# Visual Discrimination of the 17 Plane Symmetry Groups

## Abstract

**:**

## 1. Symmetry in the Visual Arts

## 2. The 17 Plane Symmetry Groups

## 3. Discrimination versus Identification

## 4. Experiments

#### 4.1. Experimental Paradigm

#### 4.2. Observers

#### 4.3. Stimuli and Procedure

#### 4.4. Results

_{Exp.}

_{1}= 1.27; d’

_{Exp.}

_{2_target}= 1.09), but was much worse for the catch trials of Experiment 2 (d’

_{Exp.}

_{2_catch}= 0.63). Effects concerning individual symmetry groups were tested via z-transformed binomial coefficients [47]. While the discrimination between p1 and p2 was always unproblematic, only in Experiment 1 could pg as a target reliably be discriminated from p1. pm catch trials were generally responded to incorrectly, and in Experiment 1, there was a trend for pm to go unnoticed in the neighbourhood of p1 distractors, whereas p1 targets were never missed between pm distractors.

#### 4.5. Discussion

## 5. Conclusions and Outlook

## Acknowledgements

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**Figure 1.**Illustrations of the 17 plane symmetry groups with mono- and multihedral tilings. Shaded areas are translating unit cells. Black lines are axes of mirror reflection, dashed lines are axes of glide reflection. Small rhombs, triangles, squares, and hexagons denote centers of rotation around π,

^{2π}/

_{3},

^{π}/

_{2}, and

^{π}/

_{3}, respectively. Icons are in black when rotation centers lie on axes of mirror reflection, where they locally define n-hedral symmetry groups. Per group, lower case letters indicate—as explained in the preceding text—nonequivalent centers of rotation, nonequivalent axes of mirror reflection, and nonequivalent axes of glide reflection, respectively (from [35], pp. 40–42; reproduced with permission from the authors).

**Figure 2.**Stimuli from Landwehr’s Experiment 2 [44]. (

**a**) The target is p2 (top right), distractors are p1; (

**b**) The target is p1 (bottom left), distractors are p2; (

**c**) The target is pg (bottom left), distractors are p1; (

**d**) The target is p1 (top right), distractors are pg.

**Figure 3.**Further examples of camouflage stimuli, illustrating issues of optimal construction. (

**a**) A catch trial stimulus, exhibiting pairwise similarity (all tiling patterns are pg); (

**b**) A misleading experimental stimulus (the target is pg, top right, distractors are p1, but most subjects chose the picture at top left as target).

**Table 1.**The 17 plane symmetry groups and the 5 types of unit cells (together with their respective symmetry groups) that can be used to realize the groups. The + sign indicates possible constructions. Grey shaded fields denote combinations that—with certain reservations—can be studied by means of traditional experimental methodology.

Unit cell Group | Parallelogram p2 | Rectangle pmm | Square p4m | Rhomb cmm | 2 equilateral triangles p6m |
---|---|---|---|---|---|

p1 | + | + | + | + | + |

pm | + | + | |||

pg | + | + | |||

cm | + | + | + | ||

p2 | + | + | + | + | + |

pmm | + | + | |||

pmg | + | + | |||

pgg | + | + | |||

cmm | + | + | + | ||

p4 | + | ||||

p4m | + | ||||

p4g | + | ||||

p3 | + | ||||

p3m1 | + | ||||

p31m | + | ||||

p6 | + | ||||

p6m | + |

© 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/).

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**MDPI and ACS Style**

Landwehr, K.
Visual Discrimination of the 17 Plane Symmetry Groups. *Symmetry* **2011**, *3*, 207-219.
https://doi.org/10.3390/sym3020207

**AMA Style**

Landwehr K.
Visual Discrimination of the 17 Plane Symmetry Groups. *Symmetry*. 2011; 3(2):207-219.
https://doi.org/10.3390/sym3020207

**Chicago/Turabian Style**

Landwehr, Klaus.
2011. "Visual Discrimination of the 17 Plane Symmetry Groups" *Symmetry* 3, no. 2: 207-219.
https://doi.org/10.3390/sym3020207