Thursday, May 19, 2016

The golden number

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
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.
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

No comments:

Post a Comment