Perception, 2009, volume 38, pages 1428 ^ 1438 doi:10.1068/p6287 The relevance of symmetry in line length perception Pom Charras, Juan Lupia n ez Departamento de Psicolog a Experimental y de Fisiolog a del Comportamiento, Campus de Cartuja S/N, CP 18071, Granada, Spain; e-mail: pomcharras@ugr.es Received 30 September 2008, in revised form 14 May 2009; published online 6 October 2009 Abstract. The length of a whole line is overestimated in comparison to the sum of its parts (Ku«nnapas, 1955 Journal of Experimental Psychology 49 134^140). This has been considered to be true for many years, although recent studies have demonstrated that it is not always so. The perception of the length of a whole line is highly dependent on the configuration of its two parts. More precisely, whereas a whole line is perceived as longer than the sum of symmetrically bisected line parts, this overestimation decreases when compared to the sum of the lengths of asymmetrically bisected line parts. Furthermore, the extent of overestimation depends on the degree of asymmetry, so that when the two parts are greatly asymmetric in length, the whole is no longer overestimated (Wolfe et al, 2005 Perception & Psychophysics 67 967 ^979). Here, two experiments are reported in which a vertical/horizontal line length comparison task was used to investigate how line bisection affects length estimation. The results give rise to a new general principle characterising the mechanisms of visual perception: the sum of the lengths of two asymmetrically bisected parts is perceived as greater than that of two symmetrically bisected parts. Also bisection plays a critical role in length perception by preventing vertical bias. 1 Introduction In visual perception, it is generally assumed that a whole is more than the sum of its parts (Koffka 1950). Bisection of the whole into parts typically generates a bias in size estimation. For instance, the gap between two points looks smaller when a third dot is displayed at the same distance from the two boundaries (Oppel 1855 ^ see figure 1). However, if the number of `dots-in-between' increases, the gap size becomes overestimated, as observed in the Oppel ^ Kundt illusion (Kundt 1863). In other words, the gap between two points can either be overestimated or underestimated depending on the number of elements displayed between them. Indeed, bisection biases collinearity, angle, and length judgments (Greene 1998). In the case of line length estimation, the bisection effect is explicitly illustrated in the horizontal ^ vertical illusion (Coren and Girgus 1978). When a vertical line cuts a horizontal one in its middle, forming an inverted T (see figure 1), the length of the vertical dividing line is largely overestimated in comparison to that of a horizontal line of the same physical length (Fick 1851; Ku«nnapas 1955). Some studies have shown that, rather than the bisection per se, it is the symmetry of the bisection that is directly involved in length processing (Ku«nnapas 1955; Wolfe et al 2005). In the inverted T, for example, overestimation of the vertical line length arises from the symmetric bisection of the horizontal line. And, interestingly, if the horizontal line bisection is made asymmetric (a) (b) (c) Figure 1. (a) A version of the Oppel ^ Kundt illusion. In this illusory figure, observers normally report that AB appears to be longer than BC, but shorter than CD. (b) and (c) Inverted T and L configurations of the horizontal ^ vertical illusion. In both figures, the length of the vertical line is overestimated, but note that the bias is much greater in the inverted T figure.
