This site is about the Poggendorff and related illusions

 

First posted Feb 2003

Completely revised August and October 2010

Minor clarifications April 2011

 

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 Illusion blog - with a Poggendorff category

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This is not peer reviewed, professionally validated stuff.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

Graphic summaries of alignment errors with rotation of three kinds of misalignment illusion bring out a common pattern of variation. Misalignment increases with angle size, but with an effect attributable to cardinal axes in the visual field systematically giving way to contributions with markedly different characteristics. One way of modelling the overall variation is by summing contributions from three proposed effects: cardinal axes projected into the visual field, symmetry axes in the angles in the figures, and an assimilation effect. The physiological culprit might be a conflict between pre-attentive eye movement signal streams. Signalled, but unrealised eye movements could be implicated both in making alignment judgments across gaps, and also in helping to locate gravitational vertical in the visual field in relation to strong orientational cues in the figures.

 

 

Who’s the author?

I am David Phillips, which is a common name, so for identification my professional background was in the UK museum world in Nottingham and Manchester from 1968. 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=p2711and

“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

 

 

1. The “signature” of the Poggendorff family of illusions.

 

The classic Poggendorff figure has often been studied as one of a family of alignment illusion figures, some of which appear in the title banner at the head of this site. All show objective alignment between two “test” arms, or an arm and a dot target, but appear misaligned. The misalignment increases with the size of the angles involved, except in the variants like the classic Poggendorff figure, which contain both acute and obtuse angles. In these misalignment increases with size of the obtuse angle. The degree of misalignment observed also depends on the orientation of the figures, varying systematically as the test lines sweep through 360 degrees, like a hand on an analogue clock.

 

         There is experimental data for the variation in effect with rotation of various figures, thanks to a small number of researchers.

 

 

 

 

Figure 2: Misalignment (vertical axis) varying with rotation (horizontal axis) in three varieties of figure as angle size increases from front to back of each stack. Left, re-plotted from Weintraub et al. (1980, figure 1); centre re-plotted from Weintraub and Brown (1986, table 1); right from a study by the author.

 

Weintraub et al. (1980) studied a figure similar to (g) in the header banner above, for a range of angles. Their results, showing strength of misalignment varying with rotation, are replotted in summary to the left in figure 2. Weintraub and Brown (1986), studied figure (c) in the header banner, for a range of angles, and the results are replotted centrally in figure 2. I also, as an amateur researcher in 2003, measured my own observations from rotation of acute angle pairs as at figure (d) in the banner, for four angle sizes, and my results in summary appear lower right in figure 2. We will also be referring to some results from Ninio and O’Regan (1999), who studied only one size of angle, with test arm joining inducing arm at 45 degrees, but in a range of the variant figures shown in our header banner. Greene (1994), gives partial rotation results for a figure similar to (g) in our banner, for a range of angles between test and inducing arms.

        

         In the three sets of data summarised and presented graphically in figure 2 a pattern of variation emerges. Ignore for the moment the detail of the graphs, and note just the common trends that they present. The most salient feature is an increase in overall strength of illusion as angle size increases, remembering to select the obtuse angle in figures containing an acute as well as an obtuse angle. The increase runs from front to back of each stack of graphs here, and is familiar in earlier research on Poggendorff related figures. It is also supported by Greene (1994). Scores for the smallest angles, at the front of the graphs, even drop into negative territory.

 

But then there is a more subtle trend in the stacks. At smaller angles in the two sets of data from Weintraub and co-workers, shown at the front of the graphs left and centre, we see a fall and a rise in scores within each quadrant (rotation through 90 degrees). To the rear of each stack, showing results for the largest angles, there is only rise to a maximum within each quadrant. In my own data, to the lower right, we can see the same trend from front to back of the graph, though not to the extent of eliminating the initial dip in scores within each quadrant. But that turns out to be quite consistent with the other results, since there were no very large angles in my study. The largest was 115 degrees, and indeed there is good agreement in overall shape between that rear curve and the one for a 120 degree angle, in red in the data to the far left. The two kinds of component curve, typical of the smallest and largest angles respectively, were recognised in the data in all the studies just cited.

 

         The steady trend in each stack of graphs as scores rise with increasing angle size, from curves showing a dip and a rise in each quadrant, to ones presenting just a rise to a maximum at a test arm orientation of 45 degrees, seems on the basis of the data available to date to be a signature of the Poggendorff type illusions. One goal for any explanation of the illusions would then be to account for the signature. I shall show that one way of accounting for it would be by summing a small number of component curves in each study, each varying systematically, and generated according to a simple rule from axes projected into the visual field, or from salient orientation cues in the figures.

 

 

2. Rival explanations of the Poggendorff illusions

 

We should begin by locating my proposal in the context of other theories about these illusions. For a start, not all specialists agree that all the misalignments seen in the header banner belong to the same family of illusions (Morgan 1999). Even if the variants are all rightly thought of as related, there is no agreement that they are all primarily due to one common brain process, rather than a combination of different processes (Hotopf 1981).

 

         Probably the leading culprit implicated in the misjudgments has been 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 ambiguities in the depth judgments involved (Gillam 1971; Gregory 1998). These are at first glance very plausible explanations. But looked at in detail, they seem to me to pose problems. My reservations about depth processing and the Poggendorff illusion appear 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 (Ninio and O’Regan 1999). We certainly do misjudge angles, but it seems to me that we can convincingly demonstrate that Poggendorff effects cannot be attributed to these misjudgments. (Wenderoth 1981; Phillips 2006)

 

         In other studies, misalignments arise 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, or because the orientation of the projected traverse is misjudged (for detailed summary review with references see Greene, 1994, page 666). That last option, a slight rotation of the invisible agent in every variety of these figures, the traverse across the gap, is the one I favour.

