Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2016 | 85 | 4 |
Tytuł artykułu

Fibonacci or quasi-symmetric phyllotaxis. Part II: botanical observations

Treść / Zawartość
Warianty tytułu
Języki publikacji
Historically, the study of phyllotaxis was greatly helped by the simple description of botanical patterns by only two integer numbers, namely the number of helices (parastichies) in each direction tiling the plant stem. The use of parastichy numbers reduced the complexity of the study and created a proliferation of generalizations, among others the simple geometric model of lattices. Unfortunately, these simple descriptive method runs into difficulties when dealing with patterns presenting transitions or irregularities. Here, we propose several ways of addressing the imperfections of botanical reality. Using ontogenetic analysis, which follows the step-by-step genesis of the pattern, and crystallographic analysis, which reveal irregularity in its details, we show how to derive more information from a real botanical sample, in particular, about its irregularities and transitions. We present several examples, from the first explicit visualization of a normal Fibonacci parastichy number increase, to more exotic ones, including the quasi-symmetric patterns detected in simulations. We compare these observations qualitatively with the result of the disk-packing model, presenting evidence for the relevance of the model.
Słowa kluczowe
Opis fizyczny
Article 3534 [21p.], fig.,ref.
  • UMR 7057 Universite Paris Diderot - CNRS, Batiment Condorcet, CC 7057, 10 rue Alice Domon et Leonie Duquet, 75013 Paris, France
  • Department of Mathematics, Smith College, Northampton, MA 01063, USA
  • 1. Golé C, Dumais J, Douady S. Fibonacci or quasi-symmetric. Part I: why? Acta Soc Bot Pol. 2016;85(4):3533.
  • 2. Hotton S, Johnson V, Wilbarger J, Zwieniecki K, Atela P, Golé C, et al. The possible and the actual in phyllotaxis: bridging the gap between empirical observations and iterative models. J Plant Growth Regul. 2006;25:313–323.
  • 3. Fierz V. Phyllotactic patterns in cones of conifers. Acta Soc Bot Pol. 2015;84(2):261–265.
  • 4. Guédon Y, Refahi Y, Besnard F, Godin C, Vernoux, T. Pattern identification and characterization reveal permutations of organs as a key genetically controlled property of post-meristematic phyllotaxis, J Theor Biol. 2013;338:94–110.
  • 5. Hamant, O, Heisler MG, JÖnsson H, Krupinski P, Uyttewaal M, Bokov P, et al. Developmental patterning by mechanical signals in Arabidopsis. Science. 2008;322:1650–1655.
  • 6. Douady S, Couder Y. Phyllotaxis as a self organizing iterative process, Part III: the simulation of the transient regimes of ontogeny. J Theor Biol. 1996;178:295–312.
  • 7. Zagórska-Marek B. Phyllotaxis triangular unit; phyllotactic transitions as the consequences of the apical wedge disclinations in a crystal-like pattern of the units. Acta Soc Bot Pol. 1987;56:229–255.
  • 8. Atela P, Golé C. Rhombic tilings and primordia fronts of phyllotaxis [Preprint]. 2007 [cited 2016 Dec 30]. Available from:
  • 9. Plantefol L. La théorie des hélices foliaires multiples. Paris: Masson; 1948.
  • 10. Meicenheimer RD. Role of parenchyma in Linum usitatissimum leaf trace patterns. Am J Bot. 1986;73(12):1649–1664.
  • 11. Zagórska-Marek B. Phyllotaxic diversity of Magnolia flowers. Acta Soc Bot Pol. 1994;62(2):117–137.
  • 12. Sadoc JF, Rivier N, Charvolin J. Phyllotaxis: a non conventional crystalline solution to packing efficiency in situations with radial symmetry [Preprint]. 2012 [cited 2016 Dec 30]. Available from:
  • 13. Rivier N, Sadoc JF, Charvolin J. Phyllotaxis: a framework for foam topological evolution The European Physical Journal E. 2016;39:7.
  • 14. van Iterson G. Mathematische und mikroskopisch-anatomische Studien über Blattstellungen nebst Betrachtungen über den Schalenbau der Miliolinen. Jena: Gustav Fischer Verlag; 1907.
  • 15. Adler I. A model of contact pressure in phyllotaxis. J Theor Biol. 1974;45:1–79.
  • 16. Douady S. The selection of phyllotactic patterns. In: Jean RV, Barabé D, editors. Symmetry in plants. Singapore: World Scientific; 1998. p. 335–358. (Series in Mathematical Biology and Medicine; vol 4).
  • 17. Atela P, Golé C, Hotton S. A dynamical system for plant pattern formation: a rigorous analysis. Journal of Nonlinear Science. 2002;12:641–676.
  • 18. Lagesse C, Bordin P, Douady S. A spatial multi-scale object to analyze road networks. Netw Sci (Camb Univ Press). 2015;3(1):156–181.
  • 19. Hofmeister W. Allgemeine Morphologie der Gewächse. In: du Bary A, Irmisch TH, Sachs J, editors. Handbuch der Physiologischen Botanik. Leipzig: Engelman; 1868. p. 405–664.
  • 20. Couder Y. Initial transitions, order and disorder in phyllotactic patterns: the ontogeny of Helianthus annuus: a case study. Acta Soc Bot Pol. 1998;67(2):129–150.
  • 21. Douady S, Couder Y. Phyllotaxis as a self organizing iterative process, Part II: the spontaneous formation of a periodicity and the coexistence of spiral and whorled patterns J Theor Biol. 1996;178:275–294.
  • 22. Zagórska-Marek B, Szpak M. Virtual phyllotaxis and real plant model cases. Funct Plant Biol. 2008;35:1025–1033.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.