head inclination into the visual field, in the form of cardinal axes, as a reference against which to assess creature and feature axes, in the form of what I have been calling figure axes.
Sensation evoked by unrealised saccades might help account for a subjective effect reported in aesthetic discussions. It has often been pointed out that images can seem to possess dynamic two-dimensional properties, for example of balance, or attraction between image elements, as if the observer were somehow experiencing a physical field of force due to picture elements, acting across the graphic surface (ref 1). For example, Bridget Riley, writing about Paul Klee, writes of “plastic energy of form in the picture plane - which I call dynamism ...” (ref 27, pp 17, 18). For psychologists, subjective observations of that kind have for a long time been compromised by earlier association with Gestalt theories that are now untenable. These theories proposed that real fields of force act within the brain, across “isomorphic” projections of the field of view. Numerous isomorphic projections of the visual field within the brain have since been identified, but no evidence has emerged for forces acting across them. (ref 15, p5). Might aesthetic field of force effects instead be explained by the signalling of virtual saccades?
Such accounts of visual experience are anecdotal, but may be worth attention. Recall the recently discovered "mirror neurones", neural channels associated with movement, but which are aroused in the brains of observers, without movement being carried out, when monkeys or humans see movement in others (ref 23). That discovery has vindicated decades old subjective descriptions by art theorists of "empathetic" projection of ourselves into visual scenes, such as Karl Groos's notion of "inner mimicry" (ref 21, pp 117-127), or a remark by Kenneth Clark that, in response to movements represented in sculpture, "through art we can relive them in our own bodies" (ref 3). It is likely that there is some physiological basis for the “field of force” reports.
The benefits of laying in a dynamic visual scaffolding might come at a small price, if programming of saccades that are not necessarily subsequently performed is also involved in alignment judgments. That seems to me subjectively a possibility. In visual alignment tasks, do we not "take aim", as it were, across the gap using unrealised eye movements almost in the way a darts player tentatively lines up dart and target with preliminary, uncompleted arm movements? Suppose the precise visual trajectory must cross a gap in a figure that is also intersected by interfering cardinal and figure axes. The “aiming” trajectory could then be inhibited and become aligned slightly with the interfering axes. The target is seen as rotated away from the dominant interfering axis. Alternatively, if experimental observers are presented with a one-armed Poggendorff and required to place a dot where the other arm should be attached, the target attachment points are placed as if rotated towards alignment with the dominant axis. In either case, a conflict arises between target position as indicated by virtual saccade orientation, and as specified by retinal co-ordinates.
Is a scheme like this neurophysiologically plausible? There is no neurological evidence for it. In favour of the proposal, some kind of conflict arising between potential saccade signal streams seems a possibility. The frontal eye field and the superior colliculus are the major, linked junctions through which preliminary saccade signalling from a rather distributed system seems correlated, prior to the firing of actual saccade circuits. The FEF, in the frontal cortex, facilitates conscious attention shifts. The superior colliculus seems a more likely candidate for a role in Poggendorff effects. It forms part of the more ancient, unconscious brain, located on the upper brain stem, and taking a signal feed direct from the retina, as well as feed-back from V1. That chimes with the apparently pre-attentive character of Poggendorff misalignments (ref 25A). It also incorporates coarse-grained maps of the visual field, which would be compatible with the “centre of gravity” axial endpoints proposed, and seems implicated in the correlation of visual, and other sensory fields, as well as in the coding of realised and potential saccade targets (refs 19A, 21A, 22A).
However, a serious problem for the proposal is just how axes would be represented in terms of signalling. The saccade system computes signals for shifts from the fixation point to target points, to rotate the eyes in order to bring the target into the fovea. The system is not known (as far as I am aware) to compute in anticipation saccades between points in the visual field that are not actually fixated, which my proposal would seem to require. The requirement cannot be met by rapid scanning, because Poggendorff misalignments, as mentioned above, appear even in figures fixed as after-images on the retina. The proposal would seem to require that the saccade system can, after all, either sketch out whole scanning sequences, or alternatively that there exists some facility in early processing that links paired targets as markers of axes. For example, we can certainly attend to points in the visual field that are not fixated (ref 22A). May we also be able to apprehend pairs of such targets, indicating an axis? If so, could that evoke a signal that could interfere with alignment saccades? Something of the kind may not be impossible, but remains entirely speculative.