Bisection, symmetry, and length perception 1429 by displacing the vertical line to the left or to the right, the overestimation of the vertical line length decreases as a function of the degree of asymmetry (see Charras and Lupia n ez, in press; Ku«nnapas 1955; Wolfe et al 2005). The present study, which was motivated by this finding, focuses on the role of bisection and of the symmetry of bisection in line-length perception. The controversy regarding the horizontal ^ vertical illusion is also due to the fact that bisection is not wholly responsible for the vertical overestimation. Indeed, the horizontal ^ vertical illusion is not only a consequence of the comparison between dividing and divided lines; it is also due to the manifestation of the vertical bias (Avery and Day 1969; Finger and Spelt 1947; Ku«nnapas 1955; Miller and Al-Attar 2000; Raudsepp 2001; Raudsepp and Djupsjo«backa 2005; Yang et al 1999). This is easily shown by using L-shaped configurations in which none of the lines is bisected, and yet vertical lines are still slightly overestimated. Consequently, it is assumed that bisection and the vertical bias produce additive effects that eventually lead to the massive vertical overestimation observed in the inverted T figure (Avery and Day 1969; Prinzmetal and Gettleman 1993; Renier et al 2006; Wolfe et al 2005). The current study mainly focuses on the interactions between bisection, symmetry of bisection, and the vertical bias. Although the vertical bias seems to be systematically involved in the comparison of vertical and horizontal lines, a recent finding has suggested that the vertical bias is disrupted when both the horizontal and vertical lines are bisected. To illustrate this, observers who were presented with a cross shape (see configuration SS in figure 2) during a line-length comparison task, showed unbiased length perception (Charras and Lupia n ez, in press). Vertical lines were not overestimated in comparison to horizontal ones, thus suggesting that bisection can prevent the manifestation of the vertical bias. Experiment 1 SnB SA2 SA1 SS Vertical displacement axis Horizontal displacement axis Experiment 2 nbnb AA 2 AA 1 SS Horizontal line at fixation Vertical line at fixation Figure 2. Examples of two-line configurations within each group of bisection category in experiments 1 and 2. Note that in the vertical displacement axis of experiment 1, the horizontal line was always symmetrically bisected; whereas in the horizontal displacement axis, it is the vertical line that was symmetrically bisected. In experiment 2, the 1st series represents the drawings in which the horizontal line was displayed at fixation, whereas in the 2nd series the vertical line remained centrally presented in every configuration. As we controlled for hemispheric differences between left and right and for up/down vertical position, every configuration (but SS) refers to two figures. For instance, in the SnB configuration of the vertical displacement axis, the vertical line could be displayed below the horizontal as represented here, or above it. And in the horizontal displacement axis, the vertical line was displayed either at the left horizontal end as represented here, or at the right end.
1430 P Charras, J Lupia n ez The following two experiments aim at systematically testing the vertical and bisection biases, providing new empirical findings that support our assumption. We are revisiting the interaction between the bisection and vertical biases, using a large set of two-line configurations (see figure 2), in order to achieve two goals: first, to explore how the a/symmetry of the bisection affects length estimation (experiment 1) and, second, to prove that bisection disrupts the vertical bias (experiment 2). 2 Experiment 1 Inverted T figures in which the vertical line is displaced along the horizontal one have been used to demonstrate that the vertical overestimation is proportional to the degree of asymmetry. Consequently, when the horizontal line is symmetrically divided, the vertical overestimation reaches its maximum and decreases with the extent of asymmetry (Ku«nnapas 1955; Wolfe et al 2005). More recently, configurations in which the two lines are bisected have been used to suggest that the length of asymmetrically bisected lines is overestimated compared to that of symmetrically bisected lines, and this happens independently of the horizontal or vertical orientation of the lines (Charras and Lupiän ez, in press). We aim here to further investigate the overestimation of the length of asymmetrically bisected lines as compared to that of symmetrically bisected lines using a large variety of two-line configurations. Line orientation (vertical versus horizontal) and the type of bisection were manipulated orthogonally (see figure 2). 2.1 Method 2.1.1 Participants. Thirteen undergraduate students (two male) from the University of Granada participated in the experiment in exchange for course credits. Their ages ranged from 18 to 25 years (mean 19 years). They all had normal or corrected-to-normal vision, and were naive to the purpose of the experiment and the illusory characteristics of the drawings. 