The Oppel–Kundt Illusion and Its Relation to Horizontal-Vertical and Oblique Effects

Abstract

The Oppel–Kundt illusion consists in the overestimation of the length of filled versus empty extents. Two experiments explored its relation to the horizontal-vertical illusion, which consists in the overestimation of the length of vertical versus horizontal extents, and to the oblique effect, which consists in poorer discriminative sensitivity for obliquely as opposed to horizontally or vertically oriented stimuli. For Experiment 1, Kundt’s (1863) original stimulus was rotated in steps of 45° full circle around 360°. For Experiment 2, one part of the stimulus remained at a horizontal or vertical orientation, whereas the other part was tilted 45° or 90°. The Oppel–Kundt illusion was at its maximum at a horizontal orientation of the stimulus. The illusion was strongly attenuated with L-type figures when the vertical part was empty, but not enhanced when this part was filled, suggesting that the horizontal-vertical illusion only acts on nontextured extents. There was no oblique effect.

Keywords

visual illusions spatial anisotropy oblique effects

The visual illusion that is named after Oppel (1860–1861) and Kundt (1863) consists in the overestimation of extents that are filled with items relative to otherwise equal, empty extents. The illusion can also be observed in the haptic-tactile sense modality (Collier & Lawson, 2016) and in 3D space (Deregowski & McGeorge, 2006) but will here be considered for vision and the Euclidean plane only. For these conditions, the prototype illustration of the illusion, a forerunner of which was introduced by Wundt (1898), is a horizontal, continuous line, the endpoints of which as well as one half of which are marked with short, orthogonal, symmetrically bisected, equally spaced ticks (Mikellidou & Thompson, 2014).¹ The illusion also works when the horizontal line is absent (Robinson, 1972) and the ticks replaced by dots (Coren & Girgus, 1978)—which is in fact the original stimulus used by Kundt (1863). In what follows, I will provide two sets of observations intended to explore the relation of the Oppel–Kundt illusion to the horizontal-vertical illusion, first described by Fick (1851), and to the so-called oblique effect (Appelle, 1972).

Fick (1851) had studied an apparent, visual horizontal-vertical anisotropy with rectangles and squares, but his observations did not unequivocally replicate (Houck et al., 1972; Sleight & Austin, 1952). Sanford (1898) is usually credited for having introduced the L as a more appropriate stimulus (Finger & Spelt, 1947). Amounts of illusion—an overestimation of vertical extents relative to horizontal ones—seen with this figure have been found to vary considerably, depending, among other things, on conditions of lighting (Avery & Day, 1969) and the connectivity of the lines (Cai et al., 2017). Findings were somewhat clearer with the ⊥ stimulus, popularized by Titchener (1901), although with this figure, bisection is introduced as an additional illusion-inducing factor (Künnapas, 1955; Landwehr, 2014; Mamassian & de Montalembert, 2010; Mikellidou & Thompson, 2013). However, by presenting individual horizontal or vertical lines in complete darkness and having observers judge their length by the method of absolute magnitude estimation, Verrillo and Irvin (1979) found that the putative illusion as such does not exist at all—except in the context of at least one second line.

The “oblique effect” (Appelle, 1972; p. 266) refers to observers’ superior discriminative performance with stimuli that display an overall horizontal or vertical orientation when compared with oblique orientations.² Applying the concept to illusions requires to understand that even if observers fall prey to errors, they may nonetheless be able to discriminate stimulus parameters quite well (Morgan et al., 1990). Both aspects of observers’ performance can be read off a psychometric function: The error, or response bias, is indicated by the displacement of the point of subjective equality (PSE) relative to the coordinate system’s origin, and the observer’s sensitivity is indicated by the slope of the function, or, alternatively, by a conventionalized threshold measure (just noticeable difference [JND], typically half the x axis distance between the 25% and 75% points of the function; Macmillan & Creelman, 2005).

An oblique effect for the Oppel–Kundt illusion would mean that observers show poorer performance in the discrimination of the lengths of the two parts of the figure when the whole stimulus is obliquely oriented. On the basis of the existing literature, no reasonable prediction for the bias can be made; in psychophysical experiments, sensitivity and bias are typically unrelated (Macmillan & Creelman, 2005). However, with reference to the differential sensitivity of the upper and lower halves of the retina (Lehmann & Skrandies, 1979; Levine & McAnany, 2005), a more specific prediction for vertically and obliquely oriented stimuli can be derived: When the patterned part of the figure is at the bottom, projecting to the more sensitive upper half of the retina, performance should improve.

In order to relate the Oppel–Kundt illusion to the horizontal-vertical illusion, it is necessary to bend the figure in its visual midpoint so that one part is horizontal and the other one vertical. When the patterned part of the figure—the length of which is usually overestimated—is vertically oriented, then, due to the additional effect of the horizontal-vertical illusion, the response bias should increase. Conversely, when the patterned part is horizontal, the Oppel–Kundt and the horizontal-vertical effects work against each other, so that the bias should decrease.

