15 Sept 2016 : Adil Mughal

Phyllotaxis, disk packing and Fibonacci numbers

Speaker : Dr Adil Mughal – Aberystwyth University, Wales

Venue    : Thu 15 Sept 2016, 3pm (Murdoch University, Senate Room)

Phyllotaxis (the arrangement of buds or branches on a stem, or  flowerets on a flower) has long been debated [1, 2], particularly in regard to the widespread occurrence of spiral structures that are related to the Fibonacci sequence [3-7]. Over the years hundreds of papers, and several books, have attempted to provide explanations of this phenomenon using models of varying complexity, sophistication and ad hoc inventiveness.

Here we off er a theoretical model which relates the problem to disk packings, extending previous work [8, 9] that seeks explanations in that way. Our method
is to adapt the closely related problem of the dense packing of hard disks on a cylinder [10{12], where helical symmetry arises naturally, to the present case of
buds on a gradually enlarging stem.

The figure shows the evolving arrangement of buds on a “bullet shaped” surface (i.e. the stem) at an initial time T1, and subsequent times T2 and T3. Towards the top the arrangement is characterised in phyllotactic notation [l = m+n; m; n] by the structure [1; 1; 0]. With increasing diameter this structure evolves into more complex arrangements – i.e. [2; 1; 1] followed by [3; 2; 1]. This is precisely the rule of progression in the Fibonacci sequence. The images on the left show the pattern “rolled out” onto the plane while the corresponding figures on the right show the arrangement wrapped seamlessly onto the stem.

Buds are introduced at the top of a “bullet-shaped” surface – roughly representative of a plant stem, see Fig (1) – and migrate downwards, while conforming to three principles: dense packing, homogeneity and continuity. Typical results are presented in a video. We show that spiral structures characterised by the Fibonacci sequence (1,1,2,3,5,8,13…), as well as related structures, occur naturally under such rules.

A. M. acknowledges fi nancial support through Aberystwyth University Research Fund.

[1] H. Airy, Proceedings of the Royal Society of London 21, 176 (1872).
[2] D. Hofstadter and C. Teuscher, Alan Turing: Life and legacy of a great thinker (Springer Science & Business Media, 2013).
[3] L. Levitov, JETP letters 54, 542 (1991).
[4] L. Levitov, EPL (Europhysics Letters) 14, 533 (1991).
[5] S. Douady and Y. Couder, Physical Review Letters 68, 2098 (1992).
[6] P. Atela, C. Gole, and S. Hotton, Journal of Nonlinear Science 12, 641 (2002).
[7] M. Pennybacker and A.C. Newell, Physical Review Letters 110, 248104 (2013).
[8] G.J. Mitchison, Science (1977).
[9] G. Van Iterson, Mathematische und mikroskopisch-anatomische Studien uber Blattstellungen: nebst Betrachtungen über den Schalenbau der Miliolinen, Ph.D. thesis, TU Delft, Delft University of Technology (1907).
[10] A. Mughal, H. Chan, and D. Weaire, Physical Review Letters 106, 115704 (2011).
[11] A. Mughal, H. Chan, D. Weaire, and S. Hutzler, Physical Review E 85, 051305 (2012).
[12] A. Mughal and D. Weaire, Physical Review E (2014).

Leave a Reply

Your email address will not be published. Required fields are marked *