# Geometrical Visual Illusions Revisited: The Curse of Symmetry, the Cure of Sighting, and Taxing Task Demands

## Abstract

**:**

## 1. Introduction

## 2. The Curse of Symmetry

^{π}/

_{4}± n ⋅

^{π}/

_{2}with n = 0, 1, 2, or 3 preserves the figure’s symmetry group, its connectivity, and its orthogonality, but it eliminates the illusion [44]. By having observers look at an illuminated L in the dark while being recumbent, Avery and Day also proved that the horizontal-vertical illusion is yoked to retinal, not geophysical coordinates, so that there is no vestibular or other proprioceptive component to it. 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 [45] demonstrated that the illusion only occurs in configurations or some kind of context [46]. Tilting only one of the lines of an L retains the symmetry group and seems to attenuate the illusion with acute angles but to enhance it with obtuse angles [47,48]. Shortening one of the lines of an L preserves orthogonality, but destroys the symmetry group and, if sufficiently extreme, the illusion; see [46] who combined this measure with dissecting the L and presenting its individual lines sequentially and centered in different or the same quadrants of a computer screen. There are many different ways in which an L can be dissected into two separate lines, but the following two seem of prime importance. Either the two lines are moved away from the symmetry axis by equal amounts in opposite directions, or one line is moved along its own direction. In the first case, symmetry and orthogonality are retained, but in the other cases, orthogonality has been isolated. If, in the first mentioned case, the lines are tilted at equal angles, symmetry will have been isolated (An alternative way to retain and isolate symmetry is to move one or both lines along the arc of a circle that connects the two lines’ endpoints). If, in the second mentioned cases, one line is tilted relative to the other one, all original geometrical singularities will have been eliminated, and there should be no illusion anymore.

## 3. The Cure of Sighting

#### 3.1. Method

#### 3.1.1. Participants

#### 3.1.2. Apparatus

^{−2}; CIE-coordinates: x = 0.312; y = 0.332; Weber contrast between stimulus elements 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.

#### 3.1.3. Stimuli and Responses

#### 3.2. Results

_{Stimuli}= 1.223, p < 0.319, η

_{p}

^{2}= 0.120; F(1, 9)

_{Orientation}= 14.993, p < 0.004, η

_{p}

^{2}= 0.625, and also a significant interaction, F(2.920, 26.278) = 8.119, p < 0.001, η

_{p}

^{2}= 0.474, which came about because of the deviating results for the Oppel–Kundt stimulus. Deviation contrasts showed that, for the Müller–Lyer figure, the effect was slightly greater than the overall mean effect, F(1, 9) = 5.544, p < 0.043, η

_{p}

^{2}= 0.381, but simple and repeated contrasts showed that the effects for the T and for the Ebbinghaus figure did not differ in size from the one for the Müller-Lyer (both Fs < 1). Generally, there was a trend for the angular settings to correspond to the length calibration of the linear extents. For five participants, this trend was quite pronounced, but for the other participants, responses were mixed or even indifferent (Figure 5 shows means of all observers). In contradistinction to Figure 4, which refers to observers’ accuracy (i.e., observers’ ability to hit a predefined criterion), the data plotted in Figure 5 may be interpreted to reflect observers’ sensitivity (i.e., observers’ ability to discriminate between the calibration of the stimuli; see Section 4. for further discussion).

_{p}

^{2}= 0.103, but a detailed look at the data revealed that two observers responded to the Oppel–Kundt stimulus as predicted, two other observers responded to the T opposite to the prediction, and another two observers, respectively, responded to the Müller–Lyer or the Ebbinghaus stimuli contrary to the hypothesis.

