# The Influence of Perception on the Distribution of Multiple Symmetries in Nature and Art

## Abstract

**:**

## 1. Introduction

## 2. Evolutionary Considerations

## 3. Multiple Symmetry in Nature and Art

#### 3.1. Flowers

#### 3.2. Human Designs

## 4. Models of Human Symmetry Perception

#### 4.1. The Transformational Approach

#### 4.2. The Bootstrap Model

#### 4.3. The Holographic Approach

## 5. Multiple Symmetry Perception

## 6. Discussion and Conclusions

## References

- Allen, G. The origin of the sense of symmetry. Mind
**1879**, 4, 301–316. [Google Scholar] [CrossRef] - Hargittai, I. Symmetry: Unifying Human Understanding; Pergamon: New York, NY, USA, 1986. [Google Scholar]
- Shubnikov, A.V.; Koptsik, V.A. Symmetry in Science and Art; Plenum: New York, NY, USA, 1974. [Google Scholar]
- Weyl, H. Symmetry; Princeton University Press: Princeton, NJ, USA, 1952. [Google Scholar]
- Grammer, K.; Thornhill, R. Human (Homo sapiens) facial attractiveness and sexual selection: The role of symmetry and averageness. J. Comp. Psychol.
**1994**, 108, 233–242. [Google Scholar] [CrossRef] [PubMed] - Johnstone, R.A. Female preferences for symmetrical males as a by-product of selection for mate recognition. Nature
**1994**, 372, 172–175. [Google Scholar] [CrossRef] [PubMed] - Møller, A.P. Female swallow preference for symmetrical male sexual ornaments. Nature
**1992**, 357, 238–240. [Google Scholar] [CrossRef] [PubMed] - Møller, A.P. Bumblebee preference for symmetrical flowers. Proc. Natl. Acad. Sci. USA
**1995**, 92, 2288–2292. [Google Scholar] [CrossRef] [PubMed] - Swaddle, J.; Cuthill, I.C. Preference for symmetric males by female zebra finches. Nature
**1993**, 367, 165–166. [Google Scholar] [CrossRef] - Thornhill, R.; Gangestad, S.W. Human fluctuating asymmetry and sexual behavior. Psychol. Sci.
**1994**, 5, 297–302. [Google Scholar] [CrossRef] - Watson, P.J.; Thornhill, R. Fluctuating asymmetry and sexual selection. Trend. in Ecol. Evolut.
**1994**, 9, 21–25. [Google Scholar] [CrossRef] - Pashler, H. Coordinate frame for symmetry detection and object recognition. J. Exp. Psychol. Hum. Percept. Perform.
**1990**, 16, 150–163. [Google Scholar] [CrossRef] - Vetter, T.; Poggio, T. Symmetric 3D objects are an easy case for 2D object recognition. Spatial Vision
**1994**, 8, 443–453. [Google Scholar] - Driver, J.; Baylis, G.C.; Rafal, R.D. Preserved figure-ground segregation and symmetry perception in visual neglect. Nature
**1992**, 360, 73–75. [Google Scholar] [CrossRef] [PubMed] - Leeuwenberg, E.L.J.; Buffart, H.F.J.M. The perception of foreground and background as derived from structural information theory. Acta Psychol.
**1984**, 55, 249–272. [Google Scholar] [CrossRef] - Machilsen, B.; Pauwels, M.; Wagemans, J. The role of vertical mirror symmetry in visual shape detection. J. Vision
**2009**, 9, 1–11. [Google Scholar] [CrossRef] [PubMed] - Kanizsa, G. Seeing and thinking. Acta Psychol.
**1985**, 59, 23–33. [Google Scholar] [CrossRef] - van Lier, R.J.; van der Helm, P.A.; Leeuwenberg, E.L.J. Competing global and local completions in visual occlusion. J. Exp. Psychol. Hum. Percept. Perform.
**1995**, 21, 571–583. [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] - Tyler, C.W. Human Symmetry Perception and Its Computational Analysis; Tyler, C.W., Ed.; VSP: Zeist, The Netherlands, 1996; pp. 3–22. [Google Scholar]
- van der Helm, P.A.; Leeuwenberg, E.L.J. Goodness of visual regularities: A nontransformational approach. Psychol. Rev.