2.1.2 Apparatus and stimuli. Participants sat in a dimly lit room in front of a 17-inch computer screen at a distance of about 57 cm. A PC computer running E-Prime software (Schneider et al 2002) controlled the presentation of the stimuli, timing operations, and data collection. As a fixation point, a white asterisk was displayed at the centre of the screen. The target stimulus was a figure made of a horizontal and a vertical line presented in the frontoparallel plane. Horizontal lines were always presented centrally at fixation, whereas the vertical line was displaced along either the horizontal or the vertical axis forming different configurations. Both lines were 0.15 deg thick. The length of the horizontal line remained constant (4.85 deg) throughout the experiment. The length of the vertical line varied from 65% to 135% of the length of the horizontal line in 8 steps [65% (which corresponds to 3.16 deg), 75%, 85%, 95%; and 105%, 115%, 125%, 135% (which corresponds to 6.55 deg)]. In the vertical displacement axis, the vertical line was shifted along the vertical axis from bottom to top of the horizontal line with 7 positions: extreme left, extreme right, central, and two more equidistant intermixed positions on each side. While the horizontal line in this condition was always symmetrically divided, the vertical line could be either not bisected (nb), asymmetrically bisected (A, note that the degree of asymmetry was manipulated: number 1 refers to slightly asymmetric and number 2 to strongly asymmetric), or symmetrically bisected (S). As a consequence, the manipulation gave rise to 4 bisection categories of line configurations: SnB, SA2, SA1, and SS, in which the horizontal line was always symmetrically bisected (see figure 2). In the vertical displacement axis, the vertical line was shifted from the left endpoint to the right of the horizontal line (7 positions). Thus, the vertical line was symmetrically bisected in every configuration (S), while the horizontal line was either not bisected (nb), asymmetrically bisected (A2 and A1), or symmetrically bisected (SS).
Bisection, symmetry, and length perception 1431 2.1.3 Procedure. On each trial, a fixation point was displayed on the screen for 1000 ms, followed by a two-line configuration target, which was displayed for 300 ms only. The target stimulus was then masked by a square filled with diagonally aligned lines, and this was presented for 2700 ms or until a response was produced. Participants performed a line-length comparison task (vertical versus horizontal) and were asked to respond by pressing one of two keys depending on whether they saw the vertical or the horizontal line as longer. The `vertical' and `horizontal' responses were assigned to either the Z or the M key on the computer keyboard, and this response mapping was counterbalanced across participants. 2.1.4 Design. The experiment had a 2 (vertical line displacement axis)64 (bisection category)68 (vertical line length) within-participants design. As detailed above, the vertical line was displaced along the vertical or the horizontal axis. As a consequence, the horizontal line was symmetrically bisected in the vertical displacement axis, whereas in the horizontal displacement, the vertical line was symmetrically divided. The bisection categories emerged from the vertical line position and led to 4 different categories: SnB, SA2, SA1, and SS, in which SnB refers to symmetric versus not bisected, SA to symmetric versus asymmetric, and SS to symmetric versus symmetric. Since we used the method of constant stimuli, one of the stimuliöthe horizontal lineöwas of equal length in all the trials; but the length of the second stimulus öthe vertical lineöcould take on 8 values (from 65% to 135% of the horizontal line length). The percentage of `vertical longer' responses was recorded and pooled as a function of vertical line length (8 levels). After that, the data for every vertical line length were normalised to compute the slope and the just noticeable difference (JND), and the point of subjective equality (PSE) of the vertical line. The PSE represents the length of the vertical line at which it is perceived to be longer than the horizontal one 50% of the times (ie in the other 50%, the horizontal line is perceived as longer). The JND is the minimal difference in line length that is perceptually detectable between two stimuli and the slope represents the steepness of the curve. Both the slope and the JND index the ability to discriminate small differences, while the PSE corresponds to the visual distortion. In order to make the data easier to understand, PSE was converted into a percentage of illusory extent. Thus, for example, a PSE of 38.4 corresponds to a 20% overestimation [ie bias ˆ 1 (38:4=48:3)], whereas a PSE of 58.0 corresponds to a 20% underestimation [ie bias ˆ 1 (58:0=48:3)]. Participants were shown a total of 14 configurations, randomly mixed within the same block of experimental trials. They gained practice first with 28 trials at the beginning of the session and then performed 784 experimental trials. 14 trials per stimulus level were presented. Participants could take a short break after every 56 trials. Participants were instructed to respond as accurately as possible within the allowed time and no feedback about response accuracy was given. 2.2 Results The illusory extents derived from the PSE and the slope were submitted to repeatedmeasures ANOVAs with 2 (displacement axis) and 4 (type of bisection) as within-participants variables. The analysis of illusory extents showed mainly an effect of displacement axis (F 112, ˆ 69:89, p 5 0:00001), which underlines that both vertical and horizontal lines are overestimated when they are asymmetrically or not bisected. Importantly, displacement axis interacted significantly with bisection category (F 336, ˆ 27:78, p 5 0:00001), revealing that the perception of line length is not biased in the SS category but is highly altered in the 3 remaining categories (SA1, SA2, SnB; see figure 3).
