*`The Annunciation', Leonardo da Vinci*

Since the days of old, artists as well a geometricians have known that there is a special, aesthetically pleasing rectangle with width 1, length x and the following property: When a square of side 1 is removed, the rectangle that remains has the same proportions as the original rectangle. This ratio is known as the golden section.

The origin of the golden section dates to the early 11th century.
In 1202, the mathematician Leonardo of Pisa, also called Fibonacci,
published an influential treatise, *Liber abaci*. It contained the
following recreational problem: "How many pairs of rabbits can be produced
from a single pair in one year if it is assumed that every month each pair
begets a new pair which from the second month becomes productive?"
Straightforward calculation generates the following sequence:

The second row represents the first 12 terms of the sequence
now known by Fibonacci's name, in which each term (except the
first two) is found by adding the two terms immediately preceding;
in general, *x*_{n} = *x*_{n
- 1} + *x*_{n - 2}*, *a relation
that was not recognized until about 1600.

Over the years, especially in the middle decades of the 20th
century, the properties of the Fibonacci numbers have been extensively
studied, resulting in a considerable literature. Their properties
seem inexhaustible; for example, *x*_{n+1}
*x*_{n-1}*
*= *x*_{n}^{2} + (-1)^{n}.
Another formula for generating the Fibonacci numbers is attributed
to Édouard Lucas:

The ratio ({sqroot 5} + 1) : 2 = 1.618 . . . , designated as
, is known as the
**golden ** number; the ratio ({sqroot 5} - 1) : 2, the reciprocal
of , is equal to 0.618
. . . . Both these ratios are related to the roots of *x*^{2}
- *x* - 1 = 0, an equation derived from the
Divine Proportion of the 15th-century Italian mathematician
Luca
Pacioli, namely, *a*/*b* = *b/*(*a* + *b*),*
*when *a* < *b, *by setting *x* = *b*/*a.
*In short, dividing a segment into two parts in mean and
extreme proportion, so that the smaller part is to the larger
part as the larger is to the entire segment, yields the so-called
**Golden **
Section,* *an important concept in both ancient
and modern artistic and architectural design. Thus, a **rectangle **
the sides of which are in the approximate ratio of 3 : 5 (
^{-1} = 0.618 . . .), or 8 : 5 (
= 1.618 . . .), is presumed to have the most pleasing proportions,
aesthetically speaking.

Raising the **golden ** number to successive powers generates the sequence that
begins as follows:

In this sequence the successive coefficients of the radical
{sqroot 5} are Fibonacci's 1, 1, 2, 3, 5, 8, while the successive
second terms within the parentheses are the so-called
Lucas sequence: 1, 3, 4, 7, 11, 18. The Lucas sequence shares
the recursive relation of the Fibonacci sequence; that is,
*x*_{n}
= *x*_{n-1} + *x*_{n-2}.

If a
**golden **
**rectangle** ABCD is drawn and a square ABEF is removed,
the remaining **rectangle ** ECDF is also a **golden ** **rectangle**. If
this process is continued and circular arcs are drawn, the curve
formed approximates the logarithmic spiral, a form found in
nature
(see Figure above). The
logarithmic spiral is the graph of the equation *r* = *
k*^{}*,
*in polar coordinates, where *k* =
^{2/}. The Fibonacci
numbers are also exemplified by the botanical phenomenon known
as
phyllotaxis. Thus, the
arrangement of the whorls on a pinecone
or pineapple, of petals on a sunflower, and of branches from
some stems follows a sequence of Fibonacci numbers or the series
of fractions

*Excerpt from the Encyclopedia Britannica without permission.*