 

         Only a rotation of the invisible traverse seems to me to account for a paradoxical version of the Poggendorff figure. Consider figure 3.

 

 

 

Figure 3: A Poggendorff paradox. The apex of the upper, downward pointing angle appears to move at one and the same time to the right in relation to the lower left angle, and to the left in relation to the lower right angle. Rotation of the invisible aiming tracks across the gap would account for the paradox.

 

 

The apex of the upper central angle is the target point for two of the lower test arms, so that it seems paradoxically to be shifted at one and the same moment to the left for one test arm and to the right for the other. The same goes for the two lower target points. Appearances would be explained if the gap between the opposing parallels was illusorily reduced, but that does not seem to be the culprit in Poggendorff illusions. No other systematic geometric distortion of the whole figure could account for all the simultaneous misalignments in the figure. However if the invisible projected track from each test arm to target was separately rotated away from the target point, the paradox would readily be accounted for.

 

         I am going to consider evidence that in all the variations of the illusion we will consider, rotation of the invisible, gap-crossing alignment tracks arises primarily from conflict between alignment judgments on the one hand, and salient orientation cues involved in our sense of visual balance on the other.

 

 

3. A note on error scoring conventions

 

At this point we need to pause to think in more detail about the measurements of the effects observed. Unfortunately, almost every study adopts different conventions. For example the variants of the figures in our header banner can be in the handedness shown, or mirror reflected. Rotation can be clockwise or anti-clockwise in either handedness and may be expressed in radians or degrees, at different intervals, clockwise or anti-clockwise, with zero at twelve o’clock in one study, or three o’clock in another, etc. Different terms are often then adopted to label the variations in handedness or rotation in graphic presentations. For anyone without the capacity to visualise even two dimensional mathematical variation clearly, (and that for sure includes the author), the interplay of rotational and mirror symmetries can be confusing.

 

         However one assumption makes the task much simpler: as a rule of thumb, similar misalignment effects tend to be seen in mirror or rotationally symmetrical figures, such as the symmetrical variants as shown at figure 4(a). Pooling results from such variants does involve loss of some information. For example, figure 4(b) shows a figure presenting misalignment with rotation of a one-armed Poggendorff figure, redrawn from Weintraub and Krantz (1971, their fig.5).

 

 

Figure 4: In the study on this site, results are pooled for mirror symmetrical versions of the same figure, as to the left. General trends are revealed more clearly, but at the cost of some detailed variation. Variation from quadrant to quadrant can be seen in the results at right in polar co-ordinates, redrawn from Weintraub and Krantz (1971, figure 5),

 

Scores are shown in polar co-ordinates, and the “propeller blades” of the figure show results from each quadrant that are similar, but by no means identical. The researchers in that study noted that the variation, especially in the lower quadrant, almost certainly does have a physiological basis. However, if the assumption is made that such variations are probably minor anomalies, they can be seen as masking much more overall, symmetrical effects, which emerge only when we pool the data within quadrants. For a radical example see Greene (1994).

 

         The claim is fundamental for my study, so herewith another example in support of it. For a single armed Poggendorff figure, with the alignment target a target dot on the opposing parallel, both Ninio and O’Regan (1999) and Wintraub et. al. (1980) made measurements of variation in misalignment with rotation when the test arm is at an obtuse angle of 135 degrees. Ninio and O’Regan (1999, figure 8) compare their results with Weintraub et al’s graphically, for the figures in each handedness, showing fair agreement. But if we pool the results from each handedness and translate them into the conventions I am using, as detailed below, the agreement becomes much more striking, as shown in my figure 5.

 

 

  Figure 5: Pooled results for mirror reflected versions of a one test armed figure with test arm angle at 135 degrees, re-plotted from (red curve) Weintraub et al. (1980, figure 1); and (blue curve) Ninio and O’Regan (1999, figure 8).

 

I suspect that the most marked remaining difference in scores after pooling the results, with test arm orientation at 90 degrees, may be due to a difference in experimental set-up: Weintraub et.al’s experimental observers looked at the figures through a lone ranger mask, to constrain head position, whereas Ninio and O'Regan's observers enjoyed something more like free inspection.

 

         For all my figures therefore I have wherever possible pooled all the results from symmetrical figures. Sometimes, in my tabulations, different data points in the same graph may even be based on different numbers of observations, “read off’ from tables or graphs in the literature. That can be noted in the detailed numerical data given in the Appendix 1. Since I’m not applying statistical tests, for any particular data point I’ve reckoned the more scores going into the pot the better.

 

         I’ve then translated the different conventions as to handedness, rotation and scoring, into one scheme: as shown in figure 6.

 

 

 

 

Figure 6: Scoring conventions in this study: (a) test arms would have always to rotate clockwise to align with inducing arms (ignoring the acute angle in figures showing obtuse and acute angles); (b) figures always rotate clockwise; (c) alignment errors are measured in degrees of angular deviation from objective alignment; (d) illusory counter-clockwise shifts in arm position are scored as positive, and clockwise shifts as negative.