1432 P Charras, J Lupia n ez We conducted a series of t tests that revealed that the illusory extents in the SS condition were not significantly different from zero (t 12 ˆ 0:64, p ˆ 0:53, ns, in the vertical displacement axis and t 12 ˆ 1:53, p ˆ 0:15, ns, in the horizontal displacement axis). For the SnB, SA2, and SA1 categories, the t tests showed that the length of the line which was asymmetrically or not bisected, independently of its horizontal or vertical dimension, was overestimated (t 12 ˆ 5:81, p 5 0:00001, t 12 ˆ 6:28, p 5 0:0001, and t 12 ˆ 2:59, p ˆ 0:02, in the vertical displacement axis for the SnB, SA2, and SA1 categories, respectively, and t 12 ˆ 5:42, p ˆ 0:0001, t 12 ˆ 5:55, p ˆ 0:0001, and t 12 ˆ 6:41, p 5 0:0001, in the horizontal displacement axis for the SnB, SA2, and SA1 categories, respectively). To investigate whether the extent of asymmetry was proportional to the extent of length overestimation, we conducted planned comparisons for the conditions SA2 and SA1. We observed that for the two axis displacement conditions, the length of lines with a large asymmetric bisection was more overestimated than that of slightly asymmetric lines (F 112, ˆ 53:97, p 5 0:00001, for the vertical displacement and F 112, ˆ 9:03, p ˆ 0:01, for the horizontal displacement). This result emphasises the role of bisection symmetry in length perception and provides direct evidence that the degree of asymmetry determines the amount of length overestimation (see also Charras and Lupia n ez, in press). We also performed planned comparisons to explore whether the overestimation was greater when lines were strongly asymmetrically bisected (SA2) as compared to not bisected (SnB). They revealed that length overestimation was similar in the SnB and SA2 conditions (F 5 1, ns for the vertical displacement, and F 112, ˆ 4:05, p ˆ 0:07, for the horizontal displacement). Vertical line length bias=% of illusory exetnt 50 40 30 20 10 0 10 20 30 40 50 SnB SA2 SA1 SS Bisection category Displacement of the vertical line along vertical axis along horizontal axis Figure 3. Vertical line length bias in experiment 1 as a function of bisection category and vertical line displacement axis (vertical versus horizontal). Note that values above 0 mean that the length of the vertical line was overestimated, whereas values below 0 mean that it was underestimated. The bars represent 1 SE. The repeated-measures analysis of the slope showed a main effect of displacement axis, indicating that precision was higher for the vertical than for the horizontal displacement (F 112, ˆ 24:07, p ˆ 0:0003). We also observed a main effect of bisection category (F 112, ˆ 22:78, p 5 0:0001), revealing that the slope is much higher for the categories SS and SA1 than for SA2 and SnB (see table 1). The interaction displacement axis 6bisection category did not reach significance (F 336 ˆ 2:32, p ˆ 0:09).,
Bisection, symmetry, and length perception 1433 Table 1. Experiment 1. Means and standard errors (SE) of point of subjective equality (PSE), plus the deviation transformed into percentage, and means and SEs of slope as a function of displacement axis (vertical versus horizontal) for the configurations SnB, SA2, SA1, and SS. Displace- SnB SS ment axis PSE SE % deviation slope SE PSE SE % deviation slope SE Vertical 39.02 3.305 19.21 0.077 0.007 47.85 1.454 0.93 0.092 0.008 Horizontal 60.52 3.945 25.29 0.057 0.006 50.38 2.814 4.31 0.092 0.008 SA2 SA1 PSE SE % deviation slope SE PSE SE % deviation slope SE Vertical 41.83 2.134 13.39 0.072 0.007 46.5 1.436 3.73 0.100 0.005 Horizontal 59.52 4.187 23.24 0.061 0.007 54.95 2.541 13.77 0.078 0.007 2.3 Discussion The relevance of symmetry in line-length perception has been reported earlier in the literature (Ku«nnapas 1955; Wolfe et al 2005), but these researchers did not compare symmetry versus asymmetry in bisection. Here, using configurations in which both lines were bisected, we have observed, for the first time to our knowledge, a bias according to which the length of asymmetrically bisected lines is overestimated as compared to that of symmetrically bisected lines. As stated in the introduction, bisection and vertical biases have additive effects on line-length perception (Avery and Day 1969; Ku«nnapas 1955; Richter et al 2007), both being responsible for the massive vertical overestimation observed in the inverted T version of the horizontal ^ vertical illusion. In the present experiment we also investigated the interaction between the bisection and the vertical bias. Importantly, we found that length