# Opposition and Identicalness: Two Basic Components of Adults’ Perception and Mental Representation of Symmetry

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## Abstract

**:**

## 1. Introduction

#### From Mirrors to Mirror Symmetry

## 2. Study 1

#### 2.1. Materials and Method

#### 2.1.1. Participants

#### 2.1.2. Materials

- (1)
- How would you define the relationship between two symmetrical shapes?
- (2)
- Draw a clear example of your idea of two symmetrical shapes.
- (3)
- Draw another clear example (radically different from the first two) of your idea of two symmetrical shapes.
- (4)
- Which of the following three definitions best describes your idea of symmetry?
- (a)
- Two identical shapes
- (b)
- Two opposite shapes
- (c)
- Two identical and opposite shapes

#### 2.1.3. Procedure

#### 2.1.4. Statistical Analysis

#### 2.1.5. Results

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_{(7, 105)}= 77.189, p < 0.0001, see top graph in Figure 5). As post-hoc tests revealed, responses referring exclusively to the sameness of the two symmetrical shapes (either in general, or specifying that they were the same in terms of shape and/or size—see examples of the descriptions under the category b in Table 1) were significantly more frequent than all of the other response categories except for those responses which made exclusive and explicit reference to a specular configuration (i.e., category c in Table 1; EST = 9.074, SE = 0.309, z-ratio = 2.934, p = 0.093).

- (a)
- Exclusive references to Sameness were significantly more frequent than references to both Sameness and Opposition (categories b versus f in Table 1: EST = 1.292, SE = 0.334, z-ratio = 3.863, p = 0.003, d = 0.376);
- (b)
- The two most frequent types of description (i.e., categories b and c in Table 1), which together amount to 61% of the total number of responses, do not explicitly refer to opposition;
- (c)
- Only one response (i.e., less than 1%) mentioned the opposition component exclusively (category e in Table 1: “Two symmetrical shapes are two opposite shapes”);
- (d)
- Overall Opposition, in one way or another (i.e., categories e, f, and g in Table 1) was mentioned in only 20 out of the 105 descriptions collected (i.e., 19.04%).

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_{(2, 109)}= 48.769, p < 0.0001; see bottom graph in Figure 5). “Identical and Opposite” was more frequently chosen as compared to exclusive references to Opposition (EST = 4.564, SE = 0.741, z-ratio = 6.159, p < 0.0001, d = 0.601) but “Identical and Opposite” was also more frequently chosen as compared to exclusive references to Identicalness (EST = 1.250, SE = 0.284, z-ratio = 4.398, p < 0.001, d = 0.429).

^{2}

_{(2, 86)}= 123.448, p < 0.0001). As shown in Figure 7 (top graph), participants more frequently drew configurations displaying a Vertical mirror axis than a Horizontal mirror axis (EST = −3.855, SE = 0.392; z-ratio = −9.843, p < 0.0001, d = −1.061), which in turn was more frequently used than an Oblique mirror axis (EST = 1.835, SE = 0.685, z-ratio = 2.678, p = 0.022, d = 0.288). As shown in the central graph in Figure 7 (which shows the interaction between Drawing and Orientation: χ

^{2}

_{(2, 87)}= 14.677, p = 0.0006), this distribution held for both the first and the second drawings. However, configurations displaying a horizontal mirror axis tended to be drawn more frequently in the second drawing as compared to the first (EST = −1.666, SE = 0.592; z-ratio = −2.816, p = 0.07, d = 0.274).

^{2}

_{(4, 56)}= 57.357, p < 0.0001). As confirmed by the post-hoc tests, the most frequent orientation was vertical in both the first and second drawings (EST = 3.152, SE = 0.58, z-ratio = 5.352, p < 0.0001, d = 0.715) despite the fact that participants had been explicitly told in the instructions that the second drawing should present a radically different example of symmetry to the first drawing.

^{2}

_{(2, 86)}= 17.758, p < 0.0001), with no interaction with Drawing (χ

^{2}

_{(2, 86)}= 2.412, p = 0.299). As shown in Figure 8, the drawings were based on asymmetrical shapes in the majority of cases: asymmetrical shapes (either Asym 1 or Asym 2) constituted around 75% of the total, including both the first and second exemplars. Perfectly symmetrical shapes, i.e., shapes that minimized the opposition component, accounted for less than 25% of the configurations.

