Abstract
This paper studies the transition of phyllotactic patterns by a group-theoretic approach. Typical phyllotactic patterns are represented here as dotted patterns on a cylinder, where the cylinder is regarded as the stem of a plant and the dots are points where leaves branch from the stem. We can then classify the symmetries of the alternate and opposite phyllotaxis into four types of groups, and clarify sequences of symmetry-breaking among these groups. The sequences turn out to correspond to transition paths of phyllotactic patterns found in the wild. This result shows the usefulness of classification of phyllotactic patterns based on their group symmetries. Moreover, the breaking of reflection symmetry is found to be an important rule for real phyllotactic transitions.
Similar content being viewed by others
References
Baglivo, J. A. and J. E. Graver (1983). Incidence and Symmetry in Design and Architecture, Cambridge: Cambridge University Press, pp. 259–265.
Bell, A. D. (1991). Plant Form, UK: Oxford University Press.
Bernasconi, G. P. and J. Boissonade (1997). Phyllotactic order induced by symmetry breaking in advected Turing patterns. Phys. Lett. A 232, 224–230.
Carr, D. J. (1998). Systems of phyllotaxis in the genus Eucalyptus in relation to shoot architecture, in Symmetry in Plants, R. V. Jean and D. Barabe (Eds), Singapore: World Scientific, pp. 33–59.
Crawford, J. D. and E. Knobloch (1991). Symmetry and symmetry breaking bifurcations in fluid dynamics. Ann. Rev. Fluid Mech. 23, 341–382.
Douady, S. and Y. Couder (1992). Phyllotaxis as a physical self-organized growth process. Phys. Rev. Lett. 68, 2098–2101.
Douady, S. and Y. Couder (1996). Phyllotaxis as a dynamical self organizing process. Part I: the spiral modes resulting from time-periodic iterations. J. Theor. Biol. 178, 255–274; Phyllotaxis as a dynamical self organizing process. Part II: the spontaneous formation of a periodicity and the coexistence of spiral and whorled patterns. J. Theor. Biol. 178, 275–294; Phyllotaxis as a dynamical self organizing process. Part III: the simulation of the transient regimes of ontogeny. J. Theor. Biol. 178, 295–312.
Golubitsky, M. and I. Melbourne (1998). A symmetry classification of columns, in Bridges—Mathematical Connections in Art, Music, and Science, R. Sarhangi (Ed.), pp. 209–223.
Golubitsky, M., I. Stewart and D. G. Schaeffer (1985). Singularities and Groups in Bifurcation Theory, New York: Springer.
Jean, R. V. and D. Barabe (1998a). Phyllotaxis—the way ahead, a view on open questions and directions of research. J. Biol. System. 6, 95–126.
Jean, R. V. and D. Barabe (Eds), (1998b). Symmetry in Plants, Singapore: World Scientific.
Kumazawa, M. (1979). Plant Organs, Japan Shokabo, pp. 220–254 (in Japanese).
Maekawa, F. (1957). Form and evolution of phyllotaxis in higher plants. Kagaku 27, 231–235 (in Japanese).
Meinhardt, H. (1984). Models of pattern formation and their application to plant development, in Positional Controls in Plant Development, P. W. Barlow and D. J. Carr (Eds), Cambridge: Cambridge University Press, pp. 1–32.
Rutishauser, R. (1998). Plastochrone ratio and leaf arc as parameters of a quantitative phyllotaxis analysis in vascular plants, in Symmetry in Plants, R. V. Jean and D. Barabe (Eds), Singapore: World Scientific, pp. 171–212.
Saunders, P. T. (Ed.), (1992). Collected Works of A. M. Turing: Morphogenesis, Amsterdam: North-Holland.
Schwabe, W. W. (1984). Phyllotaxis, in Positional Controls in Plant Development, P. W. Barlow and D. J. Carr (Eds), Cambridge: Cambridge University Press, pp. 403–440.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yamada, H., Tanaka, R. & Nakagaki, T. Sequences of symmetry-breaking in phyllotactic transitions. Bull. Math. Biol. 66, 779–789 (2004). https://doi.org/10.1016/j.bulm.2003.10.006
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1016/j.bulm.2003.10.006