 

My test arms would have always to rotate clockwise to align with inducing arms, in figures presenting only acute or obtuse angles, as shown in figure 6(a). In figures presenting both obtuse and acute angles, it’s the obtuse angle whose test arm would have to rotate clockwise to meet the inducing the inducing arm. Rotation of the figures is always clockwise, as in the segment of rotating figures at figure 6(b). Zero is always with test arms pointing at twelve o’clock (vertical), and the angle of rotation of the test arm is shown along the horizontal axis in every graph. Misalignment error is measured as an angle, as at figure 6(c) in degrees, and comprises the vertical axis in every graph. Finally, positive and negative measurements of misalignment follow the convention shown in figure 6(d). Whenever an apparent misalignment suggests a clockwise shift in test arm position, the shift is scored as negative. Whenever test arms present an anti-clockwise shift, the result is scored as positive. But remember that the rule would only apply with figures in the handedness we have settled on, with test arms (considering only the obtuse angles in full figures) clockwise from inducing arms.

 

 

4. The Role of Cardinal Axes in Poggendorff type illusions

 

         Let us set aside for the moment any suggested physiological basis for the misalignments we see in these figures. As we observe misalignment varying with rotation in these figures, so compelling is the effect that it can seem as if we were observing little objective physical systems. Let us first consider them as if they were just that, seeking only for now to characterise and measure the effects we observe.

 

         First consider the cardinal axes running through the figures, the vertical and horizontal axes of the observer’s visual field. In figure 7(a) it can be seen that vertical axis alone might be responsible for a small misalignment seen when, as in the figure, a stimulus consists simply of aligned line segments at 22.5 degrees from vertical. Since the effect can be slight, the direction in which illusion is usually reported is indicated by arrows in this figure.

 

 

Figure 7: The Poggendorff without parallels, with arrows indicating the direction of the slight shift usually reported. Misalignment is attributed to the effect of cardinal axes in the visual field.

 

 

         In this form the illusion is called either the Zehender illusion or the Poggendorff without parallels. A line at 67.5 degrees shows a similar shift, but now attributable to the horizontal axis. It is as if each segment had been just a touch translated out of alignment, away from the nearest axis. That is, they appear nudged out of position, but with no change in orientation, so that each appears just fractionally further from the nearest cardinal axis than it objectively is.

 

         To represent variation in misalignment with rotation graphically, we begin by isolating the contribution that would be made just by the vertical cardinal axis.

 

 

Figure 8: Diagram of variation of the extent and direction of test line shift, attributed to the vertical axis in the observer’s visual field, observed with rotation of a pair of opposed acute angles.

 

As diagrammed in figure 8, with the test lines vertical they appear pretty much aligned, but once the test arm has rotated a small way clockwise from the vertical axis, the arm appears pushed away from the axis in a clockwise direction. The vertical axis will continue to affect misalignment in that direction, but with decreasing force, until the test arms have rotated to 90 degrees. Thereafter the effect reverses, only weakly at first, but producing a growing anti-clockwise shift as the test lines rotate towards vertical again.

 

         Let us make a graph of that effect, bearing in mind that we are scoring a clockwise shift in negative degrees of error and an anti-clockwise shift as positive.

 

 

Figure 9: The misalignment observed attributed to the vertical axis in figure 8, graphically shown as the blue curve, with the reciprocal effect from the horizontal curve added in red. Summing the curves gives the yellow curve, which resembles the results from small angles in figure 2.

 

The effect of the vertical axis is shown by the blue line in figure 9: From zero misalignment at vertical, the effect of the axis as the arms begin to rotate clockwise is at first to induce a strong clockwise shift. Scores accordingly fall rapidly to a strong negative maximum, whilst test arms are still near the vertical axis, returning more gradually to zero as they approach 90 degrees. Thereafter, from just after 90 degrees through to 180 degrees, the effect of the vertical axis reverses, to affect misalignment in the direction we are scoring as positive, but this time rising slowly, reaching a maximum as arm orientation approaches vertical and then dropping sharply to zero as the test arms align with vertical.

 

         Now we need to add a similar sequence for the counter effect of the horizontal axis, represented in red. This acts in opposition to the effect of the vertical axis, but always with effect changing slowly as effect from the vertical axis rises rapidly, and vice versa, as seen in figure 9. When we now sum the two axes, we obtain a rather characterful yellow curve. Note that it is intriguingly reminiscent of the curves for smaller angles in figure 2. It dips gently into negative, rises slowly into positive territory, dips sharply into deeper negative through the centre of the graph, and rises gently back into positive scores.

 

         That curve fits some professional experimental data quite well. There is a variation of the Zehender figure, which consists in misalignment not between two line segments, but between just one line segment and a target dot. As measured by Ninio and O’Regan (1999) both the version with parallels and the line and dot variant give very similar results (their figure 5b). Response to the line and dot version with rotation was also measured by Weintraub and Brown (1986). The results from both studies (coloured blue and red respectively) are replotted in fig 10 together with the model (in yellow). The good agreement seen between observations and model depends on one refinement: in the model, the two axes, vertical and horizontal, no longer make equal contributions. I have had to multiply the error attributed to the vertical axis in figure 9, (where the maximum error was arbitrarily set at 1) by 1.15, and to multiply the error attributed to the horizontal axis in figure 9 by 1.65.

 

 

Figure 10: Results for a line and dot figure with rotation re-plotted from Ninio and O’Regan (1999, figure 5b) in blue, Weintraub and Brown (1986, table 1) in red and our model (yellow). The vertical axis contribution as shown in figure 9 has been multiplied in the model by 1.15, and the horizontal axis contribution by 1.65.