^{2}

_{(1, 57)}= 38.572, p < 0.0001). As shown in Figure 9, participants more frequently positioned the shapes with their internal axis of symmetry orthogonal with respect to the mirror axis rather than parallel to it (EST = 3.434, SE = 0.516, z-ratio = 6.657, p < 0.0001, d = 0.882). This means that they chose a configuration that made the opposite orientation of the two shapes evident.

^{2}

_{(1, 43)}= 1.490, p < 0.222), or interacting with Drawing, (χ

^{2}

_{(1, 43)}= 0.043, p = 0.835).

^{2}

_{(4, 57)}= 9.353, p = 0.05). A second GLMM was then carried out to assess any association between the Iconic Pair levels and the responses to question 4 (Identical; Identical and Opposite; Opposite). In this case, too, the interaction between the responses to question 4 and the Iconic Pair levels turned out to be significant (χ

^{2}

_{(4, 57)}= 27.312, p < 0.0001).

## 3. Study 2

#### 3.1. Materials and Method

#### 3.1.1. Participants

#### 3.1.2. Materials

#### 3.1.3. Procedure

#### 3.1.4. Statistical Analysis

#### 3.1.5. Results

^{2}

_{(1, 70)}= 1.861, p = 0.172). It should be noted that participants were not directly asked to choose between a convergent configuration versus a divergent configuration but between a pattern showing identicalness and a pattern showing opposition (either divergent or convergent).

^{2}

_{(1, 70)}= 7.422, p = 0.006). Convergent patterns were selected more frequently than divergent patterns (EST = 0.536; SE = 0.197; z-ratio = 2.724; p = 0.006; d = 0.326).

## 4. Discussion

#### Potential Impact and Limitations of the Study

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The point by point transformation underlying the geometry of mirror symmetry (around a vertical axis). “Take any line l perpendicular to E and any point p on l (…). Reflection in E is that mapping of space upon itself, S: p → p’, that carries the arbitrary point p into its mirror image p’ with respect to E” ([45], pp. 4–5).

**Figure 2.**Some errors emerged in tasks which required participants to predict the location and direction of motion in a reflection: (

**a**) when the “real” person moved parallel to a vertical mirror on a wall, many people expected her reflection to appear at the farther edge of the mirror walking towards the “real person”; (

**b**,

**c**) when the “real person” moved at an angle towards a mirror, some people expected the reflection to move along the same trajectory with an opposite orientation (

**b**) or along the same trajectory but with an identical orientation (

**c**).

**Figure 3.**Mirror symmetry applied to configurations that have different symmetrical structures. On the left: shapes which are symmetrical along the axis parallel to the mirror minimize the perception of contrariety (which remains relative only to the position of the shapes, i.e., one to the left and the other to the right of the mirror axis) and maximize perception of identicalness. On the right: shapes which are symmetrical only with respect to the axis which is orthogonal to the mirror axis but which are asymmetrical with respect to the axis parallel to the mirror axis make the opposite orientation easier to see (for farther explanations, see text).

**Figure 4.**The differences in perceptual impact of rotating the black shapes (original position: 0°) by, respectively, 20° and 90° angles with respect to the “mirror axis” (the solid vertical line). The dashed lines indicate the two internal orthogonal symmetry axes. The shapes in the first row (

**Sym**) are symmetrical with respect to both their internal axes (indicated by small dashes); the shapes in the second row (

**Asym 1**) are symmetrical with respect to one axis (small dashes) and asymmetrical with respect to the other axis (large dashes) and the shapes in the third row (

**Asym 2**) are asymmetrical with respect to both internal axes (large dashes). For a further explanation, see the text.

**Figure 5.**Effect plot of the proportional use of each of the various response categories for question 1 (

**top graph**) or chosen from among the three alternatives in question 4 (

**bottom graph**). Proportions are reported on a logit link scale (as computed by the GLMMs described in the main text). Error bars represent a 95% confidence interval.

**Figure 6.**Some of the drawings done by the participants as examples of their idea of symmetry (in response to questions 2 or 3).