General Method

Participants

Twenty psychology undergraduates took part in the experiments, 10 per experiment (independent samples). The number of participants had been decided upon a priori in order to achieve statistical power of 1 – β ≥ .95 for a repeated-measures analysis of variance (rmANOVA) with α = β ≤ .05, and f ≥ 1 (Faul et al., 2007). Written, informed consent was obtained from all participants, and they were treated in accordance with the Declaration of Helsinki (World Medical Association,1964/2013). All participants had normal or corrected-to-normal vision, and all served in partial fulfillment of a class requirement.

Apparatus

The essential part of the apparatus was a touch-sensitive computer screen (size: 59.6 × 33.5 cm; resolution: 2,560 × 1,440 pixels; response time: 3 ms) that was used for both stimulus presentation and response registration. The screen was oriented frontoparallel at a distance of 44 cm from the observer. Stimuli, drawn as small black dots (diameter: 1.25 mm; visual angle: 1 arcmin), were presented within a circular, light gray window (diameter: 28.5 cm; plane visual angle: 35.9°; luminance: 228 cd m^–2; CIE-coordinates: x = 0.312; y = 0.332; Weber contrast between stimulus and background: C_W = –0.998); the rest of the screen was dark (0.355 cd m^–2), and there was only faint, indirect illumination of the room.

Stimuli and Responses

The specific stimuli used will be described for each experiment separately, but in general, five different lengths were used for the two parts of the Oppel–Kundt figure—6, 6.5, 7, 7.5, and 8 cm (7.8°, 8.4°, 9.1°, 9.7°, and 10.4° visual angle)—and factorially crossed. The cases in which the parts were equally long were deselected, but participants were not informed about this. Stimuli were presented for 2 s. The observer’s response started the next trial after a delay of 200 ms.

The response format was two-alternative forced choice. Observers had to compare the empty and the filled parts of the stimulus and say whether the empty part had been longer or shorter than the filled part. Responses were delivered by gently touching response buttons that appeared on the computer screen after the stimulus had been turned off. Effective response registration was signaled to participants by the buttons shining light blue.

Data Analysis

Data were analyzed by fitting psychometric functions with cumulative Gaussians. Parameters were computed via the binary logistic regression routine of SPSS™. PSEs were found at the 50% points of the necessarily symmetric functions for the longer and shorter judgments which had been plotted against an abscissa defined by the difference in length between the empty and the filled parts of the Oppel–Kundt figure (points were found numerically by transferring the data to customized software in Mathematica™). Difference thresholds (JNDs) were read off the psychometric functions from half the difference between the 25% and 75% points of the functions. Both sets of data were further analyzed by means of rmANOVAs.

Experiment 1

Although the optimum number of dividers has never been firmly established, the Oppel–Kundt illusion is known to decrease when this number is smaller than 7 or larger than 14 (Mikellidou & Thompson, 2014; Robinson, 1972).³ Following Spiegel (1937), we used 11 dots for the patterned part of the figure and one additional dot to mark the other end of the empty part.⁴ The whole figure was rotated full circle around 360° in steps of 45° (see Figure 1 for sample stimuli). Together with the length variation, this made for 160 unique trials which were repeated once in a new random order to gather a sufficient number of trials for the estimation of the individually fitted psychometric functions.

Figure 1.

Sample stimuli as used in Experiment 1.

For the analysis of possible effects of the rotation of the Oppel–Kundt figure on observers’ response bias, the stimuli were grouped into three categories: horizontal, vertical, and oblique. The rmANOVA revealed an overall effect of rotation category, F(1.612, 14.510) = 5.565, p < .021, η_p² = .382. Repeated contrasts showed that the illusion was significantly greater for the horizontally oriented figures when compared with the vertically oriented ones, F(1, 9) = 7.885, p < .020, η_p² = .467. The other comparisons were not significant (see Figure 2). Means and standard deviations were M_h = 0.88 cm, SD_h = 0.50 cm, M_v = 0.59 cm, SD_v = 0.41 cm, M_o = 0.73 cm, SD_o = 0.44 cm. Referenced to the average lengths of the parts of the Oppel–Kundt figure, the means correspond to percent amounts of illusion (overestimation of the length of the filled part) of 12.6%, 8.4%, and 10.4%, respectively.

Figure 2.

Results of Experiment 1, concerning response bias. The benchmark line corresponds to the mean amount of illusion across all conditions.

To test for the existence of an oblique effect, the JNDs were also subjected to an rmANOVA. There was no effect, F(1.970, 17.730) = 0.069, p < .932, η_p² = .008. To further test whether the position of the filled part of the Oppel–Kundt figure in the lower or upper retinal hemifield would yield an effect, the stimuli were grouped into three alternative categories: horizontal, filled part below the empty part, and filled part above the empty part. The rmANOVA did not show any effect, F(1.453, 11.621) = 0.214, p < .741, η_p² = .026. Although there were considerable differences between observers with regard to sensitivity, the mean JND was always below the minimum difference of the two parts of the Oppel–Kundt figure as used in the experiment (0.5 cm; visual angle: 40 arcmin); the grand mean was 0.33 cm (visual angle: 25 arcmin), SD = 0.10 cm.

