Greenwood press

FIBONACCI SEQUENCE
35

a1=1
a2=1
a
n
=a
n−1
+a
n−2
t
n
=

1+
√
5
2

2
−

1−
√
5
2

2
√
5
.
recursive
explicit
The family tree for a male bee.

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squares (left-hand picture below), the vertices provide links for tracing a loga-
rithmic spiral (middle picture). The spiral (which expands one golden ratio dur-
ing each whole turn) appears in the chambered nautilus (right-hand picture).
The Fibonacci numbers appear in the branching of plants, and counts of spi-
rals in sunflower seeds, pine cones, and pineapples. In one particular variety of
sunflower, the florets appear to have two systems of spirals, both beginning at the
center. There are fifty-five spirals in the clockwise direction, and thirty-four in
the counterclockwise one. The same count of florets in a daisy show twenty-one
spirals in one direction and thirty-four in the other. A pine cone has two spirals
of five and eight arms, and a pineapple has spirals of five, eight, and thirteen. The
spiral also appears in animal horns, claws, and teeth.
On many plants, the number of petals on blossoms is a Fibonacci number.
Buttercups and impatiens have five petals, iris have three, corn marigolds have
thirteen, and some asters have twenty-one. Some species have petal counts that
may vary from blossom to blossom, but the average of the petals will be a Fibo-
nacci number. Flowers with other numbers of petals, such as six, can be shown
to have two layers of three petals, so that their counts are simple multiples of a
Fibonacci number. In the last few years, two French mathematicians, Stephane
Douady and Yves Couder, proposed a mathematical explanation for the Fibo-
nacci-patterned spirals in nature. Plants develop seeds, flowers, or branches from
a meristem (a tiny tip of the growing point of plants). Cells are produced at a con-
stant rate of turn of the meristem. As the meristem grows upward, the cells move
outward and increase in size. The most efficient turn to produce seeds, flowers,
or branches will result in a Fibonacci spiral.
In 1948, R. N. Elliott proposed investment strategies based on the Fibonacci
sequence. These remain standard tools for many brokers, but whether they are a
never-fail way of selecting stocks and bonds is open to debate. Some investors
think that when Elliott’s theories work, it is because many investors are using his
rules, so their effects on the stock market shape a Fibonacci pattern. Neverthe-
less, a substantial number of brokers use Elliott’s Fibonacci rules in determining
how to invest.
In computer science, there is a data structure called a “Fibonacci heap” that
is at the heart of many fast algorithms that manipulate graphs. Physicists have