# Affine Geometry, Visual Sensation, and Preference for Symmetry of Things in a Thing

## Abstract

**:**

## 1. Introduction

#### 1.1. Affine Geometry and Visual Sensation

#### 1.2. Reflection and Rotational Shape Symmetry

**x**,

**y**,

**z**), is an important factor in visual recognition [22,23,24]. Reflection or mirror symmetry is detected quickly [25,26] in foveal and in peripheral vision [27]. Vertical mirror symmetry facilitates face recognition by human [28] and non-human primates [29], and is used by the human visual system as a second-order cue to perceptual grouping [30].

#### 1.3. Nature-Inspired Design and the Symmetry of “Things in a Thing”

## 2. Materials and Methods

#### 2.1. Subjects

#### 2.2. Stimuli

#### 2.3. Task Instructions

#### 2.4. Procedure

## 3. Results

_{left}= 4.92 vs. M

_{right}= 5.07), or the order (first vs. second) in which a judgment was formed in response to a figure pair (M

_{first}= 5.02 vs. M

_{second}= 4.95).

#### 3.1. Aesthetic Ratings

#### 3.2. Preference Judgments

## 4. Discussion

## 5. Conclusions

## Conflicts of Interest

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**Figure 1.**The importance of curve symmetry for human endeavour dates back to the dawn of building shelter and to vernacular architecture (

**left**). Similar geometry is currently used in contemporary free-form architecture (

**middle**), which has been much inspired by the Spanish architect Gaudi, who largely exploited symmetry for the design of the hall and archways of the Sagrada Familia in Barcelona (

**right**).

**Figure 2.**Projective geometry permits generating symmetric curves from ellipses by affinity with concentric circles. Each such curve may be perceived as a single thing or as a multiple of one and the same thing in a complex shape or object, as shown here. This perception is grounded in biology in the sense that most natural objects can be represented as images of symmetrically curved shapes with the Euclidean properties of ellipses. Symmetric curves yield visual and tactile sensations of curvature which increase exponentially with the aspect ratio of the curves.

**Figure 3.**Fractal geometry and affine geometry share principles of projection in Euclidean space, as illustrated in this example here. Fractal trees, inspired by nature, may be defined as complex wholes where every part repeats itself across multiple fractal iterations, producing “symmetry of things in a thing”. In the 2D fractal mirror-tree shown here, concentric circles and affine projection are the mathematical basis for describing structural regularities with vertical reflection (mirror) symmetry, which has been identified as a major determinant of the visual attractiveness of image configurations.

**Figure 4.**Fifteen images of fractal mirror trees were designed using some of the principles of transformation shown in Figure 2 and Figure 3. The first five trees (

**top**) possess perfect “symmetry of things in a thing” across the vertical axis. In the next set of five (

**middle**), the smallest of fractal details is missing on the right. The remaining five trees (

**bottom**) are asymmetrical. It is noted that in these tree structures here, only the symmetrical ones (

**top**) appear perceptually complete.

**Figure 5.**30 image pairs with 20 presentations for figures of a given type (10 times on the left, and 10 times on the right). The image pairs were displayed in random order, and each pair was displayed twice in an individual session, yielding 60 preference judgments from each of the 30 subjects.

**Figure 6.**Average aesthetic ratings on a scale between zero and ten are shown as a function of the figure type.

**Table 1.**Results from the one-way repeated measures ANOVA on aesthetic ratings for each of the three figure types: Number of observations (N) per figure type, means (M) and standard errors (SEM), and the F value with probability limits (p) are given. Effect sizes, t values and the corresponding probability limits are given at the bottom.

One Way Repeated Measures Analysis of Variance AESTHETIC RATINGS | |||
---|---|---|---|

Treatment | N | M | SEM |

Symmetrical | 30 | 6.347 | 0.222 |

Detail missing | 30 | 4.487 | 0.258 |

Asymmetrical | 30 | 3.053 | 0.308 |

Source of Variation | DF | F | p |

Between Subjects | 29 | - | - |

Between Treatments | 2 | 64.323 | <0.001 |

Residual | 58 | - | - |

Total | 89 | - | - |

Comparison | dM | t | p |

Symmetrical vs. Asymmetrical | 3.293 | 11.311 | <0.001 |

Symmetrical vs. Detail missing | 1.860 | 6.388 | <0.001 |

Detail missing vs. Asymmetrical | 1.433 | 4.923 | <0.001 |

**Table 2.**Results from the one-way ANOVA for repeated measures of preference judgments for each of the three figure types.

One Way Repeated Measures Analysis of Variance PREFERENCES | |||
---|---|---|---|

Treatment | N | M | SEM |

Symmetrical | 30 | 8.942 | 0.163 |

Detail missing | 30 | 4.542 | 0.094 |

Asymmetrical | 30 | 1.517 | 0.145 |

Source of Variation | DF | F | p |

Between Subjects | 29 | - | - |

Between Treatments | 2 | 195.399 | <0.001 |

Residual | 58 | - | - |

Total | 89 | - | - |

Comparison | dM | t | p |

Symmetrical vs. Asymmetrical | 7.425 | 31.299 | <0.001 |

Symmetrical vs. Detail missing | 4.400 | 18.547 | <0.001 |

Detail missing vs. Asymmetrical | 3.025 | 12.751 | <0.001 |

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**MDPI and ACS Style**

Dresp-Langley, B.
Affine Geometry, Visual Sensation, and Preference for Symmetry of Things in a Thing. *Symmetry* **2016**, *8*, 127.
https://doi.org/10.3390/sym8110127

**AMA Style**

Dresp-Langley B.
Affine Geometry, Visual Sensation, and Preference for Symmetry of Things in a Thing. *Symmetry*. 2016; 8(11):127.
https://doi.org/10.3390/sym8110127

**Chicago/Turabian Style**

Dresp-Langley, Birgitta.
2016. "Affine Geometry, Visual Sensation, and Preference for Symmetry of Things in a Thing" *Symmetry* 8, no. 11: 127.
https://doi.org/10.3390/sym8110127