**1996**, 103, 429–456. [Google Scholar] [CrossRef] - Wagemans, J. Characteristics and models of human symmetry detection. Trends Cogn. Sci.
**1997**, 1, 346–352. [Google Scholar] [CrossRef] - Barlow, H.B.; Reeves, B.C. The versatility and absolute efficiency of detecting mirror symmetry in random dot displays. Vision Res.
**1979**, 19, 783–793. [Google Scholar] [CrossRef] - Giurfa, M.; Eichmann, B.; Menzel, R. Symmetry perception in an insect. Nature
**1996**, 382, 458–461. [Google Scholar] [CrossRef] - Horridge, G.A. The honeybee (Apis mellifera) detects bilateral symmetry and discriminates its axis. J. Insect Physiol.
**1996**, 42, 755–764. [Google Scholar] [CrossRef] - van der Helm, P.A. Weber-Fechner behaviour in symmetry perception? Atten. Percept. Psychophys.
**2010**, 72, 1854–1864. [Google Scholar] [CrossRef] [PubMed] - Møller, A.P. Fluctuating asymmetry in male sexual ornaments may reliably reveal male quality. Anim. Behav.
**1990**, 40, 1185–1187. [Google Scholar] [CrossRef] - Thompson, D.W. On Growth and Form; University Press: Cambridge, UK, 1942; (Original work published 1917). [Google Scholar]
- Enquist, M.; Arak, A. Symmetry, beauty and evolution. Nature
**1994**, 372, 169–172. [Google Scholar] [CrossRef] [PubMed] - Heywood, V.H. Flowering Plants of the World; Batsford: London, UK, 1993. [Google Scholar]
- Endress, P.K. Floral phyllotaxis and floral evolution. Bot. Jahrb.
**1987**, 108, 417–438. [Google Scholar] - Giurfa, M.; Dafni, A.; Neal, P.R. Floral symmetry and its role in plant-pollinator systems. Int. J. Plant Sci.
**1999**, 160, S41–S50. [Google Scholar] [CrossRef] [PubMed] - Neal, P.R.; Dafni, A.; Giurfa, M. Floral symmetry and its role in plant-pollinator systems: Terminology, Distribution, and Hypotheses. Annu. Rev. Ecol. Syst.
**1998**, 29, 345–373. [Google Scholar] [CrossRef] - Horridge, G.A. Visual discrimination of radial cues by the honeybee (Apis mellifera). J. Insect Physiol.
**2000**, 46, 629–645. [Google Scholar] [CrossRef] - Washburn, D.K.; Crowe, D.W. Symmetries of Culture: Theory and Practice of Plane Pattern Analysis; University of Washington Press: Seattle, WA, USA, 1988. [Google Scholar]
- Wynn, T. Archaeology and cognitive evolution. Behav. Brain Sci.
**2002**, 25, 389–402, 432–438. [Google Scholar] [CrossRef] - Hardonk, M. Cross-cultural Universals of Aesthetic Appreciation in Decorative Band Patterns. Ph.D. Thesis, Radboud University, Nijmegen, The Netherlands, 1999. [Google Scholar]
- Forstner, D. Die Welt der Symbole; Tyriola Verlag: Innsbruck, 1961. [Google Scholar]
- Labat, R. Manuel D’épigraphie Akkadienne: Signes, Syllabaire, Idéogrammes, 6th ed.; Imprimerie Nationale: Paris, France, 1988. [Google Scholar]
- Boselie, F. The golden section and the shape of objects. Empir. Stud. Arts.
**1997**, 15, 131–141. [Google Scholar] [CrossRef] - Garner, W.R. The Processing of Information and Structure; Erlbaum: Potomac, MD, USA, 1974. [Google Scholar]
- Palmer, S.E. Human and Machine Vision; Beck, J., Hope, B., Rosenfeld, A., Eds.; Academic Press: New York, NY, USA, 1983; pp. 269–339. [Google Scholar]