 

         It is as if either the variant of the illusion, or the experimental set-up, had aroused the cardinal axis system, but with the contribution from each axis set at a different level. We will see below that in matching the data from experiments tabulated in figure 2 with models, I found that for each of the three sets of data I had to fix the relative contributions from horizontal and vertical axes at a different ratio. However the ratio for each of the three sets of data could then be kept roughly constant as I increased the contributions from each axis with increasing angle size. For Weintraub et al (1980), in which the maxima in each quadrant appear approximately equal, I set the contributions from each axis as equal; for Weintraub and Brown (1986), where maxima tend to be greater in the first quadrant, I set the horizontal axial contribution as greater; and for my own observations, with maxima strikingly higher in the second quadrant, the vertical axis had to be greater. I suspect the latter might be a quirk of my own vision, since the data in that study came only from my own observations.

 

 

 

 

Figure 11: The cardinal axis contribution to the models in figures 13, 15 and 17 (below) to match with data re-plotted from Weintraub and Brown (1986), to the left, Weintraub et al. (1980), centre, and the author’s 2003 study, right.

 

Detailed numerical results as mentioned above are given in appendix 1, but in figure 11 we can see a graphic display of the cardinal axis contributions, which we will apply below in the models to match with the three sets of data summarised in figure 2. Note that in all three cardinal axis models we see how effect rises gently with angle size, but with the characteristic shape of each set of curves dependent on the ratios between vertical and horizontal axis contributions for that set.

 

         Variation in these cardinal axis models is symmetrical about the zero line. That was also the case for the observed results for the Poggendorff without parallels (figure 9). But now note that in fig 2, showing graphs for experimental data from more complex figures, though the curves for small angles still resemble those for the Poggendorff without parallels, they are somehow lifted at least in part into positive territory. Cardinal axes effects alone clearly cannot account for the extent and variation of illusion seen.

 

 

5. Other agencies may also contribute - "figure axes" between opposed angles and an effect assimilating the alignment track to inducing line orientation.

 

The additional, positive effect we need could come from strong orientational cues that are to be discovered in the figures themselves, and which can therefore be at any orientation, depending on the rotation of the figures. Two of these cues, diagrammed top left in figures 12, 14 and 16, are the symmetry axes of the acute and obtuse angles. (This is a change from the way I characterised these axes in earlier versions of the site). These we will call the figure acute or figure obtuse angle axes.

 

         How then might cardinal and figure angle axes 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 gaps. Let's propose a simple rule, consistent with the observations we have already made of the cardinal axis effect in the Poggendorff without parallels: wherever the projected alignment intersects with either a cardinal or a figure angle axis, the effect is to push the test line segments out of alignment, as if pushing them apart them away from that axis, without changing their orientation. As before, we ignore the mechanism for now, and just observe the effect. As we will see, cardinal and figure axes may then either co-operate to enhance misalignment, or compete to reduce it.

 

An additional agency is required to model the higher scores observed when figures present either parallel inducing line segments, as at (g) and (h) in the banner at the head of the site, or when a single inducing line extends either side of the junction with the test arm, as at (f) in the banner. Misalignment scores are not as high from figures with only one inducing arm as from those with parallel inducing lines, but that apart, with rotation they track scores from full figures quite faithfully. (Ninio and O’Regan 1999, their figure 8). In either of those cases, the figures then present salient orientation along the inducing lines.

 

In earlier versions of this site, I attributed the additional misalignment observed when figures acquire strong orientation to the effect of an axis normal to the inducing lines. I now believe that an assimilation effect is more likely (see below, section 11), rotating the projected track from test arm to target into slightly closer alignment with the inducing lines. Once again we ignore the mechanism for now, but just note the effect. As with the cardinal and angle axes, it results in an overshoot in target position, so that the effect is always positive in the scheme we are using. However like the axial contributions the assimilation effect can either enhance or oppose the effect of the cardinal axes.

 

In what follows, we will first see how well the proposed axial and assimilation agencies match data from experiments. I will then make the case that these agencies are better candidates than others proposed as the origin of these illusions, and that they could have a plausible physiological basis.

 

 

6. Comparing data from Weintraub and Brown (1986) with a model from summed axial and assimilation effects

 

         Let us start with figures that do not involve the assimilation effect, to see how the figure and cardinal angle axis components might combine. These are the simplest of the figures for which, thanks to Weintraub and Brown (1986, table 1), we have comprehensive rotational results.

 

 

 

 

 

Figure 12: Modelling the contribution from the figure angle axis for comparison with experimental data re-plotted from Weintraub and Brown (1986, table 1). The resulting models are shown in detail in figure 13, below. Figure (blue) and cardinal (yellow) axes can oppose one another, as to the top left, or act in consort, as top centre. The horizontal cardinal axis is omitted at upper left and centre for clarity, since its contribution would be small. The cardinal axis is omitted altogether when the test transit is at around 45 degrees, because at that orientation the vertical and horizontal cardinal components would cancel out. The effect of the figure angle symmetry axes is greatest when angles point horizontally, or vertically up or down, as shown by the coloured band top right, so that maximum contributions march across each quadrant with increasing angle size, as shown below.

 

As noted, in figure 12, at top left, we diagram three stages in rotation of a forty degree angle, and the effect on alignment of one arm with a target dot. Note the vertical cardinal axis in yellow in the left and central figures. The yellow arrow attached to the target arm then indicates the direction in which, under my proposal, the arm would be induced to shift by that axis. There would also be an effect from the horizontal axis, but it would be much less at this orientation, and I have omitted it here for clarity. I have omitted the cardinal axis contribution entirely from the figure with the test transit at about 45 degrees, because at that test arm orientation vertical and horizontal cardinal axis contributions would nearly cancel out.