Another objection to the proposal is that evolution would surely have eliminated such a miscalibration in the saccade system. However, the perceived misalignments may even confer a survival advantage. One recent probabilistic study suggests that, when embedded in real scenes, misjudged poggendorff arrays are more likely to correspond to real configurations in depth than would more accurate alignment judgments (ref 18A). A miscalibration in the visual system arising by chance might therefore have been selected through evolution.
7. Related figures
A final attraction of the scheme suggested is that it also potentially accounts for the related Tolanski type illusions. Of course, these may arise from quite different processes, as may misalignments from angle oppositions and from classic Poggendorff figures. However a scheme that might apply to all these target misjudgements has obvious attractions.
In Tolanski figures, as at (A) in figure 17, the position of targets of a vertical or horizontal traverse, across parallels presented obliquely, is illusorily rotated towards the inclination of the parallels, and so away from the axis across them. In the figure here objective vertical runs from the white interval in the lower line to the centre point between white intervals in the upper line, but tends to appear rotated clockwise, with the upper left white interval apparently vertically above the lower interval. 
The scheme is also consistent with a related effect, noted by chance during research into a different illusion (ref: 29, pp 17, 20), shown at (B) and (C) in figure 17. (I’ve found no further discussion of this figure, so I will call it for its discoverers Walker and Shank’s illusion). In the figure originally reported, similar to the one at (B) here, the strictly horizontal line joining the apexes of the angles appeared to ten out of twelve observers to run slightly upwards to the right. The effect is tenuous, but would be expected to be much weaker than the illusion seen in Tolanski figures according to my proposal. In Walker and Shank’s figure, only short sections of parallel lines are opposed to one another, so as to offer rival, oblique figure axes, confusing the judgment of horizontal alignment. At the same time, figure axes between the angles would be almost aligned with horizontal, and barely in this case interfere with judgment of it. But from the original study cited, a small but robust effect is clearly there. To my eye it is slightly enhanced when curves replace angles, and the figure is reduced to the outer pair of lines, as at 17(C). The persistence of effect in that case would be consistent with the idea that, if my account is sound, it is not be parallelism as such that gives rise to a figure axis normal to the parallels in Poggendorff type figures, but rather a more general figure orientational axis, to which the parallels strongly contribute.
The top two figures in Figure 18 are very subtle effects also consistent with the proposals. At figure 18i, ignore for the moment everything except the oblique line at the top of the figure, and the transit xy. The transit is objectively parallel to the oblique line above it, but appears just slightly steeper. (That effect was first published, I believe, in ref 12). Then at figure 18ii ignore for now all but the arc of dots at (a) and the target dot at (b). Imagine the arc of tiny dots as a concave mirror, and then imagine taking a shot at where its focus would be. Objectively, the focus is at (b), but to my eye it appears just slightly higher.
Figure 18iii shows some Poggendorff paradoxes. Imagine extending the arms of the central lower angle. They appear to misalign with their target arms in the usual Poggendorff way. But now note that the apex of that central lower angle is moved by the illusion at one and the same time to the left in relation to the upper right triangle, and to the right in relation to the upper left triangle. Those misalignments would arise if the gap between the triangles was reduced, but that’s been ruled out in earlier Poggendorff studies. No Euclidean transformation of one row in relation to the other could produce the result. However, what we see is consistent with rotation of the invisible transit tracks across the space. Figure 18iv presents the same paradox with Tolanski figures. And now go back to figures 18i and ii. The subtle effects in these, presented the same configuration, show the same paradox. All the figures seem to me consistent with the scheme proposed above. However it should be noted that constancy scaling effects, consistent with depth processing theories of the Poggendorff illusion, can also present results that are paradoxical from a geometric point of view.
Finally for now, there is another figure which is discussed in connection with the Poggendorff illusion, the so-called corner Poggendorff. I’ve discussed that along with some other bent line illusions at
http://www.opticalillusion.net/optical-illusions/subtle-bent-line-illusions.
My take on it is that it is presents two separate illusions, a misangulation illusion, plus when seen at a steep orientation a Poggendorff effect, but between two different opposed angles, one acute and the other obtuse. The proposal here would be consistent with that. But others disagree (ref 24).
