Mathematical Biosciences 100:161-199 (1990), Elsevier, New York
Creating Phyllotaxis: The Dislodgement Model
Frank M. J. van der Linden


Assuming that neither the Fibonacci sequence nor any numerical ratio or angular deflection is specified in the genetic material of a plant cell, there must be an arranging mechanism effecting the sequence mentioned. Considering the ubiquity of the Fibonacci numbers in nature, embracing many species of flora, we expect a very simple geometrical law to be responsible. Success in finding such a law does not constitute a proof, but it is at the least an indication that we should look here with mathematician’s rather than biologist’s eyes.
The idea may seem self-evident. However, in the literature it has not yet been honored as the basis for constructing the phyllotaxis in centric, planar models. It is shown here that for the construction of a phyllotactic structure, no special angles or distances need be defined; natural growth functions can be used; planar, cylindrical, conical, and paraboloid constructions are possible within the same model; and constructions leading to accessory sequences and multijugate sequences can also be carried out.

The ordered structure of a plant

A given plant has an underlying ‘ordered structure’ that runs on through the entire individual [5]. This structure can be considered the bearer of all forms of appendages that the plant can produce. When the plant develops a flower, this will be placed and built up according to the plant’s own individual structure.
A flower or fruit will enable the plant to reproduce. It is important for the plant to produce large numbers of offspring, since much of the reproductive material will be lost. Reproductive units are small in size and large in number. The underlying structure of a plant will become clearly visible where these units are arranged closely together, in the flower head, for example. The sunflower has a flower head (capitulum) with hundreds, often over a thousand, florets, which can produce as many seeds (achenes). These seeds are arranged according to a spiral system. From the center outwards, congruous spirals run both to the left and to the right. ‘On the way out’, spirals will be replaced by others, with their number increasing abruptly. The ring in which spirals go over into other spirals is clearly demonstrable.
In the year 1202, Fibonacci (Leonardo di Pisa) described a numerical sequence with opening terms 1 and 2. Each term is the sum of the previous two terms, which are natural numbers. Thus it runs 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... . A large proportion of the seed-bearing plants have an ordered structure having to do with congruous spirals. The number of spirals turning to the left and to the right are consecutive terms in the sequence, which is called the Fibonacci sequence or the main sequence. The most important accessory sequence, the Lucas sequence (named after Eduard Lucas and described in 1877), also occurs, although considerably less often. It starts with the terms 1 and 3. Accessory sequences have different opening terms from those of the main sequence, but the same rule of continuation (see Section Other sequences). Plants without spiral arrangements either have a different structure or form a special group within the ordered structures mentioned [1]. They may show, for example, opposite leaf arrangements, whorls, umbels. The ordered structure is called phyllotaxis.

(1) R.V. Jean, Phyllotactic pattern generation: a conceptual model, Ann. Bot. (London) 61:293-303 (1988)
(2) A. Lindenmayer, Development algorithms: lineage versus interactive control mechanisms, in Development Order: Its Origins and Regulation, S. Subtelny and P.B. Green, Eds., Alan R. Liss, N.Y., 1982, pp. 219-245
(3) B.B. Mandelbrot, The Fractal Geomery of Nature, Freeman, N.Y., 1983, pp. 151-165
(4) P. Prusinkiewicz and J. Hanan, Lindenmayer Systems, fractals and plants, Lect. Notes, Uni. of Regina, Canada, 1988, pp. 23-28
(5) O. Schüepp, Geometrical constructions on phyllotaxis, 16 mm movie, no. 10997
(6) d’Arcy W. Thompson, On Growth and Form, Cambridge University Press, London, 1917, pp. 759-766


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The introduction and the references list of the Dislodgement theory are shown.