- Palmer, S.E.; Hemenway, K. Orientation and symmetry: Effects of multiple, rotational, and near symmetries. J. Exp. Psychol. Hum. Percept. Perform.
**1978**, 4, 691–702. [Google Scholar] [CrossRef] [PubMed] - Royer, F.L. Detection of symmetry. J. Exp. Psychol. Hum. Percept. Perform.
**1981**, 7, 1186–1210. [Google Scholar] [CrossRef] [PubMed] - Wagemans, J.; van Gool, L.; Swinnen, V.; van Horebeek, J. Higher-order structure in regularity detection. Vision Res.
**1993**, 33, 1067–1088. [Google Scholar] [CrossRef] - Viola, C.M. Grundzüge der Kristallographie; W. Engelman: Leipzig, Germany, 1904. [Google Scholar]
- Jenkins, B. Component processes in the perception of bilaterally symmetric dot textures. Percept. Psychophys.
**1983**, 34, 433–440. [Google Scholar] [CrossRef] [PubMed] - Wagemans, J.; van Gool, L.; d’Ydewalle, G. Detection of symmetry in tachistoscopically presented dot patterns: Effects of multiple axes and skewing. Percept. Psychophys.
**1991**, 50, 413–427. [Google Scholar] [CrossRef] [PubMed] - van der Helm, P.A.; Leeuwenberg, E.L.J. A better approach to goodness: Reply to Wagemans (1999). Psychol. Rev.
**1999**, 106, 622–630. [Google Scholar] [CrossRef] - Treder, M.S.; van der Vloed, G.; van der Helm, P.A. Interactions between constituent single symmetries in multiple symmetry. Atten. Percept. Psychophys.
**2011**. In Press. [Google Scholar] [CrossRef] - van der Helm, P.A.; Leeuwenberg, E.L.J. Holographic goodness is not that bad: Reply to Olivers, Chater, and Watson (2004). Psychol. Rev.
**2004**, 111, 261–273. [Google Scholar] [CrossRef] - van der Vloed, G.; Csathó, Á.; van der Helm, P.A. Symmetry and repetition in perspective. Acta Psychol.
**2005**, 120, 74–92. [Google Scholar] [CrossRef] - van der Helm, P.A.; Leeuwenberg, E.L.J. Accessibility, a criterion for regularity and hierarchy in visual pattern codes. J. Math. Psychol.
**1991**, 35, 151–213. [Google Scholar] [CrossRef] - van der Helm, P.A.; Treder, M.S. Detection of (anti)symmetry and (anti)repetition: Perceptual mechanisms versus cognitive strategies. Vision Res.
**2009**, 49, 2754–2763. [Google Scholar] [CrossRef] [PubMed] - Treder, M.S.; van der Helm, P.A. Symmetry versus repetition in cyclopean vision: A microgenetic analysis. Vision Res.
**2007**, 47, 2956–2967. [Google Scholar] [CrossRef] [PubMed] - Csathó, Á.; van der Vloed, G.; van der Helm, P.A. Blobs strengthen repetition but weaken symmetry. Vision Res.
**2003**, 43, 993–1007. [Google Scholar] [CrossRef] - Csathó, Á.; van der Vloed, G.; van der Helm, P.A. The force of symmetry revisited: Symmetry-to-noise ratios regulate (a)symmetry effects. Acta Psychol.
**2004**, 117, 233–250. [Google Scholar] [CrossRef] [PubMed] - Attneave, F. Some informational aspects of visual perception. Psychol. Rev.
**1954**, 61, 183–193. [Google Scholar] [CrossRef] [PubMed] - Attneave, F. Symmetry, information, and memory for patterns. Am. J. Psychol.
**1955**, 68, 209–222. [Google Scholar] [CrossRef] [PubMed] - Barlow, H.B. Current Problems in Animal Behaviour; Thorpe, W.H., Zangwill, O.L., Eds.; Cambridge University Press: Cambridge, UK, 1961; pp. 331–360. [Google Scholar]
- Barlow, H. The exploitation of regularities in the environment by the brain. Behav. Brain. Sci.
