Answer 2:
For along time, it had been noticed that these
numbers were important in nature and we still do
not completely understand why. You also have to be
aware that in the case of the flower petals the
Fibonacci numbers are only verified on
average while in the case of the sunflower,
pineapple or pinecone the numbers are exact.
To understand why the number of petals
correspond on average to a Fibonacci number we
first have to look at a sunflower head. When you
look at the head you notice two series of curves
one winding in one sense and one in another; the
number of spirals not being the same in each
sense. Why is the number of spirals in general
either 21 and 34, either 34 and 55, either 55 and
89, or 89 and 144? They all belong to the
Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55,
89, 144, but the explanation is also linked to
another famous number, the golden mean. In
many cases, the head of a flower is made up of
small seeds which are produced at the center, and
then migrate towards the outside to fill
eventually all the space (as for the sunflower but
on a much smaller level). Each new seed appears at
a certain angle in relation to the preceding one.
For example, if the angle is 90 degrees, that is
1/4 of a turn, the result after several
generations is four straight lines starting from
the center in a 90 degree angle.
Of course, this is not the most efficient way
of filling space. In fact, if the angle between
the appearance of each seed is a portion of a turn
which corresponds to a simple fraction, 1/3, 1/4,
3/4, 2/5, 3/7, etc. (that is a simple rational
number), one always obtains a series of straight
lines. If you want to get a spiral, it is
necessary to choose a portion of the circle which
is an irrational number (or a non simple
fraction). If this latter is well approximated by
a simple fraction, one obtains a series of curved
lines (spiral arms) which even then do not fill
out the space perfectly.
In order to optimize the filling, it is
necessary to choose the most irrational number
there is, that is to say, the one the least well
approximated by a fraction. This number is exactly
the golden mean. The corresponding angle, the
golden angle, is 137.5 degrees. (It is obtained by
multiplying the nonwhole part of the golden mean
by 360 degrees and, since one obtains an angle
greater than 180 degrees, by taking its
complement). With this angle, one obtains the
optimal filling, that is, the same spacing between
all the seeds (see sunflower head).
When the angle is exactly the golden mean, and
only this one, two families of spirals (one in
each direction) are then visible: their numbers
correspond to the numerator and denominator of one
of the fractions which approximates the golden
mean : 2/3, 3/5, 5/8, 8/13, 13/21, etc. These
numbers are precisely those of the Fibonacci
sequence (the bigger the numbers, the better the
approximation) and the choice of the fraction
depends on the time laps between the appearance of
each of the seeds at the center of the flower.
This is why the number of spirals in the
centers of sunflowers, and in the centers of
flowers in general, correspond to a Fibonacci
number. Moreover, generally the petals of flowers
are formed at the extremity of one of the families
of spiral. This then is also why the number of
petals corresponds on average to a Fibonacci
number.
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