Appendix
Experimental methods
For figures 5 and 6 - the tests I tried on myself
The experiments were done with the graphics package Appleworks on a Powerbook with a screen approximately 11 inches by 8. A separate file was made for one handedness of each angle pair, with angle arms just over one inch long on the screen, and test arms at 0° (vertical). The gap between test points was just over two inches. A chart was prepared to assign a number to each of the 18 orientations of each angle pair, (at 10° intervals from -5° through to 165°), and to the 18 mirror orientations of each pair. 144 paper slips were prepared, each bearing a number corresponding to each unique combination of orientation and handedness. These were then selected one by one, by lucky dip. For each test adjustment, the computer file for the relevant angle pair was first called up and copied. In the copy file, using the marquee tool, one or other angle (informally varied) was set out of alignment. The image was then mirror reversed if necessary, and set to rotate to the orientation specified by the selected slip number. A possible problem arose because in Appleworks lines near to cardinal orientations are very jagged, but informal tests suggested that this did not seem to give rise to any "twisted cord" type orientation illusion. The test lines were then adjusted to apparent alignment, using the marquee tool, once again with informal variation in which angle was moved. Following the methods of Weintraub (ref 31), the distance between testpoints was not controlled, though the attempt was made to keep it at approximately two inches, on the assumption that angular error remains constant with variation in separation. This avoided the need for distracting position references on the screen, which was also uncluttered because in Appleworks the outlines of the marquee tool vanish whilst selections are moved, so that only a small arrow offered any visual distraction. The file was then printed and closed without saving, and the printed sheets set aside without being inspected. A tick was then put on the numbered slip, and the slip was returned to the lucky dip until it had accumulated three ticks, when it was set aside. Three tests were therefore made for every orientation of each angle and its mirror partner. The experiment was run over several days, with not more than thirty adjustments in a single session. (I find that acute angle judgments are strangely taxing at certain orientations, and suspect that performance may fall off for observers who soldier through longer sequences in one session, however conscientiously). When the tests seemed complete, the errors were measured by hand and tabulated. In tabulating the data, it was found that a few orientations had accidentally been tested too many times and a few tests had been omitted. The absent tests were therefore done as a final sequence, but the extra tests were retained to feed into the average for their orientation, to avoid a decision as to which test result to discard. The averages are these:
Experimental methods
For figures 2 and 12 - from some older experiments with a sample of observers
These were four of ten images prepared, along with a further ten mirror images, each isolated in the middle of an A4 portrait orientation sheet. The gap between the ends of the test arms was 2.75 inches in each case. The sheets were placed in a presentation folder of transparent envelopes, so that the order of presentation could be informally shuffled for each observer. There were fifteen observers, with normal or corrected to normal vision, naive to the purposes of the study and aged from mid-teens to mid fifties. Observations were made with the observer seated at a table with the folder propped up so that line of sight was approximately at right angles to the plane of the sheet. Observers were told (truthfully) that all the test bars were in objective alignment, and asked to decide first whether they appeared out of alignment, and if so to which side of the opposing bar each would project. They then referred to a comparison strip, to decide whether the misalignment was in a direction being scored as positive or negative. One reference strip is shown as figure 20, but all were prepared so that the reference images were always of exactly the same size, orientation and handedness as the test images.
The observers' final task was to estimate the extent of the shift. A traversing bar was substituted for the usual line, in order to offer an easy index of misalignment. An apparent shift that misaligned a side of one bar to the centre of the bar opposite was scored as 1 or -1, a greater shift so that a bar seemed just its own thickness out of alignment was a 2 or -2, whilst a translation that would leave a gap between bars after projection was a 2.5 or -2.5. Observers were invited to opt for scores at plus or minus 0.5 or 1.5 if they wished. For each mirror pair of images an average of the scores for each observer was taken, and then an average of the observers' scores. These scores were therefore arbitrary units, not measures of misalignment in degrees or millimetres, but for the purposes of the comparisons on this website they have been roughly converted into error in degrees, by measuring on the reference strip the error in degrees corresponding to the numerical scores.
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(These include sources I have particularly benefited from as background, in addition to references specifically cited).
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© David Phillips, first posted 15th Feb 2003, last revised May 2009
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