 

Now note the figure acute angle axis, in blue at each orientation, running through the symmetry axis of the angle. Once again coloured arrows attached to the test arms indicate the direction of proposed shift due to this axis. Two important points emerge. First, when the test arm is just anti-clockwise from the vertical axes the effects of the cardinal and figure axes are in competition, pushing in different directions, but, once the arm has rotated through vertical, they collaborate. Second, whereas the cardinal axis can evoke a clockwise or an anti-clockwise shift in the test arm position, the effect of the figure axis, with the angle at this handedness, will always be clockwise, whatever the rotation.

 

         By trial and error in matching models and data, I have found the best fit across all the sets of data when making the assumption that the effect of figure acute and obtuse angle axes is maximum when angle symmetry axes are aligned with either vertical or horizontal (so when the angles point straight up, down or horizontal). Now note the segment of the array of rotating figures shown upper right in figure 12. The coloured band shows how the orientations at which the angles point directly vertical march across the array from upper right to lower left with increasing angle size. There would be a similar band for angles pointing horizontal if the array were extended to the left, to an arm orientation at zero. In the coloured model graphs of effect below, therefore, maximum marches from right to left with increasing angle size within each quadrant, at the same time as overall effect increases.

 

         In figure 13 We can now compare the data in Weintraub and Brown’s (1986) study, which we saw summarised graphically as the central nest of graphs in figure 2, with the model.

 

 

Figure 13: Models in red, with data in blue replotted from Weintraub and Brown (1986, table 1), for individual angle sizes. Curves of similar colour in the stacks of model contributions at top right from cardinal (top) and figure axes (below) are summed for each angle size. For the smallest angle (top left) the cardinal axis contribution is added in yellow: at this small angle size, results only briefly depart from those for the Poggendorff without parallels.

 

As a reminder, at top right in figure 13, I diagram the component graphs of the axial contributions in this case, cardinal axes (from figure 11) and figure acute axes (from figure 12). In the individual comparison graphs for each angle size in figure 12, experimental data is the blue line, with the model in red. (Numerical data is given in appendix 1).

 

         Note that the graph for a 22.5 degree angle (upper left) has an extra curve, in yellow. This is the model curve for the Poggendorff without parallels, seen above in figure 10 (in also yellow in that figure) and is symmetrical about the zero line. It’s intriguing that for much of the rotation of the test arm, experimental scores (in blue) for this very small angle remain similar to those for the Poggendorff without parallels, hugging the yellow model line. Only just anti-clockwise from vertical does the figure angle axis seem aroused, with the data from observation rising briefly to meet the red model line.

 

         The agreement between models and data across all angles is more ragged than I would like, but the data itself is less regular than in the other cases we will look at. Overall, both model and data follow what I have called the signature of these illusions, with the initial dip in scores in each quadrant progressively ironed out as the figure acute angle axis comes to dominate cardinal axes at larger angle sizes.

 

 

7. Comparing data from my own observations with a model from summed axial and assimilation effects

 

The data in Weintraub and Brown’s (1986) study was of course from the observations of a proper sample of observers. By contrast my own study of misalignment between the arms of pairs of rotating acute and obtuse angles, in 2003, was based only on my own observations. For each data point, I made six observations, over a couple of weeks, half of them with figures in one handedness, half in another. My experimental procedures are detailed in Appendix 2. We have seen the results summarised graphically to the lower right in figure 2.

 

 

 

Figure 14: Effects of cardinal and figure angle axes are as described for figure 12 (above), but as applied to the author’s 2003 study of opposed angles. An assimilation effect (curve at rear of stack) has then to be added for the largest angle, 115 degrees.

 

         In the diagram of combined axial effect top left in figure 14 I have as before omitted for clarity from the left and centre figures the small contribution at these test arm orientations from the horizontal cardinal axis. Also as before I have omitted the cardinal axes entirely from the right hand figure, because with the test transit near to 45 degrees vertical and horizontal cardinal axis contributions would nearly cancel one another.

 

However, for the largest angle in my study, 115 degrees, we have to bring in the proposed assimilation effect. With angles of over 90 degrees, segments of the inducing arms lie opposite to one another, as can be seen by looking at the angle pairs in the rotating array top right in figure 14. As mentioned above, the figures now begin to acquire an orientational emphasis aligned with the parallel segments, which is absent in the opposed acute angle figures, but becomes much stronger as obtuse angle size increases. Adding in an effect assimilating the alignment track to the figure orientation component, so that it “overshoots” the target and produces a contribution to misalignment always in the same positive direction as from the figure angle axes, can account for the higher scores we observe in these figures.

 

As with the figure angle effect, misalignment from the supposed assimilation effect grows with obtuse angle size. However, for any particular obtuse angle, whereas with the figure angle effect I found the best match with data by setting the maximum misalignment whenever the angles were pointing due vertical or horizontal, maximum scores from the effect attributed to assimilation matched data best when test arms are at 45 or 135 degrees.

 

         The agreement of models and data, shown in figure 15, is not too bad.

 

 

Figure 15: Data from the author’s 2003 study compared with models. Curves of similar colour in the stacks of model contributions at top right are summed for each angle size. For the 115 degree angles, an effect (shown at the rear of the stack top right, with maxima with test arms at 45 and 135 degrees) attributed to assimilation of the alignment track to the overall orientation of the figures has to be added in, as well as the effect of the figure angle axis.

