Phyllotaxis
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.
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Archimedean spiral
