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 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 .
See also Daisy ,
Fibonacci
Number ,
Golden Angle ,
Spiral
Explore with Wolfram|Alpha
References 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. Church,
A. H. The
Relation of Phyllotaxis to Mechanical Laws. Church, A. H. On
the Interpretation of Phenomena of Phyllotaxis. Conway,
J. H. and Guy, R. K. "Phyllotaxis." In The
Book of Numbers. 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. Coxeter,
H. S. M. "The Golden Section and Phyllotaxis." Ch. 11 in
Introduction
to Geometry, 2nd ed. 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. Douady, S. and Couder, Y. "Phyllotaxis as
a Self-Organized Growth Process." In Growth
Patterns in Physical Sciences and Biology et
al. ). New York: Plenum, 1993. Gardner, M. Mathematical
Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from
Scientific American. Hargittai, I. and
Pickover, C. A. (Eds.). Spiral
Symmetry. Hunter, J. A. H.
and Madachy, J. S. Mathematical
Diversions. 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. Jean, R. V. Phyllotaxis:
A Systematic Study in Plant Morphogenesis. Naylor, M. "Golden, Math. Mag. 75 ,
163-172, 2002. Livio, M. The
Golden Ratio: The Story of Phi, the World's Most Astonishing Number. Pappas, T. "The Fibonacci Sequence &
Nature." The
Joy of Mathematics. Prusinkiewicz, P. and Lindenmayer, A. The
Algorithmic Beauty of Plants. Steinhaus,
H. Mathematical
Snapshots, 3rd ed. Stevens,
P. S. Patterns
in Nature. Stewart, I. "Daisy,
Daisy, Give Me Your Answer, Do." Sci. Amer. 200 , 96-99, Jan. 1995. Thompson,
D. W. On
Growth and Form. Trott,
M. Graphica
1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael
Trott. 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. Referenced on Wolfram|Alpha Phyllotaxis
Cite this as:
Weisstein, Eric W. "Phyllotaxis." From
MathWorld https://mathworld.wolfram.com/Phyllotaxis.html
Subject classifications
, 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.
, and 
Flowers: A Spiral Story." Math. Mag. 75,
163-172, 2002.Livio, M.