perception in the SS condition was not affected by the vertical bias. Moreover, the length of asymmetrically bisected vertical lines was not more overestimated than that of asymmetrically bisected horizontal lines. This pattern of data may be taken as evidence that when the two lines are bisected, either symmetrically or asymmetrically, the vertical bias is overridden, so that in this case the bisection and vertical biases interact with each other, instead of being additive, as observed in the inverted T configuration. However, in the SnB configurations, in which the horizontal line was symmetrically divided, although we expected the length of vertical not-bisected lines to be more overestimated than that of horizontal not-bisected lines, the results showed no significant difference in illusory extent between these configurations (F 112, ˆ 1:59, ns). More surprisingly, the overall data show an unexpected horizontal overestimation that might arise from the fact that the horizontal line, unlike the vertical one, was always centrally presented. This experimental manipulation may have led to a response bias, as has happened in many studies in which a two forced-choice response was used for discrimination (Spence et al 2001). Experiment 2 was in part designed to solve this issue: we used two series of configurations, in one of them the horizontal line remained at fixation, while in the other one it was the vertical line that remained at fixation. 3 Experiment 2 The main objective of experiment 2 was to further investigate whether bisection, symmetric or asymmetric, disrupts the manifestation of the vertical bias. Instead of using biased comparisons, like symmetric versus asymmetric, we compared here only
1434 P Charras, J Lupia n ez similar bisections. In other words, we used two-line configurations in which both lines were either symmetrically bisected (SS), slightly asymmetrically bisected (AA1), strongly asymmetrically bisected (AA2), or not bisected (nbnb). According to our hypothesis, the vertical bias should produce an effect only in the nbnb condition. Also, as noted above, in order to eliminate the contribution of responses biases for either the horizontal or the vertical line presented at fixation, this factor was manipulated: in one series of figures the vertical line was always displayed at fixation, in the other series of figures the horizontal line was at fixation. 3.1 Method 3.1.1 Participants. Twelve undergraduate students (one male) from the University of Granada participated in this experiment in exchange for course credits. Their ages ranged from 18 to 25 (mean 21) years. They were all naive to the objective of the experiment and they all had normal or corrected-to-normal vision. 3.1.2 Apparatus, stimuli, and procedure. Apparatus was similar to that used in experiment 1. Participants were always presented with two-line configurations and were asked to indicate which line was the longest. Horizontal and vertical stimuli had the same length and width as in experiment 1. During the experiment, participants were presented with two series of 7 drawings that were randomly mixed within the experiment. In the horizontal-at-fixation condition, the horizontal line position was invariably presented at fixation, and the vertical line was displaced from top-left to bottom-right. In the vertical-at-fixation condition, the vertical line was always displayed at fixation while the horizontal line position was manipulated diagonally from top-right to bottomleft (see figure 4). Configurations belonged to one of the four possible categories: symmetric ^ symmetric (SS), asymmetric1 ^ asymmetric1 (AA1), asymmetric2 ^asymmetric 2 (AA2), and not bisected ^ not bisected (nbnb). The procedure was identical to that of experiment 1 and the design was also very similar; the factors (2, line at fixation)6(7, line position)6(8, line length) were manipulated within-participants. As in experiment 1, for the sake of simplicity in the analyses, the 7 line positions were converted into the 4 described earlier bisection categories. Vertical line length bias=% of illusory extent 50 40 30 20 10 0 10 20 30 40 Horizontal line at fixation Vertical line at fixation 50 nbnb AA2 AA1 SS Bisection category Figure 4.Vertical line length bias as a function of bisection category and of the line that remained at fixation in experiment 2. Note that values above 0 mean that the length of the vertical line was overestimated, whereas values below 0 represent an overestimation of the length of the horizontal line. The bars represent 1 SE.
