Shape can be considered an important factor in visual perception (Livingstone & Hubel, 1987, 1988), perception and representation of object shape through vision are basic to thought, action, and learning (Baker & Kellman, 2018). This suggests that research on visual illusions cannot ignore the role of shape. The Delboeuf illusion is a visual phenomenon was first described by the Belgian philosopher Franz Joseph Delboeuf. In this illusion, when two circles (test figures) of equal radius are presented next to each other and surrounded by concentric circles (inducers) of different radii, the test figure surrounded by a slightly larger inducer is overestimated or underestimated (Murray, 2012; O’Halloran & Weintraub, 1977; Weintraub & Cooper, 1972; Weintraub & Schneck, 1986). The left and right test figures are also called the target and probe, respectively (see Figure 1A). If the target’s inducer is much larger than the target, the target’s size will be underestimated (i.e., size contrast); in contrast, if the target’s inducer is only slightly larger than the target, the target’s size will be overestimated (Mruczek, Blair, Strother, & Caplovitz, 2017). In the present study, we explored only the overestimate of the illusion. The Delboeuf illusion has been extensively studied among humans (Shoshina & Shelepin, 2014), and studies have shown that viewers’ size perception is strongly influenced by the luminance (Daneyko, Zavagno, & Stucchi, 2014) and color (McClain et al., 2014) of the background and stimulus. The Delboeuf illusion has been shown to influence food selection behavior in animals such as chimpanzees (Pan troglodytes), monkeys, domestic dogs, bantams, and fish (guppies [Poecilia reticulate]); food intake in humans (Byosiere et al., 2017; Lucon-Xiccato, Santacà, Petrazzini, Agrillo, & Dadda, 2019; Nakamura, Watanabe, & Fujita, 2014; Parrish & Beran, 2014; Parrish, Brosnan, & Beran, 2015); and is also found in insects (Howard, Avarguès-Weber, Garcia, Stuart-Fox, & Dyer, 2017). Both animals and humans are susceptible to geometrical illusions. Animal research may shed light onto the evolutionary and environmental influences of perception (Byosiere et al., 2017); however, human research on attribute perceptions of illusion, such as shape is essential to clarify the mechanism and factors that have not yet been explored.
Previous research has largely ignored the role of shape in the Delboeuf illusion. The shape of the test figures or inducers might have a strong effect on the illusion (Roberts, Harris, & Yates, 2005; Rose & Bressan, 2002; Surkys, Bertulis, & Bulatov, 2006). Among the few researchers who have examined this topic, Weintraub & Schneck (1986) created new models by changing the shape of the inducers from a circle to a discontinuous circle, a discontinuous line (four-fragment concentric cases), a square, or several forms around the circle and then compared them with the Delboeuf illusion models of different sizes. They found that the magnitude of the illusion differed while the inducer is a circle or square. In Surkys et al.’s (2006) study, the test circle and the inducers had the same shape, changed simultaneously; the results showed that the strength of the illusion differed between squares and circles, indicating that shape might influence the illusion’s magnitude. Weintraub and Schneck (1986) and Surkys et al. (2006) suggested that their findings could be explained by the theory of contour attraction, which proposes that the proximal contours of the test figure and the inducer have a perceptual effect of interattraction, which would lead to overestimating the size of the test figure, whereas the interattraction diminishes as the distance between the test and inducer increases, thus, distal contours perceptually repel after a certain distance. That is, the magnitude of the Delboeuf illusion is affected by the distance between the test and inducer; the greater the distance the greater the magnitude, but the less attraction within a certain distance (Kawahara, Nabeta, & Hamada, 2007; Nakamura et al., 2014; Surkys et al., 2006; Weintraub & Cooper, 1972; Weintraub & Schneck, 1986). This theory fits well with the characteristics of lateral inhibition of neurons in early visual pathways and the known properties of activation as measured with fMRI (Schwarzkopf, Song, & Rees, 2011).
Rose and Bressan (2002) researched the Ebbinghaus illusion (see Fig 1B), which is closely related to the Delboeuf illusion (Girgus, Coren, & Agdern, 1972; Mruczek et al., 2017; Picon, Dramkin, & Odic, 2019; Roberts et al., 2005), by same-shape illusions (i.e., with an identically shaped test and inducer) and different-shape illusions (i.e., the test figure was a circle and the inducers were circles, hexagons, triangles, or angular shapes). The results showed that inducer shape had a significant effect; same-shape illusions were significantly larger than different-shape illusions with circles or triangles as the test stimuli. The effect of the distance between the test figure and the inducer in the Ebbinghaus illusion showed many commonalities with the Delboeuf illusion, in other words, the distance between the test figure and the inducer was affected in both the Ebbinghaus and Delboeuf illusions. Roberts et al. (2005) attempted to simplify the complexity of influence factors in the Ebbinghaus illusion and found the completeness of the inducer ring was a primary factor; furthermore, smaller inducers constructed more complete annuli and consistently increased the magnitude of the illusion.
The studies by Weintraub and Schneck (1986) and Surkys et al. (2006) showed that the Delboeuf illusion occurs even when the test and inducer are not circles. Rose and Bressan (2002) showed that the shape of the inducer played a vital role, and Roberts et al. (2005) suggested that the completeness of the inducer had a major effect. Based on the contour attraction theory, completeness in the Delboeuf illusion refers to the inducer’s subjective similarity (Li, Liang, Lee, & Barense, 2019) to a circle: the more subjectively similar it is to a circle, the more complete. In other words, a regular octagon is more complete than a regular hexagon, a regular hexagon is more complete than a square, and a circle is the most complete, different shapes lead to different completeness. Actually, the completeness is due to the shape of the inducer in Delboeuf illusion.