**2001**, 24, 602–607, 652–671. [Google Scholar] [CrossRef] - Leeuwenberg, E.L.J. Quantitative specification of information in sequential patterns. Psychol. Rev.
**1969**, 76, 216–220. [Google Scholar] [CrossRef] - Leeuwenberg, E.L.J. A perceptual coding language for visual and auditory patterns. Amer. J. Psychol.
**1971**, 84, 307–349. [Google Scholar] [CrossRef] - Nucci, M.; Wagemans, J. Goodness of regularity in dot patterns: global symmetry, local symmetry, and their interactions. Perception
**2007**, 36, 1305–1319. [Google Scholar] [CrossRef] - Hamada, J.; Ishihara, T. Complexity and goodness of dot patterns varying in symmetry. Psychol. Res.
**1988**, 50, 155–161. [Google Scholar] [CrossRef] [PubMed] - Wenderoth, P.; Welsh, S. Effects of pattern orientation and number of symmetry axes on the detection of mirror symmetry in dot and solid patterns. Perception
**1998**, 27, 965–976. [Google Scholar] [CrossRef] [PubMed] - van der Vloed, G. The Structure of Visual Regularities. Ph.D. Thesis, Radboud University, Nijmegen, The Netherlands, 2005. [Google Scholar]
- Beh, H.C.; Latimer, C.R. Symmetry detection and orientation perception: Electrocortical responses to stimuli with real and implicit axes of orientation. Aust. J. Psychol.
**1997**, 49, 128–133. [Google Scholar] [CrossRef] - Gurnsey, R.; Herbert, A.M.; Kenemy, J. Bilateral symmetry embedded in noise is detected accurately only at fixation. Vision Res.
**1998**, 38, 3795–3803. [Google Scholar] [CrossRef] - Lee, T.S.; Mumford, D.; Romero, R.; Lamme, V.A.F. The role of the primary visual cortex in higher level vision. Vision Res.
**1998**, 38, 2429–2454. [Google Scholar] [CrossRef] - Sally, S.; Gurnsey, R. Symmetry detection across the visual field. Spatial Vision
**2001**, 14, 217–234. [Google Scholar] [PubMed] - Joung, W.; Latimer, C. Tilt aftereffects generated by symmetrical dot patterns with two or four axes of symmetry. Spatial Vision
**2003**, 16, 155–182. [Google Scholar] [CrossRef] - Joung, W.; van der Zwan, R.; Latimer, C.R. Tilt aftereffects generated by bilaterally symmetrical patterns. Spatial Vision
**2000**, 13, 107–128. [Google Scholar] - van der Zwan, R.; Leo, E.; Joung, W.; Latimer, C.; Wenderoth, P. Evidence that both area V1 and extrastriate visual cortex contribute to symmetry perception. Curr. Biol.
**1998**, 8, 889–892. [Google Scholar] [CrossRef] - Koffka, K. Principles of Gestalt Psychology; Routledge & Kegan Paul: London, UK, 1935. [Google Scholar]
- van der Helm, P.A. Simplicity versus likelihood in visual perception: From surprisals to precisals. Psychol. Bull.
**2000**, 126, 770–800. [Google Scholar] [CrossRef] - Birkhoff, G.D. Aesthetic Measure; Harvard University Press: Cambridge, MA, UK, 1933. [Google Scholar]
- Boselie, F.; Leeuwenberg, E.L.J. Birkhoff revisited: Beauty as a function of effect and means. Amer. J. Psychol.
**1985**, 98, 1–39. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Evolutionary factors relevant to a high perceptual sensitivity to symmetry. The dashed arrows do not indicate direct causation—they merely indicate that their starting terms provide survival value to their end terms which must have been caused by something else.