 

The higher scores in the right hand quadrant in each graph required the vertical component of the cardinal axis contribution to be set substantial higher than the horizontal axis component. That may be an idiosyncrasy of my own vision. Overall both data and models seem to fit the trends observed in the data from experiments by Weintraub and co-researchers.

 

 

8. Comparing data from Weintraub et al.'s study with a model from summed cardinal and figure axes

 

The results from Weintraub et al. (1980) are graphically summarised, as we have seen, in figure 2, top left. The variant of the illusion they studied is closest of the figures studied here to the classic Poggendorff illusion. They used a figure with only one test arm and a target dot on the opposed parallel inducing line, because some observers, asked to judge alignment in classic Poggendorff figures, find the task impossible because they see the test arms as lying at different angles. Using one test arm and a target dot avoids the problem, and has been the figure of choice for a number of researchers. It gives results close to those for two-test-armed versions. For example Ninio and O’Regan compared data for one-armed and classic figures and reported good agreement (1999, their figure 9).

 

         With these single test arm figures, as with classic Poggendorff figures, figure acute as well as figure obtuse angle axes come into play, as diagrammed upper left in figure 16.

 

 

 

 

 

Figure 16: Effects of cardinal and figure axes as described for figures 12 and 14 (above), but for Weintraub and Brown (1986). A negative effect from the acute angle has now to be added in to the mix. As shown lower right, it has a maximum negative effect at larger acute angle sizes, where obtuse angles and their effects are smaller. The additional assimilation contributions also required once figures present parallel inducing lines are shown lower left.

 

Cardinal axis effects have been treated as described above for figures 12 and 14. Unfortunately even so the diagrams of figure angle effect at upper left now become cluttered if not positively demented. However by following the principle established, that the coloured arrows show the effect of each similarly coloured axis, the combined effects that will be summed in the model, and the variation between competition and collaboration, can be disentangled. As with the figure angle scores seen before, we find the best match between data and model if the figure angle contribution is maximum when the angles point due vertical or due horizontal. The maxima, as seen in the array of rotations to top right in figure 16, therefore once again march across each quadrant with increasing angle size from right to left.

 

But now note that in all the figures that present obtuse and acute angles together in the same figure, we need to allow for competition between the acute and obtuse angles. First the acute angle effect must always oppose that of the obtuse angles. Second, it rotates with the effect of the obtuse angles, so that it too is at a maximum when the angles point due horizontal or vertical. But it also increases with angle size, like the obtuse angle effect. Since the acute angles are of course greatest when the obtuse angles are smallest, that means that their opposing effect is strongest just when the effect from the obtuse angles is least. Recall that the effects of the figure axes that we have seen in the figures so far considered, presenting only one angle type at a time, always lift the curves further into positive territory. In figures presenting both obtuse and acute angles, the largest acutes when angles point vertically or horizontally can even reverse the usual direction of the classic Poggendorff effect, driving scores back down below the zero line. That can be seen in the graph for the 105 degree angle in figure 17, below.

        

The twin parallels of these figures also present strong orientational emphasis. As with the 115 degree figure in my own data (see figure 15), the higher overall scores that arise once such an emphasis is apparent can be accounted for if we add in an assimilation effect, rotating the alignment track towards the inducing line orientation so that it “overshoots” the target, and acting in consort with the obtuse angles. As before, this proposed assimilation effect matches the data best if it set at a maximum when test arms are at 45 or 135 degrees.

        

 

        

 

 

Figure 17: Data replotted from Weintraub et al. (1980, figure 1) in blue compared with models in red. Curves of similar colour in the stacks of contributions are summed: from the cardinal axis effect at upper left; from figure obtuse (positive effect) and figure acute angles (negative effect) at upper right; and from an assimilation effect (positive), below. The assimilation effect is maximal with test arms at 45 and 135 degrees.

 

         The match between models and data in figure 17 is perhaps the best of our three comparisons, and again both models and data show the proposed signature shift, from dominance by the cardinal axial contribution at smaller obtuse angles to dominant figure axis effect, especially from the assimilation component, at larger angles.

 

        

9. Planned eye movements and axial agencies in Poggendorff misalignments

 

         In moving to consider possible physiological processes responsible for these proposed axial effects, 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 eye movements 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 (Evans and Marsden 1966). Nonetheless they do seem to be affected by whereabouts in a figure we fixate and how we scan (Novak 1966; Prytulak 1973; Virsu 1971; Wenderoth et al. 1978). It has therefore been suggested that they might involve neural activation associated with the pre-attentive planning of saccades.

 

         For example, some years ago I noticed that when the Poggendorff without parallels, with arms at 22.5 degrees from vertical, is adorned just with blurry blobs as in figure 18, misalignment seems enhanced.

 

 

Figure 18: Misalignment in the Poggendorff without parallels is enhanced just by adding blurry blobs to mark the approximate vertical axis.

 

That seemed an interesting example of the way that different figures can arouse the cardinal axis effect at different levels, and I did some experiments to measure the effect of the blobs, with a small sample of observers. (The result showed an increase of average misalignment from about 0.5 degrees without the blobs to around 2 degrees once they were added). One exceptionally acute observer however noted that the effect depended where in the figure he fixated. That chimed with a similar observation made earlier by another observer in a different study, to which I had paid little attention 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 19, 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.

