# Similar Symmetries: The Role of Wallpaper Groups in Perceptual Texture Similarity

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## Abstract

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## 1. Introduction

## 2. The 17 Wallpaper Groups

## 3. Methods

#### 3.1. Stimuli

#### 3.2. Participants

#### 3.3. Procedure

## 4. Results

## 5. Discussion

## References and Notes

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**Figure 3.**Histograms showing (left) the number of subsets chosen by the participants; (right) the size of the subsets made.

**Figure 4.**Similarity matrix. Each row and column represents an image, and the colour of the pixel at position $(i,j)$ indicates the number of times image i was placed in a subset with image j. Brighter colours indicate increasing similarity. The high values along the diagonal are due to each image being classed as similar to itself. The images are grouped by symmetry group, as indicated by the white lines.

**Figure 6.**Mean number of retrievals from the same wallpaper group. The dotted line shows the number of retrievals that would be expected by chance (hypergeometric distribution). Each line represents a different symmetry group.

**Figure 7.**Graph showing how similar the wallpaper groups are to each other. The different shapes represent a group’s rotations (circle = no rotations, rectangle = 1 rotation, square = 2 rotations, triangle = 3 rotations, hexagon = six rotations) and the colour represents the number of reflections (black = no reflections, dark blue = 1 reflection, light blue = two reflections, green = 3 reflections, pink = 4 reflections and red = 6 reflections). The size of the shapes represent how consistently participants grouped members from the same wallpaper group together, and the thickness of the connecting lines represents how similar the different wallpaper groups are to each other. Note: only the top quartile of inter-group similarities are shown.

**Table 1.**A summary of the 17 possible wallpaper groups. Note, rotation centres lie on reflection axes except for groups $CMM$ and $P31M$. If a group has n rotations, this means that it is invariant under rotations of $2\pi /n$.

Group | #Reflections | #Rotations | #Glides | Notes |
---|---|---|---|---|

P1 | 0 | 1 | 0 | - |

P2 | 0 | 2 | 0 | - |

PM | 1 | 1 | 0 | - |

PG | 0 | 1 | 1 | - |

CM | 2 | 1 | 2 | Glides parallel to ref. axes |

PMM | 2 | 2 | 0 | - |

PMG | 1 | 2 | 1 | Glide perpendicular to ref. axes |

PGG | 0 | 2 | 2 | - |

CMM | 2 | 2 | 0 | Rotation centres do not lie on ref. axes |

P4 | - | 4 | 0 | - |

P4M | 4 | 4 | 0 | - |

P4G | 2 | 4 | 2 | - |

P3 | - | 3 | 0 | - |

P3M1 | 3 | 3 | 3 | - |

P31M | 3 | 3 | 3 | Rotation centres do not lie on ref. axes |

P6 | - | 6 | 0 | - |

P6M | 6 | 6 | 6 | - |

© 2011 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 license (http://creativecommons.org/licenses/by/3.0/.).

## Share and Cite

**MDPI and ACS Style**

Clarke, A.D.F.; Green, P.R.; Halley, F.; Chantler, M.J.
Similar Symmetries: The Role of Wallpaper Groups in Perceptual Texture Similarity. *Symmetry* **2011**, *3*, 246-264.
https://doi.org/10.3390/sym3020246

**AMA Style**

Clarke ADF, Green PR, Halley F, Chantler MJ.
Similar Symmetries: The Role of Wallpaper Groups in Perceptual Texture Similarity. *Symmetry*. 2011; 3(2):246-264.
https://doi.org/10.3390/sym3020246

**Chicago/Turabian Style**

Clarke, Alasdair D. F., Patrick R. Green, Fraser Halley, and Mike J. Chantler.
2011. "Similar Symmetries: The Role of Wallpaper Groups in Perceptual Texture Similarity" *Symmetry* 3, no. 2: 246-264.
https://doi.org/10.3390/sym3020246