Fibonacci
             Spirals
                          in
                                Plants

The spiral structures found in sunflowers, pineapples, pine cones, etc. are are easily observed in our daily environment; and they are of considerable scientific interest--from the viewpoint of both mathematics and biology. This is a good subject matter for projects by high school students; in other words, projects in which the students investigate the spiral structures in plants The purpose of these web pages is to provide material which will help high school teachers design such projects for their students. Our main emphasis is on a mathematical discussion of the spirals. Hence our treatment is addressed primarily to high school mathematics teachers.

Spiral patterns occur in certain plants such as sunflower heads, pineapples and artichokes. Simply by counting the spirals, we see that the number of spirals with a given pitch (slope) is a Fibonacci number. Typically, two families of spirals will be visible--one clockwise and one counterclockwise. The two families have consecutive Fibonacci numbers, the shallower spirals having the lower number.

Why is the number of spirals with a given pitch a Fibonacci number? This is explained by the fact that there is an underlying generative spiral. The florets, scales or bracts lie on a spiral at angular increments of f = 360(2-s) = 137.50776... degrees, where s = (sqrt(5)+1)/2 = lim x[n+1]/x[n], where x0, x1, x2, ..., xn, ... is the Fibonacci sequence 0, 1, 1, 3, 5, 8, 13,... In other words, the botanical elements (florets, scales or bracts) are laid down at successive increments of f = 137.5... degrees during the growth of the plant. The generative spiral (as a discrete structure) is not just a mathematical fiction but a biological reality; it can be seen directly in the case of of artichokes, broccoli, and certain trees and shrubs. (The continuous curve that one uses to join the points is, of course, a mathematical fiction.) The observational phenomenon that the botanical elements lie on certain spirals is a purely geometric consequence of the fact that these objects lie on the generative spiral. This is easily illustrated by using mathematical software to draw a few of these spirals. The mathematics of these spirals is, on the whole, accessible to high school students.

Thus the observed phenomena can be described by an interesting mathematical model. This mathematical model is purely descriptive: it expresses the fact that the botanical elements are laid down at successive increments of f = 137.5... degrees during the growth of the plant. But what is the biological explanation of this? The botanical elements evolve from primordia (little clumps of cells) laid down at regular time intervals at the boundary of the apex (tip of the growing shoot). Thus the question reduces to: Why are the primordia laid down at successive increments of f? A detailed investigation of this question is beyond the scope of the present project. some discussion is provided later on in these pages. For the moment, it suffices to mention that attempts to provide an answer date back to 1868 (Hofmeister). In 1993, Douady and Couder developed a biologically plausible model whose results correspond quite well--both qualitively and quantitatively--to the observed phenomena. Although the model of Douady and Couder is a good contribution, much remains to be done. The underlying biological mechanisms are not well understood. The formation and nature of spiral structures in plants is an ongoing topic of research [5].

1. http://www.ee.surrey.ac.uk./Personal/R.Knott/Fibonacci/fib.html

2. Ian Stewart, Life's Other Secret, Wiley, 1998.

3. Trudi H. Garland, Fascinating Fibonaccis, Dale Seymore Publications, 1987 (1-800-872-1100).

4. S. Douady and Y. Couder, Phyllotaxis as a Self-Organized Growth Process, In: Growth Patterns in Physical Sciences and Biology, ed, by J. M. Garcia-Ruiz et al., Plenum Press, New York, 1993.

5. R. Jean and D. Barabé, eds., Symmetry in Plants, Word Scientific, Singapore, 1998.

For an excellent introduction to Fibonacci sequences and their applications, see Knott's website [1]. Garland's book [3] has useful material. Chap. 6 of Stewart's book [2] provides an informative discussion of spiral structures in plants.

The brief account, above, contains the following assertions.

      (1) Plants such as sunflowers, pineapples, etc., have two families of spirals; the number of spirals in each family is a Fibonacci number (the two numbers being consecutive in the Fibonacci sequence).

      (2) In the spiral of petals in an artichoke, or branches in plants such as broccoli or a rose vine, each petal or branch makes an angle of 137.5... degrees with the preceding one.

      (3) If, on a shallow spiral, one puts a point every f = 137.5... degrees, the points will lie on spirals of the sort seen in (1).

Item (3) can be taken as a descriptive model; but, in the brief account above, it is implied that the generative spiral, even when it is not directly visible, is more than a mathematical fiction. In other words:

      (4) The botanical elements are laid down at successive increments of f = 137.5... degrees during the growth of the plant.

Concerning item (4), two questions immediately arise:

      (5) Is (4) true for plants without a visible generative spiral (eg., pine cones, sunflowers)?

      (6) When (4) occurs, why does it occur? What is the biological mechanism?

Possibly (4) is a bit too simplistic. The primordia do not pop into existence one-by-one. Photomicrographs show that by the time a primordium has been formed, later primordia are beginning to emerge. In the case of pine cones, it may be possible to verify (4) simply by taking a pine cone apart. (I made an attempt but found that pine cones are really tough.) Looking at the photomicrographs in [5], Chap. , my feeling is that, in the case of sunflowers, (4) is too simplistic.

A variety of possible projects for high school students, varying in length or difficulty, can be devoted to an investigatiion of (1)-(3).

Even is one does not try to investigate (4) directly, pondering (4) makes one wonder: How do pineapples, pine cones, sun flowers, etc., grow? This is a large topic but, for example, if a pine tree is available, a first step would be to watch the formation of pine cones during the spring. It would be instructive to watch a sunflower grow. In fact, the students, or the teacher, could grow some sunflowers. There are many varieties of sunflower. Make sure that you grow one of the varieties with yellow florets, that show the spirals in all their glory.

A typical project has two parts:

     Part 1  field work, in which the students investigate some examples exhibiting the phenomena (1)-(2),
     Part 2  a mathematical investigation of the spirals.

Part 1 of the project provides the student with the opportunity to examine the truth of (1) and (2). In my own experience, out of 15 or 20 sunflowers, only 2 or 3 have shown deviations from the predicted results; these were sunflowers that had gone to seed and showed serious dislocations in their spiral structure. In counting the spirals of a mature sunflower, it is necessary to stay near the periphery of the seedhead. Perhaps a third or a quarter of the pineapples I have examined have showed dislocations which made it difficult or impossible to count the steeper spirals (there are supposed to be 13); but, without exception, there were 8 shallow spirals. Some heads of broccoli are not suitable for examining (2) but the majority provide good examples of (2). In my experience artichokes always satisfy (2) to a good approximation.

Having experienced the truth of (1)-(2), the student will want a biological explanation. After engaging in Part 2, the student will understand that the spirals one sees are a purely geometric consequence of the fact that the biological elements lie on the biologically more fundamental generative spiral. A detailed explanation of the 137.5 degree property of the generative spiral is beyond the scope of our presentation, but, at least, in going through Part 2 the student will have pursued the question to a certain level of basic science.

As preparation for Part 1, all the student needs to know is:

   (i)  The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, ...; x0 = 0, x1 = 1, x[n+2] = x[n+1] + xn. By a Fibonnaci number is meant any number occuring in the Fibonacci sequence.
   (ii)  A certain angle f plays a basic role in the project. In degrees it is 360(2-s), where s = (sqrt(5)+1)/2. Thus f is approximately 137.5 degrees. The student could be told s = lim x[n+1]/x[n]; but, for Part 1, this is optional.

Next Page