 

 

figure 19: Deliberate scanning between the blobs, switching fixation between blobs at about one second intervals, strikingly reduces misalignment for the author when the blobs are within the acute angles, as to the right

 

         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 a heroic 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." (Cameron and Steele 1905).

 

         The idea that some kind of effect associated with prominent saccade targets is at work also seems consistent to my eye with appearances in figure 20.

 

 

Figure 20: Even without deliberate scanning, misalignment appears to the author reduced when the axis between acutes is emphasised, at (b) though not to the extent of reversal of effect, as at (c) in the absence of parallels. At (e), with vertical emphasis added to the test figure, misalignment is enhanced. The effects seem consistent with a role for eye movement planning.

 

Here the strength of misalignment seems affected, in finely balanced versions of the Poggendorff figure, just by adjusting the salience of saccade targets. In these examples no deliberate scanning is involved. In the figure at 20(a), with the obtuse angles weighted, the usual Poggendorff shift appears. Weighting the acute angles at 20(b) seems to leave the test lines pretty much aligned. Yet this is not because at 20(b) the figure has in effect been reduced just to opposed acute angles. At this orientation these tend to give a small reversed misalignment, in the opposite direction to the standard Poggendorff effect, as at figure 20(c). Then at 20(d), to my eye the test lines are pretty much aligned, whereas when the figure is given a salient vertical emphasis at 20(e), once again a small reversed misalignment appears, (upper bar shifted left, lower bar right), of just about the width of the test bar.

 

 

10. Might conflict between pre-attentive eye-movement signal streams cause misalignment?

 

         But just how might planned eye movements 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 judging alignment across a gap, and on the other hand in helping to locate gravitational vertical in the visual field, in spite of varying head inclination, in concert with the vestibular system.

        

         We can imagine that eye-movement signalling may be involved in judging orientation across gaps in the visual field. 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?

 

         Imagine next that some saccade signalling may also be evoked, not as a preparation for real eye movements at all, but instead in pre-attentive mapping of the salient two-dimensional orientation features in a scene.

 

         In earlier versions of this site, I suggested a scheme in which the role of the cardinal axes in pre-attentive signalling might be to project orientation of head inclination in the visual field, whilst the role of the figure axes would be to calibrate the dominant three-dimensional axes of objects in the field in relation to visual and gravitational vertical. I have since found a demonstration that convinces me that the three-dimensional axes of objects in view are not involved in that way in these misalignment illusions. The idea also now seems to me implausible because of the difficulty of assessing whether oblique edges lie in the picture plane or recede into depth, and the complexity of defining what “up” means, in visual studies (Carriot et al., 2008)

 

         However a combination of projected-in cardinal axes, together with two dimensional orientation cues discovered in the figures, might still be a mechanism for calibrating scene orientation in relation to visual and gravitational vertical, at an early stage in visual processing. The relation of gravitational vertical to head inclination is of course established for the visual field as a whole by the vestibular system. But might we not benefit from an almost tactile sense of the orientation of dominant two-dimensional features and their axes in the visual field, relative to head inclination and gravitational vertical? Keeping track from moment to moment of the relation between gravitational vertical, head (and therefore field of view) inclination and the orientation of edges in view is a daunting real-time problem. My speculation is that the relationship might be, as it were, acted out in terms of signalled but unrealised saccades, sensed in relation to the projected-in cardinal axes. The two-dimensional scheme would then provide a basis from which the brain could proceed to inference about three dimensional configuration.

 

        Such a process, evoked by unrealised saccades, might account for some subjective effects reported in aesthetic discussions. It has often been pointed out that pictures 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 (Arnheim 1974). For example, Bridget Riley, writing about Paul Klee, writes of “plastic energy of form in the picture plane - which I call dynamism ...” (Riley 2002, 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 (Gregory 1998, p5). Might aesthetic field of force effects instead be explained by the signalling of virtual saccades?

 

         Such accounts of visual experience are anecdotal, but there is a precedent for descriptions of aesthetic experience that have later turned out to have a neurological foundation. A number of art theorists in the past described the way that we seem to project ourselves empathetically into visual scenes. Typical are Karl Groos's notion of "inner mimicry" of observed behaviour (Langfield 1920, 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" (Clark 1956). Those reports seem to have been born out by the recent discovery of "mirror neurones". These are neural channels aroused in the movement areas of the brains of observers, but without movement being carried out, when movements are merely observed in others (Motluk 2001). It is likely that there is some physiological basis for the aesthetic “field of force” reports as well.

 

Suppose then that analogous unrealised eye movement signals are involved in mapping both alignment and orientation in early processing. From the candidate components proposed above, let us first consider a conceivable mechanism involving the cardinal and figure angle axes, leaving aside for the moment the assimilation effect. Suppose that an alignment trajectory must cross a gap in a figure that is also intersected by interfering cardinal and figure axes. We imagine both alignment trajectory and axes, as proposed above, projected in by virtual, unrealised eye movements. Suppose that the alignment track requires a small extra component, as it were to “push through” the interfering cardinal and angle axes. The extra component might be experienced as an “overshoot” in target position, as usually observed in these illusions. We might then expect that, as observed, the effect of axes projected into the visual field would be greatest when they are at cardinal orientations, since it is then that they are most salient.

