A Max-Min Principle for Phyllotactic Patterns

  • Wai-Ki Ching
  • Yang Cong
  • Nam-Kiu Tsing
Part of the Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering book series (LNICST, volume 5)


An interesting phenomenon about phyllotaxis is the divergence angle between two consecutive primordia. In this paper, we consider a dynamic model based on Max-Min principle for generating 2D phyllotactic patterns studied in [2,5]. Under the hypothesis that the influence of the two predecessors is enough to fix the birth place of the new generated primordium, analysis and numerical experiments are conducted. We then propose a new measurement for evaluating the pattern uniformity (sparsity) of different divergence angles. It is found that the golden angle gives very good sparsity but there are other angles give even better sparsity under our proposed measurement.


Divergence Angle Golden Angle Max-Min Principle Phyllotactic Patterns 


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  1. 1.
    Adler, I.: A model of space filling in phyllotaxis. J. Theoret. Biol. 53(2), 435–444 (1975)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atela, P., Golé, C., Hotton, S.: A Dynamical System for Plant Pattern Formation: Rigorous Analysis. J. Nonlinear Sci. 12(6), 641–676 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Douady, S., Couder, Y.: Phyllotaxis as a physical self-organized growth process. Phys. Rev. Lett. 68, 2098–2101 (1992)CrossRefGoogle Scholar
  4. 4.
    Hofmeister, W.: Allgemeine Morphologie der Gewachse, Handbuch der Physiologishen Botanik, pp. 405–664. l Engelman, Leipzig (1868)Google Scholar
  5. 5.
    Hotton, S., Johnson, V., Wilbarger, J., Zwieniecki, K., Atela, P., Golé, C., Dumais, J.: The Possible and the Actual in Phyllotaxis: Bridging the Gap between Empirical Observations and Iterative Models. J. Plant Growth Regul. 25, 313–323 (2006)CrossRefGoogle Scholar
  6. 6.
    Jean, R.V.: Mathematical Approach to Patterns and Form in Plant Growth. Wiley, New York (1984)Google Scholar
  7. 7.
    Marzec, C., Kappraff, J.: Properties of maximal spacing on a circle related to phyllotaxis and to the golden mean. J. Theoret. Biol. 103(2), 201–226 (1983)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Steeves, R.V., Sussex, I.M.: Patterns in Plant Development. Cambridge University Press, Cambridge (1989)CrossRefGoogle Scholar
  9. 9.
    Vogel, H.: A better way to construct the sunflower head. Mathematical biosciences 44, 145–174 (1979)CrossRefGoogle Scholar

Copyright information

© ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering 2009

Authors and Affiliations

  • Wai-Ki Ching
    • 1
  • Yang Cong
    • 1
  • Nam-Kiu Tsing
    • 1
  1. 1.Advanced Modeling and Applied Computing Laboratory, Department of MathematicsThe University of Hong KongHong KongHong Kong

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