Bisection, symmetry, and length perception 1435 3.2 Results and discussion The illusory extent, derived from PSE, and the slope were submitted to repeated-measures ANOVAs with the factors (2, line-at-fixation)6(4, bisection category). To test whether the global horizontal overestimation observed in experiment 1 was due to the invariable line location, we first focused on the possible influence of line-at-fixation. As expected, a main effect of line-at-fixation was observed, revealing that the centrally presented line was slightly, but significantly, overestimated (F 111, ˆ 13:87, p ˆ 0:003) (see figure 4). What is important is that this result made it possible to shed some light on the overall great horizontal overestimation observed in experiment 1 since the horizontal line was always presented at fixation. We also observed a main effect of bisection category (F 111, ˆ 4:67, p ˆ 0:008). Corresponding planned comparisons showed that there was a significant difference between the nbnb and SS conditions (F 111, ˆ 26:34, p ˆ 0:0003). However, performance in the AA2 and AA1 conditions did not statistically differ from that in the SS condition (F 111, ˆ 2:71, p ˆ 0:12, and F 5 1, ns, respectively). This thus suggests that vertical lines were more overestimated in the nbnb than in the SS condition, but, conversely, vertical lines were perceived identical whenever they were displayed in the AA2, AA1, or SS configurations. The interaction between line-at-fixation and bisection category was significant (F 333, ˆ 3:48, p ˆ 0:03), revealing that line-at-fixation did not modulate the length perception in the SS condition. This was largely expected because in this condition the two lines were presented at fixation. We also conducted t tests that confirmed that line perception was not biased in that condition (t 11 ˆ 0:61, p ˆ 0:55, ns, for horizontal-atfixation and t 11 ˆ 0:19, p ˆ 0:84, ns, for vertical-at-fixation). As regards the slope analysis, we observed a main effect of line-at-fixation, showing that participant's responses were significantly more precise when the vertical line remained at fixation (F 111, ˆ 7:11, p ˆ 0:02). The effect of bisection category was also significant (F 333, ˆ 4:75, p ˆ 0:007), revealing that the slopes were higher in the SS, AA1, and AA2 conditions than in the nbnb condition (see table 2) (F 111, ˆ 8:22, p ˆ 0:01; F 111 ˆ 8:45, p ˆ 0:01; and F 111 ˆ 12:67, p ˆ 0:004, respectively).,, Table 2. Experiment 2. Means and standard errors (SE) of point of subjective equality (PSE) plus the deviation transformed into percentage, and means and SEs of slope as a function of line at fixation (vertical versus horizontal) for the configurations nbnb, AA2, AA1, and SS. Line at nbnb SS fixation PSE SE % deviation slope SE PSE SE % deviation slope SE Horizontal 45.32 1.722 6.17 0.080 0.006 48.97 2.26 1.39 0.098 0.007 Vertical 48.08 2.416 0.45 0.072 0.008 48.56 2.672 0.53 0.081 0.01 AA2 AA1 PSE SE % deviation slope SE PSE SE % deviation slope SE Horizontal 45.60 1.664 5.60 0.093 0.005 48.97 2.089 2.40 0.089 0.008 Vertical 49.57 3.101 2.63 0.083 0.009 48.56 2.459 2.16 0.087 0.008 4 General discussion Experiment 1 has shown that bisection, and more specifically the symmetry of bisection, plays a crucial role in line-length perception. In the SS condition, length perception was unbiased, in the sense that the vertical line length was perceived as identical to that of the horizontal when the two lines were physically identical in length. In contrast, when the lines were asymmetrically bisected, their lengths were overestimated as
