title: A method for quantifying rotational symmetry.
abstract: Here, a new approach for quantifying rotational symmetry based on vector analysis was described and compared with information obtained from a geometric morphometric analysis and a technique based on distance alone. A new method was developed that generates a polygon from the length and angle data of a structure and then quantifies the minimum change necessary to convert that polygon into a regular polygon. This technique yielded an asymmetry score (s) that can range from 0 (perfect symmetry) to 1 (complete asymmetry). Using digital images of Geranium robertianum flowers, this new method was compared with a technique based on lengths alone and with established geometric morphometric methods used to quantify shape variation. Asymmetry scores (s) more clearly described variation in symmetry and were more consistent with a visual assessment of the images than either comparative technique. This procedure is the first to quantify the asymmetry of radial structures accurately, uses easily obtainable measures to calculate the asymmetry score and allows comparisons among individuals and species, even when the comparisons involve structures with different patterns of symmetry. This technique enables the rigorous analysis of polysymmetric structures and provides a foundation for a better understanding of symmetry in nature.
authors: Departments of Biology and.
authors: Frank M
authors: FM
authors: Frey
authors: Mathematics, Colgate University, 13 Oak Drive, Hamilton, NY 13346, USA.
authors: Aaron
authors: A
authors: Robertson
authors: Departments of Biology and.
authors: Michael
authors: M
authors: Bukoski
mesh_terms: D005075:Biological Evolution; D035264:Flowers; D031315:Geranium / Q000033:anatomy & histology*; D008954:Models, Biological; D018515:Plant Leaves / Q000033:anatomy & histology*; D013045:Species Specificity
references: Adams DC, Rohlf FJ, Slice DE. 2004. Geometric morphometrics: ten years of progress following the ‘revolution’. Italian Journal of Zoology 71: 5-16.
references: Auffray J-C, Debat V, Alibert P. 1999. Shape asymmetry and developmental stability. In: Chaplain MAJ, Singh GD, McLachlan JC, eds. On growth and form: spatio-temporal pattern formation in biology. Chichester, UK: Wiley, 309-324.
references: Bookstein FL. 1991. Morphometric tools for landmark data: geometry and biology. Cambridge, UK: Cambridge University Press.
references: Dryden IL, Mardia KV. 1998. Statistical analysis of shape. Cambridge, UK: Wiley.
references: Endress PK. 1999. Symmetry in flowers: diversity and evolution. International Journal of Plant Sciences 160: S3-S23.
references: Frey FM, Davis R, Delph LF. 2005. Manipulation of floral symmetry does not affect seed production in Impatiens pallida. International Journal of Plant Sciences 166: 659-662.
references: Kent JT, Mardia KV. 2001. Shape, Procrustes tangent projections and bilateral symmetry. Biometrika 88: 469-485.
references: Klingenberg CP. 2002. Morphometrics and the role of the phenotype in studies of the evolution of developmental mechanisms. Gene 287: 3-10.
references: Klingenberg CP. 2003. Quantitative genetics of geometric shape: heritability and the pitfalls of the univariate approach. Evolution 57: 191-195.
references: Klingenberg CP, Leamy LJ. 2001. Quantitative genetics of geometric shape in the mouse mandible. Evolution 55: 2342-2352.
references: Klingenberg CP, McIntyre GS. 1998. Geometric morphometrics of developmental instability: analyzing patterns of fluctuating asymmetry with Procrustes methods. Evolution 52: 1363-1375.
references: Klingenberg CP, Barluenga M, Meyer A. 2002. Shape analysis of symmetric structures: quantifying variation among individuals and asymmetry. Evolution 56: 1909-1920.
references: Malo JE, Leirana-Alcocer J, Parra-Tabla V. 2001. Population fragmentation, florivory, and the effects of flower morphology alterations on the pollination success of Myrmecophila tibicinis (Orchidaceae). Biotropica 33: 529-534.
references: Mardia KV, Bookstein FL, Moreton IJ. 2000. Statistical assessment of bilateral symmetry of shapes. Biometrika 87: 285-300.
references: Midgley JJ, Johnson SD. 1998. Some pollinators do not prefer symmetrically marked or shaped daisy (Asteraceae) flowers. Evolutionary Ecology 12: 123-126.
references: Møller AP. 1995. Bumblebee preference for symmetrical flowers. Proceedings of the National Academy of Sciences, USA 92: 2288-2292.
references: Møller AP, Eriksson M. 1995. Pollinator preference for symmetrical flowers and sexual selection in plants. Oikos 73: 15-22.
references: Møller AP, Sorci G. 1998. Insect preference for symmetrical artificial flowers. Oecologia 114: 37-42.
references: Neal PR. 1998. Floral symmetry and its role in plant-pollinator systems. terminology, distribution, and hypotheses. Annual Review of Ecology and Systematics 29: 345-373.
references: Palmer AR, Strobeck C. 1986. Fluctuating asymmetry: measurement, analysis, patterns. Annual Review of Ecology and Systematics 17: 391-421.
references: Rohlf FJ, Slice D. 1990. Extensions of the Procrustes method for the optimal superimposition of landmarks. Systematic Zoology 39: 40-59.
references: Slice D. 2001. Landmark coordinates aligned by Procrustes analysis do not lie in Kendall's shape space. Systematic Biology 50: 141-149.
references: Walker JA. 2000. The ability of geometric morphometric methods to estimate a known covariance matrix. Systematic Biology 49: 686-696.
references: Weeks EL, Frey FM. 2007. Seed production and insect visitation rates in Hesperis matronalis are not affected by floral symmetry. International Journal of Plant Sciences 168: 611-617.
references: West EL, Laverty TM. 1998. Effect of floral symmetry on flower choice and foraging behaviour of bumblebees. Canadian Journal of Zoology 76: 730-739.
references: Zabrodsky H, Peleg S, Avnir D. 1992. Continuous symmetry measures. Journal of the American Chemical Society 115: 8278-8289.