Mathematical Biosciences 100:161-199 (1990), Elsevier, New York

Creating Phyllotaxis: The Dislodgement Model

Frank M. J. van der Linden

Abstract

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.

**References**

(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