**Figure 7.**Effect plots of the Orientation of the mirror axis in the configurations drawn by participants. Top graph: Main effect of mirror axis Orientation. Central graph: interaction between mirror axis Orientation and Drawing. Bottom graph: Main effect of a combined analysis of the two drawings done by each participant. In all plots, error bars represent a 95% confidence interval.

**Figure 8.**Effect plot of the use of symmetrical and asymmetrical shapes in the drawings done by the participants to exemplify their idea of a “symmetrical configuration” (Asym 2 = asymmetrical with respect to both the vertical and horizontal axes; Asym 1 = symmetrical around one axis and asymmetrical with respect to the other; Sym = symmetrical with respect to both the vertical and horizontal axes). Error bars represent a 95% confidence interval.

**Figure 9.**Effect plot of the Orientation (Orthogonal or Parallel) of the internal axis of symmetry of the shapes drawn by participants with respect to the mirror axis. Two examples of orthogonal configurations are shown on the left and one example of parallel configuration is shown on the right. Error bars represent a 95% confidence interval.

**Figure 10.**Mosaic plot showing the association between the three Iconic Pair levels relating to the shapes drawn by participants as exemplar configurations (in terms of symmetry/asymmetry) and the responses to question 1 (mosaic on the left) and question 4 (mosaic on the right).

**Figure 11.**The configurations used in the pair comparison task in study 2. The pairs inside the red borders are those instantiating a match between a configuration which shows only identicalness and another which shows both identicalness and opposition. The pairs inside the blue border are formed of two configurations both of which show identicalness and opposition. The configurations which are not inside a border are the pairs which are formed of two configurations, both of which only show identicalness.

**Figure 12.**Scaling of the configurations used in the pair comparison task (based on the Thurston Case V scaling).

**Table 1.**The categories used to classify the definitions of symmetry produced by the participants in study 1 (in response to question 1). Examples of each type of description and the frequency of each category are presented.

Types of Descriptions | Examples | Counts (and %) |
---|---|---|

a. Geometrical | [Shapes with corresponding points at the same distance from the axis of symmetry] | 3 (2.9%) |

b. Same | [Identical shapes] [Perfectly overlapping shapes] [Identical, coincident shapes] [shapes of the same form] [Shapes of the same form and size] | 42 (40.0%) |

c. Mirror | [Specular shapes] [Reflected shapes] | 22 (21.0%) |

d. Same + Mirror | [Shapes with same form and size, specular to each other] [Similar shapes, as if reflected in a mirror] [Specular/shapes with the same characteristics] | 11 (10.5%) |

e. Opposite | [Two opposite shapes] | 1 (1.0%) |

f. Same + Opposite | [Identical shapes, but with one reversed with respect to the other] [Same but contrary shapes] [Same and opposite shapes] [Shapes with the same features but which are inverted left to right] | 16 (15.2%) |

g. Same + Opposite + Mirror | [Reflected shapes: identical but reversed] [Specular shapes: identical but inverted] [Equal and opposite shapes, as if reflected in a mirror] | 3 (2.9%) |

h. Other | [Two shapes, one near the other] [Shapes which are parallel to each other] | 7 (6.7%) |

Total | 105 | |

Missing | (missing responses or tautological responses) | 4 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bianchi, I.; Bertamini, M.; Burro, R.; Savardi, U. Opposition and Identicalness: Two Basic Components of Adults’ Perception and Mental Representation of Symmetry. *Symmetry* **2017**, *9*, 128.
https://doi.org/10.3390/sym9080128

**AMA Style**

Bianchi I, Bertamini M, Burro R, Savardi U. Opposition and Identicalness: Two Basic Components of Adults’ Perception and Mental Representation of Symmetry. *Symmetry*. 2017; 9(8):128.
https://doi.org/10.3390/sym9080128

**Chicago/Turabian Style**

Bianchi, Ivana, Marco Bertamini, Roberto Burro, and Ugo Savardi. 2017. "Opposition and Identicalness: Two Basic Components of Adults’ Perception and Mental Representation of Symmetry" *Symmetry* 9, no. 8: 128.
https://doi.org/10.3390/sym9080128