The beautiful arrangement of leaves in some plants, called phyllotaxis, obeys a number of subtle mathematical relationships. For instance, the florets in the head of a sunflower form two oppositely directed spirals: 55 of them clockwise and 34 counterclockwise. Surprisingly, these numbers are consecutive Fibonacci Numbers. The ratios of alternate Fibonacci Numbers are given by the convergents to , where is the Golden Ratio, and are said to measure the fraction of a turn between successive leaves on the stalk of a plant: 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). A similar phenomenon occurs for Daisies, pineapples, pinecones, cauliflowers, and so on.

Lilies, irises, and the trillium have three petals; columbines, buttercups, larkspur, and wild rose have five petals; delphiniums, bloodroot, and cosmos have eight petals; corn marigolds have 13 petals; asters have 21 petals; and daisies have 34, 55, or 84 petals--all Fibonacci Numbers.

**References**

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.* New York: Dover,
pp. 56-57, 1987.

Church, A. H. *The Relation of Phyllotaxis to Mechanical Laws.* London: Williams and Norgate, 1904.

Church, A. H. *On the Interpretation of Phenomena of Phyllotaxis.* Riverside, NJ: Hafner, 1968.

Conway, J. H. and Guy, R. K. ``Phyllotaxis.'' In *The Book of Numbers.* New York: Springer-Verlag, pp. 113-125, 1995.

Coxeter, H. S. M. ``The Golden Section and Phyllotaxis.'' Ch. 11 in *Introduction to Geometry, 2nd ed.*
New York: Wiley, 1969.

Coxeter, H. S. M. ``The Golden Section, Phyllotaxis, and Wythoff's Game.'' *Scripta
Mathematica* **19**, 135-143, 1953.

Dixon, R. *Mathographics.* New York: Dover, 1991.

Douady, S. and Couder, Y. ``Phyllotaxis as a Self-Organized Growth Process.'' In *Growth Patterns in Physical Sciences and Biology*
(Ed. Juan M. Garcia-Ruiz *et al.*). Plenum Press, 1993.

Hunter, J. A. H. and Madachy, J. S. *Mathematical Diversions.* New York: Dover, pp. 20-22, 1975.

Jean, R. V. *Phyllotaxis: A Systematic Study in Plant Morphogenesis.* New York: Cambridge University Press, 1994.

Pappas, T. ``The Fibonacci Sequence & Nature.'' *The Joy of Mathematics.*
San Carlos, CA: Wide World Publ./Tetra, pp. 222-225, 1989.

Prusinkiewicz, P. and Lindenmayer, A. *The Algorithmic Beauty of Plants.* New York: Springer-Verlag, 1990.

Stewart, I. ``Daisy, Daisy, Give Me Your Answer, Do.'' *Sci. Amer.* **200**, 96-99, Jan. 1995.

Thompson, D. W. *On Growth and Form.* Cambridge, England: Cambridge University Press, 1952.

© 1996-9

1999-05-26