As mentioned in the Introduction, the spirals that one sees are a purely geometric consequence of the fact that the botanical elements lie on the biologically more fundamental generative spiral. This theme will be pursued by means of a mathematical model consisting of:

        a) a generative spiral going inward, together with points at increments of f,
        b) some of the associated secondary spirals, which we call Fibonacci spirals.

The following very schematic description of the growth of the plant helps us understand why the botanical elements lie on a generative spiral as just described. In the growth of a sunflower seedhead, a more sophisticated scheme is needed (see XXXX below), but the present scheme provides a step towards understanding the more sophisticated scheme.

                          Simplified description of the growth of the plant

Fig. 2.1

Plants which have Fibonacci spirals, or which have a visible generative spiral, grow as follows. At the growing tip is a little disk-shaped mass of cells called the apex, which is typically less than a millimeter in diameter. Thus in this diagram we are loooking down at the growing tip through a microscope. At regular time intervals, a clump of cells, called a primordium, forms at the edge of the apex. We have chosen the time scale so that the primordia appear at unit time-intervals. The primordium migrates away from the apex while evolving into a botanical element such as a branch (broccoli) or a bract (pine cone). More precisely, the growing tip leaves the evolving primodium behind on the surface of the stem. The surface of the stem is a cone with a very gradual taper. In the following diagrams, this cone has been flattened out, first by increasing the diameter of the base of the cone, then projecting onto a horizontal plane. Each evolving primordium moves along a radius line. Thus primordium number one moves horizontally to the right.
 
 

Fig. 2.2

Here is the situation at time t = 20; in other words, the situation after 21 primordia have been formed. The magnification has been greatly reduced from that of Fig. 2.1.

The little circles represent the botanical elements; they are numbered in the order of growth; i.e., the order in which they appeared at the edge of the apex. Thus the numbers increase as we go inward on the generative spiral.
 
 
 
 

Fig. 2.3a

 
 
 
 

Here is the situation after 80 primordia have been formed. The magnification has been further reduced.
 
 

Fig. 2.3b

The contiuous generative spiral is just a mathematical fiction that helps us keep track of the botanical elements. We now remove the continuous spiral from the preceding diagram in order to see better the points which represent the botanical elements.

Can you see any families of Fibonacci spirals? There is a family of spirals going counterclockwise inward that is fairly easy to see. Elements number 1, 9, 17, and 25 lie on one of these spirals. How many spirals are in this family?

Problem XXX Determine the numbers x, y, z (cf. Fig. 2.2).

Do you see the family of 13 spirals going clockwise inward?

The two families just mentioned are indicated in the following two diagrams.
 

Fig. 2.3c

 

The points in the preceding diagram lie on 8 spirals.Proceeding inward on any of the indicated spirals, if a point on the spiral has number n, then the next point has number n+8. Because of this, and also because there are 8 spirals in the family, let us say that these are the Fibonacci spirals of order 8.
 

Fig. 2.3d

 
 

The points of Fig. 2.3d also lie on 13 spirals. These are the Fibonacci spirals of order 13.
 
 

What mathematical law should we use for the generative spiral?

The spiral in Fig. 2.1 (microscopic view of the apex) is an exponential spiral r = b*expt(n - cq), I think. (I used the spiral drawing facility in Adobe Illustrator). This is in keeping with the biological reality of growth near the apex. In Fig. 2.2 and 2.3 we used an Archimedian spiral r = c(h - q). The Archimedian law was chosen on the grounds of simplicity; it is the obvious choice for a preliminary study of the principal that as the generative spiral becomes shallow (small pitch) the higher-order Fibonacci spirals become visible.

Higher-Order Spirals

When the generative spiral becomes shallower, the higher-order spirals become visible.

Fig. 2.5
 
 
 
 

For n = 200 points, the Fibonacci spirals of order 21 (counterclockwise inward) leap to the eye. The spirals of order 13 are also easy to see.
 
 
 

Fig. 2.6
 
 
 
 

The continuous Fibonacci spirals of order 13, k, and 21 are added to the previous diagram.
 

Problem XXX: Determine the number k.
 
 
 
 

Fig. 2.7
 
 
 
 

When the generative spiral becomes shallower still, even higher order Fibonacci spirals emerge:
1000 points
Here there are 34 spirals going clockwise inward, and 55 spirals going counterclockwise inward.
 
 
 
 
 
 
 
 

Fig. 2.8 
 

1800 points

The spirals of order 89 have now emerged.
 
 
 
 
 
 
 
 
 
 

When the generating spiral in Fig. 2.7 is drawn clockwise, the resulting diagram shows the pattern of spirals in the sunflower in Fig. 1.6.

Similarly, when the generating spiral in Fig. 2.8 is drawn clockwise, the resulting diagram shows the pattern of spirals in the sunflower in Fig. 1.4.
Here we draw Fig. 2.8 with clockwise generating spiral, and we have  and andand we have

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