However, Roberts et al. (2005) proposed that the similarity of test figures and inducers is not a major factor in explaining the Ebbinghaus illusion, but the effect of parallel attraction. Pollack (1964) found that attraction between non-intersecting contours was maximal when they were parallel and gradually decreased as the angle of the projected intersection of boundaries increased (Rose & Bressan, 2002). As the regular polygons become subjectively similar to a circle, the sides of the test and the inducer would or close to be parallel in the Delboeuf illusion configuration, we can even think of the concentric circles of the Delboeuf illusion as special parallel lines. Therefore, the parallel attraction would be an effect in the Delboeuf illusion with polygons. But how will the contour attraction or the parallel attraction impact the magnitude when configurations of Delboeuf illusion with circles or regular polygons?
Surkys et al. (2006) showed that the magnitude of the illusion differed by the shape of the test figure (circle or square) as well as when the inducers changed simultaneously into a circle or square. However, they did not explore whether other shapes influence illusion magnitude or whether the illusion is maintained when the inducers change shape and the test figure remains a circle, and have not taken into account the parallel attraction in the square. Weintraub and Schneck (1986) adopted configurations that combined four squares or other shapes around test circles; these configurations were closer to the Ebbinghaus illusion than the Delboeuf illusion (Foster & Franz, 2014; Hamburger, Hansen, & Gegenfurtner, 2007). Because of the use of fragmented and complex combinations, Weintraub and Schneck’s (1986) study should have made some improvements.
Furthermore, in Surkys et al.’s (2006) study, the minute of the arc of the inducers changed at discrete intervals, which caused the data to become discontinuous; this was similar to Weintraub and Schneck’s study (1986) but with different step sizes. In most previous studies, although different procedures were used, their methods had the same problem. In particular, while they all changed the size of the inducer in steps, the step size differed across studies. Additionally, in Weintraub and Schneck’s (1986) study, participants had to use multiple fixed-size graphics for comparison with the reference target stimulus and select the graphic they perceived to be the closest in size to the reference target stimulus. This could have resulted in inaccuracies when judging the magnitude of the illusion. However, neither study has examined how the magnitude of the Delboeuf illusion changes when the inducers are gradually modified to resemble a circle, nor research has considered the relation of the contour attraction and the parallel attraction in the Delboeuf illusion, though these are essential questions to explore the mechanisms of this illusion.
To fill the above gaps in the existing research, we adopted a quasi-Delboeuf configuration, in which regular polygons replace the test circles or inducers of the classical Delboeuf illusion configuration. The regular polygons were graphics incircled within the test or inducing circles (more details see Appendix). The inner polygon was also called the test figure and the outer one the inducer. The radius of the polygon was measured as its circumcircle, to ensure the distance between the test polygon and the inducer polygon be closed to the distance between circles.
Based on previous findings, we selected the following shapes as independent variables: circle, regular octagon, regular hexagon, square, and equilateral triangle. These shapes are perceived as distinct figures, however, they are equivalent in the present quasi-Delboeuf illusion configurations from the viewpoint of topology (Chen, 1982) and suitable to the subjective similarity of the configurations constructed by themselves (Li, Liang, Lee, & Barense, 2019). This series of shapes create the condition which can examine the contour attraction and the parallel attraction by the shape in the Delboeuf illusion.
In our study, participants compared the target stimulus and adjusted the size of the probe stimulus arbitrarily until they perceived them as equal. The difference value between the radii of the probe stimulus and target stimulus determined the magnitude of the illusion. We created two shape-changing situations in experiments 1 and 2. In Experiment 1, the shape of the test figure and the inducers changed simultaneously, and parallel straight lines were formed from regular polygons in illusion configurations, the results would demonstrate the effect of shape and the parallel lines on the quasi-Delboeuf illusion. In Experiment 2, the inducer shape changed from circle to equilateral triangle while the test figure remained a circle and there were no parallel straight lines in illusion configurations. Both Experiment 1 and 2 would demonstrate the effect of shape. Comparing Experiments 1 and 2 would demonstrate the effect of parallel lines in the Delboeuf illusion and the effect of the inducer’s shape. In Experiments 1 and 2, the test figures and inducers of the quasi-Delboeuf illusion were polygons, and their radii were equal to those of the circles resulted in that their areas and the distance between the test figure and the inducer decreased from the circle to the equilateral triangle. The area is a vital impact in size estimation and germane to distance directly, it is impossible to maintain the distance between the test figure and the inducer equal in varied shapes, yet maintain the areas of shapes equal is feasible. In Experiment 3, the inducer shapes changed while their areas remained equal caused that the distances between the test and the inducer increased as the shape progressed from an octagon to an equilateral triangle and were all larger than those in Experiment 2, comparing Experiments 2 and 3 would demonstrate which the inducer shape or the distance between the test figures and inducers affect more on the illusion, the distance is a vital factor of the contour attraction. Therefore, the first hypothesis: the illusion magnitudes significantly differed by shape, all three experiments would demonstrate this hypothesis; the second hypothesis: the illusion magnitudes of polygons in Experiment 1 are significantly larger than the identical shape in Experiment 2, the shapes of polygons more subjective similar to a circle the less difference quantity; the third hypothesis: the illusion magnitudes of polygons in Experiment 3 are significantly larger than these in Experiment 2, the shapes of polygons more subjective similar to a circle the less difference quantity.
In present study, we explored whether the parallel attraction or the contour attraction holds for inducers of varying shapes using polygons and how the shape of the test figures and inducers of a Delboeuf illusion configuration influence the illusion magnitude. Accordingly, we would explore the relationship between the contour attraction and the parallel attraction in the Delboeuf illusion, and the effect of shape on the magnitude of the Delboeuf illusion.