1436 P Charras, J Lupia n ez compared to symmetrically bisected lines. Importantly, the degree of asymmetry seems to be proportional to the amount of line-length overestimation. The vertical bias, commonly observed in horizontal versus vertical line-length comparison, was eliminated. Experiment 2 was designed to further explore the interaction between the vertical bias and bisection and confirmed that when the two lines were bisected in a similar way (SS, AA1, or AA2), line orientation did not affect length processing, in the sense that vertical lines were no longer overestimated. Yet, when the lines were not bisected (nbnb condition), the vertical bias produced a small illusory effect (about 5%). Thus, the present study provides evidence that bisection prevents the vertical bias and that the type of bisection leads to a bias in line-length estimation, such that the length in asymmetric bisections is overestimated as compared to that in symmetric bisections. Unfortunately, the experimental design used here does not allow us to determine whether the length of symmetrically bisected lines is underestimated, or that of asymmetrically bisected lines is overestimated, or both. However, the vertical overestimation observed in the inverted T configuration indicates that symmetry is underestimated per se. Also, it can be inferred from the proportional relation between the degree of asymmetry and the amount of overestimation that asymmetric bisection leads to an overestimation. We thus claim that the a/symmetry of bisection can produce two distinct effects: a symmetry underestimation and an asymmetry overestimation. These two effects may co-exist, as in the Oppel ^ Kundt illusion (see figure 1). Yet, it is necessary to underline that previous studies provided more consistent evidence for symmetry underestimation than for asymmetry overestimation. For instance, when asymmetry is manipulated by moving the vertical line location in an inverted T figure (see Charras and Lupia n ez, in press; Ku«nnapas 1955; Wolfe et al 2005), the results have shown that the asymmetrically bisected horizontal line was not overestimated per se. But, it is worth noting that the length of the non-bisected vertical line was very slightly, or even not, overestimated, suggesting that the vertical bias was weakened. Obviously, further research is needed to test the present assumptions and to explore to what extent these new principles of visual perception apply to any perceptual dimension or are restricted to some dimensions only. The results also raise the question of whether these overestimations or underestimations are relative or absolute. Can they only be observed in a comparison task? One can wonder whether this discrepancy between asymmetry and symmetry would also occur in a production task or in an absolute-length perception task, for instance. With respect to the vertical bias, two main theories have been put forward to account for this effect. The first one relies on depth interpretation of two-dimensional drawings (Girgus and Coren 1975; Gregory 1963; Woodworth 1938). In the horizontal ^ vertical illusion, the vertical line would unconsciously be perceived as receding in depth from the observer, and then its retinal length would be compensated for by the size-constancy scaling mechanisms and eventually would be perceived as longer than the horizontal line. The second account is based on the visual field properties (Ku«nnapas 1955, 1957; Prinzmetal and Gettleman 1993) and claims that the visual field has a horizontally aligned elliptic shape. Consequently, any vertical stimuli falling into this shape will look closer to the visual field boundaries than horizontal stimuli. This discrepancy would internally be solved by overestimating the size of vertical stimuli in comparison to horizontal ones. It is, however, worth noting that several studies demonstrated that both the depth interpretation and the visual-field properties could explain some aspects of the vertical bias (Williams and Enns 1996), but neither of them could fully account for all its manifestations (Wolfe et al 2005). Recently, Raudsepp and Djupsjo«backa (2005) proposed that the vertical bias could result from the internal representation of motor abilities (see the effort accountö Bhalla and Proffitt 1999; Guski et al 1993; Lipshits et al 2001; Yang et al 1999).
