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 89 petals--all Fibonacci numbers.

Azukawa, K. and Yuzawa, T. "A Remark on the Continued Fraction Expansion of Conjugates of the Golden Section." Math. J. Toyama Univ. 13, 165-176, 1990.

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.

Cook, T. A. The Curves of Life, Being an Account of Spiral Formations and Their Application to Growth in Nature, To Science and to Art. New York: Dover, 1979.

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 Role of Intermediate Convergents in Tait's Explanation for Phyllotaxis." J. Algebra 10, 167-175, 1972.

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

Dixon, R. "The Mathematics and Computer Graphics of Spirals in Plants." Leonardo 16, 86-90, 1983.

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. J. M. Garcia-Ruiz et al. ). New York: Plenum, 1993.

Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, 1979.

Hargittai, I. and Pickover, C. A. (Eds.). Spiral Symmetry. New York: World Scientific, 1992.

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

Jean, R. V. "Number-Theoretic Properties of Two-Dimensional Lattices." J. Number Th. 29, 206-223, 1988.

Jean, R. V. "On the Origins of Spiral Symmetry in Plants." In Spiral Symmetry. (Ed. I. Hargittai and C. A. Pickover). New York: World Scientific, pp. 323-351, 1992.

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

Naylor, M. "Golden, , and Flowers: A Spiral Story." Math. Mag. 75, 163-172, 2002.

Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, 2002.

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.

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 138, 1999.

Stevens, P. S. Patterns in Nature. London: Peregrine, 1977.

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.

Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 11 and 83, 1999.

Vogel, H. "A Better Way to Construct the Sunflower Head." Math. Biosci. 44, 179-189, 1979.

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 65-66, 1986.

CITE THIS AS:

Weisstein, Eric W. "Phyllotaxis." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Phyllotaxis.html