**Figure 2.**Natural selection mechanism, according to which a high perceptual sensitivity to symmetry is not the consequence but the cause of the symmetry preference in mate assessment. The separate functionalities of symmetry regarding genetic quality and object recognition then are factors favourable towards the survival of such a visual system.

**Figure 3.**In the plant family of monocotyledons, 80% of the mostly 2-fold and 3-fold symmetrical flowers is 3-fold symmetrical, and in the plant family of dicotyledons, 70% of the mostly 4-fold and 5-fold symmetrical flowers is 5-fold symmetrical.

**Figure 4.**Decorative bands. The inset shows a decorative band consisting of a repetition of a 1-fold symmetrical motif. The histogram covers about 600 decorative bands containing motifs with 1–8 symmetry axes [37]. The dashed line indicates the rule—with notable exceptions—that motifs with more symmetry axes occur less often.

**Figure 5.**Motifs used often in mystical art to symbolize supernatural powers; (a) The 3-fold symmetrical triqueta; (b) The 5-fold symmetrical pentagram.

**Figure 6.**Proposed perceptual structures of symmetry and repetition (see text for details); (a) The transformational approach specifies visual regularities by invariance under rotations and translations, implying a block structure for both symmetry and repetition; (b) The bootstrap model starts from shared properties of virtual lines between corresponding elements, implying a point structure for both symmetry and repetition; (c) The holographic approach specifies these regularities by invariance under growth (i.e., under expansion by symmetry pairs or repeats), implying a point structure for symmetry and a block structure for repetition.

**Figure 7.**Predicted goodness (i.e., salience, or detectability) of n-fold symmetries with $n=1-8$. The dashed line indicates the predictions $1-1/\left(2n\right)$ by the transformational approach; the dots indicate the predictions by the holographic approach.

**Figure 8.**Symmetry detection according to the bootstrap model; (a) In 1-fold symmetry, virtual lines between corresponding elements form correlation trapezoids, which serve as anchors for the detection process; (b) In 2-fold symmetry, some of these trapezoids are rectangles, which are proposed to boost the detection process; (c) In 3-fold symmetry, rectangles do not occur, so that the detection process can only start from trapezoids.

**Figure 9.**Schematic overview of Treder et al.’s [50] experimental conditions involving equally noisy 2-fold symmetries with and without correlation rectangles (the actual stimuli contained dots only; here, in both cases, correlation trapezoids are present but not highlighted); (a) Fifty percent of the elements are noise about both axes, and the other 50% are symmetrical about both axes, so that there are both correlation trapezoids and correlation rectangles; (b) Fifty percent of the elements are symmetrical about one axis but noise about the other axis, and vice versa for the other 50%, so that there are trapezoids but no rectangles.

**Figure 10.**Representational information reduction by exploiting symmetry; (a) In 2-fold symmetry, one global symmetry can be exploited to represent the pattern by the information in one symmetry half, after which the remaining local symmetry can be exploited to further reduce information; (b) In 3-fold symmetry, after exploiting a global symmetry, only one of the remaining local symmetries can be exploited to further reduce information; (c) A 3-fold symmetry can be represented more efficiently by a repetition of three local symmetries.

**Figure 11.**Schematic overview of Nucci and Wagemans’ [64] experimental conditions combining global and local symmetries (the actual stimuli were dot stimuli), and the associated holographic weights of evidence (W) for the regularities. In the second condition, each of the two flanking local symmetries has $W=0.25$, which does not sum to $W=0.50$: they constitute noise to one another, so that considering the task, this condition hardly differs from the first condition with one local symmetry flanked by noise.

**Figure 12.**Examples of Treder et al.’s [50] equally noisy 2-fold symmetrical stimuli, constructed by superimposing two perfect 1-fold symmetries in different relative orientations (absolute orientations are given in the insets); (a) Two superimposed dot patterns; (b) Two superimposed Gaussian noise patterns, filtered in the spatial frequency (SF) domain to a 1-octave low-SF band and a 1-octave high-SF band, respectively, with one octave in between (in the insets, thick lines refer to the low-SF band and thin lines to the high-SF band).

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**MDPI and ACS Style**

Van der Helm, P.A.
The Influence of Perception on the Distribution of Multiple Symmetries in Nature and Art. *Symmetry* **2011**, *3*, 54-71.
https://doi.org/10.3390/sym3010054

**AMA Style**

Van der Helm PA.
The Influence of Perception on the Distribution of Multiple Symmetries in Nature and Art. *Symmetry*. 2011; 3(1):54-71.
https://doi.org/10.3390/sym3010054

**Chicago/Turabian Style**

Van der Helm, Peter A.
2011. "The Influence of Perception on the Distribution of Multiple Symmetries in Nature and Art" *Symmetry* 3, no. 1: 54-71.
https://doi.org/10.3390/sym3010054