# 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

## References

- Hargittai, M.; Hargittai, I. Visual Symmetry; World Scientific Publishing: Singapore, 2009. [Google Scholar]
- Müller, C. Symmetrie und Ornament. Eine Analyse mathematischer Strukturen der darstellenden Kunst; Westdeutscher Verlag: Opladen, Germany, 1985. [Google Scholar]
- Washburn, D.K.; Crowe, D.S. Symmetries of Culture: Theory and Practice of Plane Pattern Analysis; University of Washington Press: Seattle, WA, USA, 1988. [Google Scholar]
- Jablan, S. Theory of Symmetry and Ornament; Matematički Institut: Beograd, Serbia, 1995. [Google Scholar]
- Barrucand, M.; Bednorz, A. Maurische Architektur in Andalusien; Taschen: Köln, Germany, 1992. [Google Scholar]
- Grünbaum, B.; Grünbaum, Z.; Shephard, G.C. Symmetry in Moorish and other ornaments. Comput. Math. Appl.
**1986**, 12B, 641–653. [Google Scholar] [CrossRef] - Burckhardt, J. Die Kultur der Renaissance in Italien—Ein Versuch; 12. Aufl. Kröner: Stuttgart, Germany, 2009. [Google Scholar]
- Grayson, C. On Painting and on Sculpture: The Latin Texts of de Pictura and de Statua by Leon Battista Alberti; Phaidon: London, UK, 1972. [Google Scholar]
- Edgerton, S.Y. The Renaissance Rediscovery of Linear Perspective; Basic Books: New York, NY, USA, 1975. [Google Scholar]
- Kubovy, M. The Psychology of Perspective and Renaissance Art; Cambridge University Press: Cambridge, UK, 1986. [Google Scholar]
- MacMillan, R.H. Pyramids and pavements: Some thoughts from Cairo. Math. Gaz.
**1979**, 251, 251–255. [Google Scholar] [CrossRef] - Hargittai, M. Symmetry, crystallography, and art. Appl. Phys. A
**2007**, 89, 889–898. [Google Scholar] [CrossRef] - Vitz, P.C. Visual science and modernist art: Historical parallels. In Perception and Pictorial Representation; Nodine, C.F., Fisher, D.F., Eds.; Praeger: New York, NY, USA, 1979; pp. 134–166. [Google Scholar]
- Vitz, P.C.; Glimcher, A. Modern Art and Modern Science: The Parallel Analysis of Vision; Praeger: New York, NY, USA, 1984. [Google Scholar]
- Kepes, G. Structure in Art and in Science; Braziller: New York, NY, USA, 1965. [Google Scholar]
- Kepes, G. Module, Proportion, Symmetry, Rhythm; Braziller: New York, NY, USA, 1966. [Google Scholar]
- Peitgen, H.O.; Richter, P.H. The Beauty of Fractals. Images of Complex Dynamical Systems; Springer: Berlin, Germany, 1986. [Google Scholar]
- Burckhardt, J.J. Zur Geschichte der Entdeckung der 230 Raumgruppen. Arch. Hist. Exact Sci.
**1967–1968**, 4, 235–246. [Google Scholar] [CrossRef] - Niggli, P. Die regelmäßige Punktverteilung längs einer Geraden in einer Ebene (Symmetrie von Bordürenmuster). Z. Kristallogr.
**1926**, 63, 255–274. [Google Scholar] [CrossRef] - Pólya, G. Über die Analogie der Kristallsymmetrie in der Ebene. Z. Kristallogr.
**1924**, 60, 278–282. [Google Scholar] [CrossRef] - Coxeter, H.S.M. Introduction to Geometry, 2nd ed.; Wiley: New York, NY, USA, 1969. [Google Scholar]
- Locher, J.L. Leven en werk van M.C. Escher; Meulenhoff: Amsterdam, The Netherlands, 1981. [Google Scholar]
- Coxeter, H.S.M.; Emmer, M.; Penrose, R.; Teuber, M.L. M.C. Escher: Art and Science; North-Holland: Amsterdam, The Netherlands, 1986. [Google Scholar]
- Escher, M.C. Regelmatige vlakverdeling; Stichting de Roos: Utrecht, The Netherlands, 1958. [Google Scholar]
- Schattschneider, D. Visions of Symmetry. Notebooks, Periodic Drawings, and Related Work of M.C. Escher; Freeman: New York, NY, USA, 1990. [Google Scholar]
