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the negative root of the equation
x
2
where
φ
is the positive root and
ϕ
=
x
+
1
(
square = successor
)thatis:
√
5
√
5
n
n
)=
(
1
+
)
−
(
1
−
)
F
(
n
2
n
√
5
being
ϕ
smaller than 1, in absolute value, for
n
sufficiently large,
F
(
n
)
approximates
/
√
5 (whence
F
n
to
φ
(
n
+
1
)
/
F
(
n
)
approximates to
φ
).
1
φ
1
1
φ
2
It is easy to verify that
−
ϕ
=
and that 1
+
ϕ
=
1
−
φ
=
2
.Thevalue1
/
φ
is
2
is frequently occurring in nature
also called
circle golden ratio
, and the angle 2
π
/
φ
structures.
The problem, which suggested to Fibonacci his sequence, was the growth of a
population of rabbits. The (ideal) rule of their reproduction establishes that each
rabbit generates one offspring (a couple of rabbits, an offspring couple), but a new-
born at some generation time, say
i
, can generate rabbits only when it becomes adult,
that is, at generation
i
2 (one step is necessary to a newborn to become an adult).
Fig. 5.10 exemplifies this kind of development by means of a tree.
+
Fig. 5.10
Five generations of Fibonacci development
Fibonacci sequence is surprisingly ubiquitous in processes of biological morpho-
genesis and development. In DNA double strands, the angle formed by two consec-
utive nucleotides is approximately
π
/
5, and the ratio between the two grooves of
DNA helix is near to
. Moreover, Fibonacci sequence and golden ratio are found
in many patterns formed by leaves and flowers during plant developments. In the
φ
