martes, 6 de noviembre de 2012


The sequence of Fibonacci numbers is given by 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …, in which each number is the sum of the two preceding numbers. This can be expressed as  with  and .
Fibonacci (whose real name was Leonardo Pisano) found this sequence as the number of pairs of rabbits  months after a single pair begins breeding, assuming 
that each pair of rabbits produces a pair of offspring when it is two months old.

As , the ratio of successive Fibonacci numbers  approaches the limit , known as the "golden ratio". The ancient Greeks regarded  as the most aesthetically pleasing proportion for the sides of a rectangle. In this Demonstration the black rectangle has sides proportional to . See how these approach the "golden rectangle" as you increase .

Commercial interests are apparently aware of the appeal of the golden ratio. Credit cards, for which you probably have begun getting offers, have almost exactly this shape.

The ancient Greeks thought that the most pleasing proportions for a rectangle were those in which the rectangle's sides were in the ratio of about 1.618 to 1. This number is called the golden section or golden ratio and a rectangle with those proportions is called a golden rectangle.

Leonardo da Vinci referred to the golden ratio as the "divine proportion". There are several features of his most famous painting, the Mona Lisa, to which a golden rectangle can be fitted. See how many of these you can find by varying the orientation, position, and size of the rectangle

Is the Golden Ratio Really the Most Beautiful Proportion? from the Wolfram Demonstrations Project by Herbert W. Franke

Since the classical age of Greece, the golden ratio  has been well known as the most balanced and therefore most aesthetically perfect proportion. In fact, it is supposedly often found in the fine arts and architecture and it is certainly important in geometry. But where is the proof that it is really the most beautiful proportion? For example, could a different value be derived on the basis of information psychology?
This Demonstration allows you to make your own contribution 
to this question. Use the slider to realize the most harmonic rectangle from your point of view. Then you might find some examples from art or science and make a comparison.

The yellow rectangle has sides whose ratio is the golden ratio . The sides of the blue rectangle are in the proportion , which the German physicist Georg Christoph Lichtenberg designated in 1796 as the most beautiful geometric ratio (Zerbe and Davidshofer, see references). Finally, the green rectangle should have the best proportions, according to a result from information psychology.

Golden Section from the Wolfram Demonstrations Project by Michael Schreiber
Partition a line segment into two segments according to the construction shown. The larger segment is the golden section of the original segment. The ratio of the original segment to its golden section is called the golden ratio.

The ratio of two successive Fibonacci numbers approaches the value of the golden ratio .

This tool can be used to determine whether two measurements occur in the golden ratio (). Regardless of the angle between the outermost arms of the divider, the golden ratio is seen between the intervals created by the outer and middle arms, and between the total span of the dividers and the larger interval.

Golden Spiral from the Wolfram Demonstrations Project by Yu-Sung Chang
This Demonstration draws an approximation to a golden spiral using a golden rectangle.

Geometric and Continued Fraction Expansion of the Golden Ratio from the Wolfram Demonstrations Project by Adam Kelchner
his Demonstration illustrates the growth of the continued fraction expansion of , the golden ratio. Also shown are the convergents of its continued fraction and a series of squares in the golden rectangle.

Hexagons and the Golden Ratio from the Wolfram Demonstrations Project by Sándor Kabai

This Demonstrations has to do with Odom's recognition of the relationship between the golden ratio and the equilateral triangle. Construct three triangles by extending the edges of an equilateral triangle.
When the 
extension is inversely proportional to the golden ratio, two vertices of each triangle are on a circle circumscribing a triangle twice as large as the original triangle.

When the extension is proportional to the golden ratio, the outside vertices of the three triangles determine a hexagon having two different edge lengths whose ratio is equal to the golden ratio. The vertices of the hexagon determine two triangles that can be found in the compound of two icosahedra or the compound of five octahedra.

Five Circles and the Golden Ratio from the Wolfram Demonstrations Project by Sándor Kabai
Place five circles of diameter  symmetrically about a given point. They intersect the unit circle at the vertices of a regular pentagon when  or , where ϕ is the golden ratio, 1.61803….

Nested Square Root Representation of the Golden Ratio from the Wolfram Demonstrations Project by S. M. Blinder

One of the simplest possible formulas involving an infinite sequence of nested square roots is . From that, it can be easily seen that . The positive root of this quadratic equation is , which is none other than the golden ratio, known to Euclid and the ancient Greeks. This is also called the golden mean, golden section, and divine proportion. This is also equal to the limiting ratio of successive Fibonacci numbers: .
Mathematica can readily compute multiply nested functions. With , the sequence of  square roots can be computed using Nest[,,]. This Demonstration lets you compute approximations to  containing up to 100 square roots. For each value of , the accuracy, expressed as the number of significant figures, is also given.

Phyllotaxis Explained from the Wolfram Demonstrations Project by Claude Fabre

Here is a visual illustration of how the angle of divergence affects the distribution of petals, leaves, florets, and so on in botany.

In the initial settings, the angle parameter is the inverse of the golden ratio, . Count the number of ghost spirals (parastichies) turning to the left and right. You find two consecutive Fibonacci numbers depending on the number of florets. Unmask this optical illusion by looking at the growth spiral.
Beyond this special golden angle the Demonstration gives all distributions of phyllotaxis. Try irrational and rational numbers for the angle parameter.

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