- Lepsky, S. M.C. Escher—Eine Randfigur der Kunstgeschichte; (2 Bde.); Doctoral Dissertation: Aachen, Germany, 1992. [Google Scholar]
- Schattschneider, D.; Emmer, M. M.C. Escher’s Legacy. A Centennial Celebration; Springer: Berlin, Germany, 2003. [Google Scholar]
- Landwehr, K. Nonperiodicity. In Psychology of Beauty and Kansei: New Horizons of Gestalt Perception; Noguchi, K., Ed.; Fuzambo: Tokyo, Japan, 2007; pp. 731–744. [Google Scholar]
- Coxeter, H.S.M. Coloured symmetry. In M.C. Escher: Art and Science; Coxeter, H.S.M., Emmer, M., Penrose, R., Teuber, M.L., Eds.; North-Holland: Amsterdam, The Netherlands, 1986; pp. 15–33. [Google Scholar]
- Shephard, G.C. What Escher might have done. In M.C. Escher: Art and Science; Coxeter, H.S.M., Emmer, M., Penrose, R., Teuber, M.L., Eds.; North-Holland: Amsterdam, The Netherlands, 1986; pp. 111–122. [Google Scholar]
- Radovic, L.; Jablan, S. Antisymmetry and modularity in ornamental art, not dated. Available online: http://www.mi.sanu.ac.rs/vismath/radovic/index.html (accessed on 4 April 2011).
- Martin, G.E. Transformation Geometry. An Introduction to Symmetry; Springer: Berlin, Germany, 1982. [Google Scholar]
- Schattschneider, D. The plane symmetry groups: Their recognition and notation. Am. Math. Mon.
**1978**, 85, 439–450. [Google Scholar] [CrossRef] - Escher, M.C. Grafiek en tekeningen; Tijl: Zwolle, The Netherlands, 1959. [Google Scholar]
- Grünbaum, B.; Shephard, G.C. Tilings and Patterns; Freeman: New York, NY, USA, 1987. [Google Scholar]
- Garner, W.R.; Hake, H.W.; Eriksen, C.W. Operationism and the concept of perception. Psychol. Rev.
**1956**, 63, 149–159. [Google Scholar] [CrossRef] [PubMed] - Bridgman, P.W. The Logic of Modern Physics; Macmillan: London, UK, 1927. [Google Scholar]
- Skinner, B.F. The operational analysis of psychological terms. Psychol. Rev.
**1945**, 52, 270–277. [Google Scholar] [CrossRef] - Hulme, O.J.; Friston, K.F.; Zeki, S. Neural correlates of stimulus reportability. J. Cognit. Neurosci.
**2009**, 21, 1602–1610. [Google Scholar] [CrossRef] [PubMed] - Macmillan, N.A.; Creelman, C.D. Detection Theory: A User’s Guide, 2nd ed.; Erlbaum: Mahwah, NJ, USA, 2005. [Google Scholar]
- Green, D.M.; Swets, J.A. Signal Detection Theory and Psychophysics; Wiley: New York, NY, USA, 1966. [Google Scholar]
- Wickens, T.D. Elementary Signal Detection Theory; Oxford University Press: New York, NY, USA, 2002. [Google Scholar]
- Elder, J.H.; Velisavljević, L. Cue dynamics underlying rapid detection of animals in natural scenes. J. Vis.
**2009**, 9, 1–20. [Google Scholar] [CrossRef] [PubMed] - Landwehr, K. Camouflaged symmetry. Perception
**2009**, 38, 1712–1720. [Google Scholar] [CrossRef] [PubMed] - Gibson, J.J. The Perception of the Visual World; Houghton-Mifflin: Boston, MA, USA, 1950. [Google Scholar]
- Stevens, P.S. Handbook of Regular Patterns. An Introduction to Symmetry in Two Dimensions; MIT Press: Cambridge, MA, USA, 1981. [Google Scholar]
- Siegel, S. Nonparametric Statistics for the Behavioral Sciences; McGraw-Hill: New York, NY, USA, 1956. [Google Scholar]
- Wagemans, J. Detection of visual symmetries. Spat. Vis.
**1995**, 9, 9–32. [Google Scholar] [CrossRef] [PubMed] - Treder, M.S. Behind the looking-glass: A review on human symmetry perception. Symmetry
**2010**, 2, 1510–1543. [Google Scholar] [CrossRef]

**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/).

## Share and Cite

**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