Bisection, symmetry, and length perception 1437 They demonstrated that fatigue or external load, for instance, influence the vertical length overestimation. We propose here to focus on a new possible account for the vertical bias coming from a very recent study which suggested that saccades are more efficient on the horizontal than on the vertical axis during visual searches (Phillips and Edelman 2008). Phillips and Edelman showed that the perceptual span which refers to the amount of processed visual information is larger during horizontal than during vertical searches. The lower efficiency during vertical searches slows down information processing. From a study of Casasanto and Boroditsky (2008, experiment 6), we know that a longer exploration area can be interpreted as a larger stimulus [see also `a theory of magnitude' proposed by Walsh (2003)]. Going back over the vertical bias, we propose that overestimation of the vertical length may be due to poorer performance in scanning the vertical line. One may then wonder why and how bisection can affect line-length perception. We think that line junction is a crucial clue for interpreting a drawing, and therefore might capture the gaze and the locus of attention. This redistribution of attention would influence the way of scanning the figure, and avoid a direct vertical versus horizontal comparison. It seems necessary to replicate this experiment whilst controlling for eye movements to test whether saccades could explain this phenomenon. Even though more research is needed to understand why bisection prevents the vertical bias, we think that it is important to take into account the present results in devising a robust theory of vertical/horizontal length estimation that would account for all the manifestations of the vertical bias. Acknowledgments. This research was supported by the Spanish Ministerio de Educaciön y Ciencia (Ministry of Education and Science) through predoctoral grant AP-2006-3911 to the first author, and a research project (SEJ2005-01313PSIC) to the second author. References Avery G, Day R, 1969 ``Basis of the horizontal vertical illusion'' Journal of Experimental Psychology 81 376^380 Bhalla M, Proffitt D R, 1999 ``Visual-motor recalibration in geographical slant perception'' Journal of Experimental Psychology: Human Perception and Performance 25 1076 ^ 1096 Casasanto D, Boroditsky L, 2008 ``Time in the mind: Using space to think about time'' Cognition 106 579^593 Charras P, Lupia n ez J, in press ``Length perception of horizontal and vertical bisected lines'' Psychological Research Coren S, Girgus J S, 1978 Seeing Is Deceiving: The Psychology of Visual Illusions (Hillsdale, NJ: Lawrence Erlbaum Associates) Fick A, 1851 De Errore Quodam Optico Asymetrica Bulbi Effecto (Marburg: J A Kochin) Finger F W, Spelt D K, 1947 ``The illustration of the horizontal vertical illusion'' Journal of Experimental Psychology 37 243 ^ 250 Girgus J S, Coren S, 1975 ``Depth cues and constancy scaling in the horizontal vertical illusion: The bisection error'' Canadian Journal of Psychology 29 59^65 Greene E, 1998 ``A test of the gravity lens theory'' Perception 27 1221 ^ 1228 Gregory R L, 1963 ``Distortion of visual space as inappropriate constancy scaling'' Nature 199 678 ^ 680 Guski R, Rudolph R, Schindauer T, 1993 Zur Funktionalita«t der ``vertikalen-ta«uschung'' Publication No. 241, Fakulta«t der Psychologie, Ruhr Universita«t, Bochum, Germany Koffka K, 1950 Principles of Gestalt Psychology (Andover, Hants: Routledge and Kegan Paul) Kundt A, 1863 `Ànnalen der Physik und Chemie. Untersuchungen u«ber AugenmaÞ und optische Ta«uschungen'' Poggendorff Annalen 120 118 ^ 158 Ku«nnapas T M, 1955 ``An analysis of the `vertical ^ horizontal illusion''' Journal of Experimental Psychology 49 134 ^ 140 Ku«nnapas T M, 1957 ``The vertical ^ horizontal illusion and the visual field'' Journal of Experimental Psychology 53 405 ^ 407
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