Experiment 2

The stimuli used in Experiment 2 were similar to the ones used in Experiment 1. The horizontal and vertical Oppel–Kundt figures were kept for a control. For the other stimuli, one half of the figure was kept at a horizontal or vertical orientation, whereas the other half was tilted 45° or 90° so as to produce triangular or L-type figures (see Figure 3 for examples).⁵ The experiment comprised four parts, defined by the orientations of the empty and the filled extents. The combination of the length variation, which was analogous to Experiment 1, and of positions and angles made for 200 unique trials per part. The 800 trials were run in two sessions with the sequence of the parts counterbalanced.

Figure 3.

Sample stimuli as used in Experiment 2. The outermost sample stimuli shown in Figure 1 also appeared in Experiment 2.

Stimulus types were ordered into five categories: horizontal, vertical, triangular, and two types of L figures—one with the empty stretch vertical and one with the filled stretch vertical. From Experiment 1, the horizontal figures were expected to yield a stronger illusion than the vertical figures. From the reasoning given in the General Introduction section, the L figures with a filled vertical were expected to yield a stronger illusion than those with an empty vertical, and possibly even a stronger illusion than any other type of figure. Figure 4 displays the results.

Figure 4.

Results of Experiment 2, concerning response bias. The benchmark line corresponds to the mean amount of illusion across all conditions.

The rmANOVA confirmed an overall effect of stimulus type, F(1.760, 15.840) = 8.211, p < .005, η_p² = .477. Simple and repeated contrasts showed a significant difference between the horizontal and the triangular, oblique-arms figures, F(1, 9) = 11.577, p < .008, η_p² = .563, a significant difference between the vertical and the L figures with empty verticals, F(1, 9) = 9.547, p < .013, η_p² = .515, and a nearly significant difference between the horizontal and the vertical figures, F(1, 9) = 4.896, p < .054, η_p² = .352.⁶ Difference thresholds did not differ between stimulus types, F(2.415, 21.734) = 0.452, p < .678, η_p² = .048, but sensitivity and bias were negatively correlated, r(50) = –.385, p < .006. Of the seven participants with high sensitivity (i.e., low thresholds), four showed a large response bias, and three a low bias. The rest of the sample showed low sensitivity (i.e., high thresholds) and a low bias. Overall, the mean difference threshold was 0.41 cm (visual angle: 32 arcmin), SD = 0.24 cm.

Discussion

Descriptively at least, the Oppel–Kundt illusion was at its maximum at a horizontal orientation of the figure, and it was markedly attenuated at a vertical orientation (Experiments 1 and 2); the oblique orientations fell in between (Experiment 1). My hypotheses concerning observers’ sensitivity were clearly falsified: There was no oblique effect, and the position of the textured or empty part of the Oppel–Kundt figure in the upper or lower retinal hemisphere did not matter (Experiment 1). The reason why we did not find an effect of the latter kind may have to do with the long stimulus presentation time, which allowed participants to scan the stimulus up and down. Tachistoscopic presentation conditions may be required to answer this question.

The fact that the Oppel–Kundt illusion was not enhanced in an L figure with a filled vertical (Experiment 2) suggests that the Oppel–Kundt and the horizontal-vertical illusions do not combine additively. Rather, the attenuation of the illusion in an L figure with an empty vertical suggests that the horizontal-vertical effect may be confined to nontextured extents. Hence, in my Experiment 2, the horizontal-vertical illusion could only subtract from the Oppel–Kundt illusion but not add to it.

A possible cause of the Oppel–Kundt illusion may be the figure’s asymmetry. Barlow and Reeves (1979) have suggested that symmetry acts as a parsing mechanism in neural information processing. Obviously, such kind of mechanism is not applicable in the case of the Oppel–Kundt figure, making the task to compare its two parts more difficult than in the case of similar parts of a figure. However, symmetry may also support illusions, as, for example, with the ⊥ (Cormack & Cormack, 1974; Landwehr, 2017). Thence, another factor to be considered for an explanation of the Oppel–Kundt illusion is the figure’s internal redundancy. Scanning the filled part will yield nearly identical neural responses across minimum eye movements. Worse, for the empty part, there are no specific responses at all. What observers would have to do is to focus exclusively on the outer dots that demarcate the ends of the two different parts of the Oppel–Kundt figure. Presenting these dots in a different color and/or presenting the intermediate dots at reduced contrast should support such focusing of attention and is here predicted to attenuate or even eliminate the illusion. Another possibility, utilizing the ticks version of the Oppel–Kundt figure, is to use ticks of different lengths as fillers (Robinson, 1972). In fact, Wackermann and Kastner (2009) found an inversion of the illusion in 2 out of 6 observers when the filler ticks were 3 times as long as the delimiting ones.

Footnotes

Acknowledgements

I thank Agnes Münch for programming experiments and Corinna Falter and Lea Schanz for gathering and transcribing data.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work presented in this article was supported by a grant of the Deutsche Forschungsgemeinschaft (LA 487/6–4: “Visuelle und haptische Täuschungen in komplexen Reizanordnungen”).

Open Practices Statement

The data and materials of the experiments reported here will be made available to qualified researchers upon request. None of the experiments was preregistered.

ORCID iD

Klaus Landwehr

Notes

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