#### 3.3. Discussion

#### 3.3.1. Do Illusions Really Go Away

#### 3.3.2. The Technique of Sighting and Seeing Things in Perspective

#### 3.3.3. Utilizing Perspective Distortions to Find States of Apparent Congruence

## 4. Observers’ Task Demands: Discrimination versus Identification

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Nine Popular and Much Researched Geometrical Visual-Illusion Figures. Note. The figure shown in (

**C**) is modelled after Kundt [15]. Oppel [16] only described the phenomenon in abstract terms and with reference to an empty versus furnished room. For empirical investigations, Kundt [15] used a horizontal dots figure and a corresponding mechanical apparatus. The figure shown in (

**D**) is usually attributed to Ponzo [17]. He, however, asserted to have copied it from the French translation of Sanford [18], who, in turn, attributed similar figures to Thiéry [19,20]. The figure shown in (

**I**) is the more common of two versions [4] (p. 67): Zöllner [21] first described a pattern of thick black lines that he had observed with a textile fabric (for which pattern, the shape and the color symmetries do not coincide), but he also noted that the illusion is just as strong with thin lines.

**Figure 2.**Canonical Modifications and Deconstructions of the L Figure. Note. The numbers of the logistic numbering system code for the symmetry operations and the similarity transformation applied. 1 = Rotation of the whole figure; 2 = Rotation of one line; 3 = Shortening or lengthening of one line; 4 = Translation of one line. Numbers in parentheses denote different cases. The letters of the inserts refer to the mathematical properties of the figures. S = Symmetry; O = Orthogonality; C = Connectivity.

**Figure 3.**Canonical Modifications and Deconstructions of the T Figure. Note. The numbers of the logistic numbering system code for the symmetry operations and the similarity transformation applied. 1 = Rotation of the whole figure; 2 = Rotation of one line; 3 = Shortening or lengthening of one line; 4 = Translation of one line. Numbers in parentheses and subscripts denote different cases. The capital letters of the inserts refer to the mathematical properties of the figures. S = Symmetry; O = Orthogonality; C = Connectivity; B = Bisection.

**Figure 4.**Mean Angular Settings and 95 % Confidence Intervals Obtained in the Experiment. Note. Positive angular settings refer to forward scrolls, making the upper half of a picture recede into depth, and negative angular settings refer to backward scrolls, making the lower half of a picture recede into depth. Angles are referenced to the vertical, frontoparallel presentation surface. The insets below the Figure show how the stimuli looked like at the respective angular settings.

**Figure 5.**Mean Angular Settings and 95 % Confidence Intervals Plotted for the Different Lengths of the Larger-Appearing Extents of Three of the Illusion Figures. Note. Plots for the smaller-appearing extents look very similar, only inverted. There are no plots for the Ebbinghaus circles because there had been no variation of the circles’ sizes with this figure.

**Table 1.**Mathematical Analyses of the Nine Visual-Illusion Figures Shown in Figure 1.

Name | Symmetry Group(s) | Other Geometrical Singularities |
---|---|---|

The L (Horizontal-vertical; 1851) | d1 | Orthogonality |

The T (17th century) | d1 | Orthogonality, bisection |

Oppel-Kundt (1860–1861; 1863) | 2 × d2 | Repetition (texture) |

Ponzo (1912) | d1 | Convergence, parallelism |

Müller-Lyer (1889) | 2 × d2 | Convergence/divergence, parallelism |

Ebbinghaus (Wundt, 1898) | 2 × cn or dn | Annularity, size |

Poggendorff (Zöllner, 1860) | c2 | Parallelism, collinearity |

Hering (1861) | d2 | Convergence, parallelism |

Zöllner (1860) | pmg | Relative tilt, multiple parallelism |

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**MDPI and ACS Style**

Landwehr, K.
Geometrical Visual Illusions Revisited: The Curse of Symmetry, the Cure of Sighting, and Taxing Task Demands. *Symmetry* **2022**, *14*, 2550.
https://doi.org/10.3390/sym14122550

**AMA Style**

Landwehr K.
Geometrical Visual Illusions Revisited: The Curse of Symmetry, the Cure of Sighting, and Taxing Task Demands. *Symmetry*. 2022; 14(12):2550.
https://doi.org/10.3390/sym14122550

**Chicago/Turabian Style**

Landwehr, Klaus.
2022. "Geometrical Visual Illusions Revisited: The Curse of Symmetry, the Cure of Sighting, and Taxing Task Demands" *Symmetry* 14, no. 12: 2550.
https://doi.org/10.3390/sym14122550