The Greeks knew since ancient times the
so-called golden section of a segment,
which is nothing but its division into two parts, so that the longer divided by
the shortest is the same as the length of the total segment divided by the
longest. Consider, for example, segment AB. Its golden division is given by the
point X if and only if AX/XB = AB/AX.
A X B
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| Leonardo: the Vitruvian Man |
For the Greeks, as for many great painters, the
golden ratio or golden section divides a segment in the most aesthetically
attractive way. The Italian mathematician Lucas Paccioli, who called this ratio
the divine
proportion, influenced Leonardo da Vinci and Albrecht Durer. In the
twentieth century, neo-Impressionist painters like Seurat have used the golden
section to define the dimensions of some of their compositions. Architects like
Le Corbusier used the golden ratio to design their works. And many books
published in the sixteenth to eighteenth centuries had the dimensions of a
golden rectangle. The golden ratio has also been used by musicians such as Erik
Satie and Debussy, and provided some mystics with food for thought.
The golden ratio has curious properties. For
example, you can build a golden rectangle whose height is the golden section of
its base. If you take from this rectangle the square whose side is equal to its
height, the smaller rectangle is also golden. This effect can be repeated
indefinitely from the new rectangle.
The relationship between the golden section of
a segment and its total length is easy to calculate geometrically. The value
obtained is:
This is the golden
ratio or golden number,
an irrational number whose value is approximately 1.6180339887..., which also
has curious properties. For example, its inverse equals
Its approximate value is 0,6180339887..., the
golden number minus 1.
Geometers found that if the radius of a circle
is divided by the side of the inscribed regular decagon, the result is gets the
golden ratio. If a circle is divided into ten equal parts and each point is joined
with those three places away on either side, a star-shaped decagon is obtained,
whose side is equal to the radius of the circle multiplied by the golden
number.
It can be seen that the golden ratio appears
frequently in geometry. We now turn to a different field, at first sight.
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| Leonardo of Pisa (Fibonacci) |
During the beginning of the thirteenth century,
the Italian mathematician Leonardo Fibonacci, also called Leonardo of Pisa,
wrote his influential treatise Liber Abaci
(Book
of the Abacus), which introduced in the Western civilization the Arabic
numerals, which actually came from India, a revolution for arithmetic, as operations
are much easier to perform than with Roman numerals. In that book, Leonardo
also proposed the following problem:
A man put a pair of rabbits in a
closed area. How many pairs of rabbits will there be after one year, assuming
that each month a pair begets a new pair, which from the second month becomes
fertile?
It seems a simple problem, but it has given Leonardo
of Pisa more fame than introducing the Arabic numerals in the West. It is easy
to compute the number of pairs of rabbits for every month, assuming that no
rabbit dies, and that once they become fertile they remain so indefinitely. The
following sequence is obtained:
1 1 2 3 5 8 13 21 34
55 89 144...
This sequence is the Fibonacci
series, so named in honor of its author, which has the property
that every term is the sum of the previous two terms, although nobody noticed this
until the French mathematician Albert Girard signaled it in 1634. Later it was
found that the series appears frequently in nature: in the spirals forming
sunflower heads; in pine cones; in the shells of snails; in the arrangement of
the tips of leaves around a stem; in the successive layers of the horns of
mammals; and in many other places. Each turn in these spiral curves contains
several components, whose numbers in successive turns form the Fibonacci
series.
What does this have to do with the golden
number? Let us see. The following formula computes the term with rank n in the
Fibonacci series without having to compute the previous terms of the series:
In this formula, A is the golden number and 1 -
A (see above) the opposite of its inverse. This expression is very curious: it
contains three times the square root of 5, which is an irrational number (one
in each of the two A, the third in the denominator), and yet the end result of
the expression is always integer, when n is a positive integer.
The Fibonacci series is related in another way to
the golden number. If every term in the series is divided by the preceding one,
a new series is obtained, whose first terms are: 1 2 1.5 1.666666666
1.6 1.625 1.615384615
1.619047619 1.617647059 1.618181818
1.617977528... It can be seen that
the terms of this series fluctuate around the golden number and approach it ever
more. Therefore, the golden ratio is the limit of the sequence formed by the
quotients of the terms of the Fibonacci series.
One last point that does not exhaust the
properties of the golden ratio and the Fibonacci series: the quadratic equation
x2-x-1=0 has as solutions the golden ratio A, and the opposite of
its inverse (1-A). The curious reader can check it out.
Manuel Alfonseca
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