Phyllotaxis
The beautiful arrangement of leaves in some plants, called phyllotaxis, obeys a number of subtle mathematicalrelationships. For instance, the florets in the head of a sunflower form two oppositely directed spirals: 55 of themclockwise 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 thestalk 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 forwillow 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 have34, 55, or 84 petals--all Fibonacci Numbers.
See also Daisy, Fibonacci Number, Spiral
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.
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