Fibonacci
Spirals
in
Plants
The spiral structures found in sunflowers, pineapples, pine cones, etc. 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.
In developing material for a high school project on the investigation of spiral structures in plants, my idea has been to present the material for an elementary project first, then add more advanced material later. For the elementary project, I have deliberately kept the mathematics to a minimum. The main variation in the length and difficulty of the project depends on the extent to which the students become actively involved in the use of mathematical software. I have discussed this issue in
The following presentation is "rough and ready". We will be improving it and
adding more material soon. We are putting this on line now in order to facilitate
communication with high school teachers who are interested in developing
projects on on the investigation of spiral structures in plants.
A typical project has two parts:
(1) field work, in which the students collect some
examples,
(2) a mathematical investigation of
the spirals.
A project of this kind provides the student with opportunities to become
(a) more mathematically observant,
(b) more scientifically observant,
and, perhaps, also more interested in the structure and biology of plants.
BRIEF ACCOUNT OF THE BIOLOGICAL PHENOMENA AND MATHEMATICAL IDEAS TO BE INVESTIGATED IN THE PROJECT
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 exact angular increments of phi, where the theoretical value of phi is 360(2-s) = 137.50076... degrees, where s = (sqrt(5)+1)/2 = lim x[n+1]/x[n], where x1, x2, ... xn,... is the Fibonacci sequence 1,1,3,5,8,13,... In other words, the botanical elements (florets, scales or bracts) are laid down at successive increments of phi 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 phi 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 phi? 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 remaions to be done. The underlying biological mechanisms are not well understood. The formation and nature of spiral structures in plants is an ongoing topis of research (
REFERENCES
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.
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.
On being told that
(*) the number of spirals in a sunflower, pineapple or pinecone is always a Fibonacci number,
(**) the branches of a head of broccoli, or petals of an artichoke, taken in order of growth, occur at increments of 137.5 degrees around the stalk,
one's natural reaction is: "Is this true? If so, what is the biological explanation?" Step 1 of the elementary project provides the student with the opportunity to examine the truth of (*) and (**). 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 (**) but the majority provide good examples of (**). In my experience artichokes always satisfy (**) to a good approximation.
Having experienced the truth of (*)-(**), the student will want a biological explanation. After going through Step 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 Step 2 the student will have pursued the question to a certain level of basic science.
THE PROJECT
The project has an elementary phase and an advanced phase. The extent to which the advanced phase is included will depend on the amount of time available and the students' proficiency in mathematics and the use of computers.
The elementary phase of the project consists of two steps. In Step 1 the students count and measure the spirals in various plants. In Step 2 the students examine the mathematical model. As preparation for Step 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 Step 1, this is optional.