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 offer 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.

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.

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