 

The cardinal and figure angle axes are, like the alignment track itself, only projected or implied in the visual field, so my proposal is that when alignment conflicts with orientation features that are implied rather than explicit, the effect is one of opponency. By contrast when we come to consider real edges, (such as those of inducing parallels in Poggendorff type illusions), we can get a good match with data if we assume that their effect is to assimilate alignment trajectories, rather than opposing them – the trajectories “line up” with the edges, rather than rotating away from them. As will be apparent from a moment’s study of figure 21, below, this assimilation effect on alignment judgments will then be just as from opposition by the implied figure angle axial effects - an overshoot in perceived target position. We might account for the observation that the assimilation effect seems greatest when test lines lie at forty five degrees if our powers of alignment, considered in isolation in the absence of interfering axes and figure elements, are at their weakest when most oblique, but increase as test arms approach vertical and horizontal.

 

         As can also be seen in figure 21, some such scheme would account for the most salient characteristic of the “signature” of the Poggendorff type illusions with which we began, the increase of misalignment with obtuse angle size. In figures with extended and parallel inducing lines, cardinal axis effect is subordinate, and we can omit it here. As obtuse angle size increases, so does both the opposition between the obtuse angle axis and the alignment track, whilst the track and inducing lines become increasingly aligned.

        

 

Figure 21: The small cardinal axes effects in play in these figures are omitted. As obtuse angle size increases, so does the angle of incidence between the alignment track from test arm to dot target and the symmetry axis of the obtuse angle. This fits an opponent process. At the same time the test track and inducing lines become increasingly aligned, consistent with assimilation process. Both processes produce a shift in apparent target position in the direction usually reported in Poggendorff type illusions

 

There is some additional evidence to suggest that the parallels alone, and even a single oblique line, can produce an assimilation of projected transit towards their own orientation, in the form of the Tolansky type illusions, seen in figure 22.

 

        

 

Figure 22: Tolansky type figures. At (a) and (c) the small central segment in the upper line is objectively vertically above the interval in the lower line (a) and the dot (c), but appears shifted to the right at (a) and to the left at (c). At (b) and (c) the intervals in the lines are objectively aligned horizontally, but can appear to rise slightly to the right. In each case assimilation of the alignment track to dominant inducing line orientation could be the cause.

 

         In the Tolansky illusion, as seen in figure 22a and 22c, two vertical targets on an oblique line are equally spaced either side of a vertical transit from a point on a lower, parallel line (22a), or an isolated dot (22c). The points on the upper lines appear shifted, as if the vertical transits had been assimilated towards the inclination of the inducing lines. Assimilation could clearly be involved, but the relationship, if any, between Tolansky and Poggendorff figures is not clear.

 

Assimilation of judgments to dominant orientation cues in the visual field is also consistent with a related effect, noted by chance during research into a different illusion (Bourdon’s illusion, researched by Walker and Shank 1987 pp 17, 20). Versions of the figure they chanced upon are shown at 22b and 22d. 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 21b 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 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 21c.

 

 

11 Problems

 

The scheme proposed is for sure not the only way that the so-called signature pattern of variation in misalignment could be modelled by summing curves. For example, even sticking with the same component curves, by trial and error I sometimes obtained a better match with sets of data in one or other study considered in isolation by keeping the cardinal axis contribution constant, and varying the figure angle axis contribution in other ways. In other matching trials, I got locally consistent results with contributions with figure axes at a constant level, or maximal at different points. Across all the sets of data the scheme suggested seemed to give the best fit, but suggestion is all it can be.

 

There is also a problem with the proposed assimilation componant in these illusions. Assimiltaion would not be the only candidate process that might affect alignment judgments and be at a maximum when test arms are at 45 and 135 degrees, in the presence of extended inducing parallels. That component of the misalignment could in truth arise from any process at all, depending only on the presence of the parallels, (or, as we noted in passing, a single inducing line extending either side of the junction with the test arm).

 

For example the inducing arms when paired or extended might evoke in the oblique alignment trajectory the depth inplications that a number of researchers believe is the main agency in these misalignments. As mentioned above, I feel that there are powerful objections to all the depth processing proposals, plausible though they are in principal. Nor does it seem to me that overall they can account for the data as well as a scheme related to balance in the visual field. For me it’s the best bet, but a bet is all it can be in the absence of more data.

 

But then a serious problem for my eye-movement signalling scheme is just how axes and orientations would be signalled neurally. 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 (Leigh 1999). 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 in anticipation whole scanning sequences, or alternatively that there exists some facility in early processing that links paired targets as markers of orientation. We can certainly attend to points in the visual field that are not fixated (Leigh and Zee 1999), so the scheme proposed may not be impossible, but it remains speculative.

 

         A final objection to the proposal is that evolution would surely have eliminated these miscalibrations in the saccade system. That is more easily answered, but with two answers as opposed to one another as they could conceivably be. One is simply that the misalignment we observe in these artificial figures is of negligible survival importance, and is therefore tolerated. The other answer is that on the contrary, the misalignments more often than not suggest three-dimensional configurations that in everyday vision are objectively correct, to the extent of conferring survival advantage (Howe et al. 2005). That would explain why a whole series of small miscalibrations of the visual system might have arisen, all contributing to a similar misalignment effect. Each could have arisen by chance through genetic errors, but then been selected for in evolution.

 

         I have considered a number of incidental Poggendorff issues, as well as some new suggestive versions of the illusion, in the Poggendorff category of my visual illusion blog.      

 

 

 

 

APPENDIX 1 - NUMERICAL DATA

 

 

 

 

APPENDIX 2

 

Experimental methods for the author’s 2003 study of opposed angles

 

         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.

 

 

 

SOURCES

(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,

completely revised August and October 2010,

minor clarifications April 2011