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In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio. 67.

The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.

The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.

These often appear in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio.

In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

The Golden Ratio: The Story of Phi, the Worlds Most Astonishing Number.

Abu Kamil (c. 850930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c.

German mathematician Simon Jacob (d.

Fibonacci numbers converge to the golden ratio; 26 this was rediscovered by Johannes Kepler in 1608.

The first known decimal approximation of the (inverse) golden ratio was stated as about 0.6180340 in 1597 by Michael Maestlin of the University of Tbingen in a letter to Kepler, his former student.

The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem.

Kepler said of these: Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio.

The first we may compare to a mass of gold, the second we may call a precious jewel.

Le Corbusiers faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as rhythms apparent to the eye and clear in their relations with one another.

They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned.

He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vincis Vitruvian Man, the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture.

He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human bodys height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system.

Le Corbusiers 1927 Villa Stein in Garches exemplified the Modulor systems application.

The villas rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.

Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders.

In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.

Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art.

Divina proportione explored the mathematics of the golden ratio.

Though it is often said that Pacioli advocated the golden ratios application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.

Pacioli also saw Catholic religious significance in the ratio, which led to his works title.

But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by Leonardos own writings.

A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.

The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).

On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4, and 6.

Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.

The golden ratio is also apparent in the organization of the sections in the music of Debussy s Reflets dans leau (Reflections in Water), from Images (1st series, 1905), in which the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position.

As a musical interval the ratio 1.618.

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In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio.

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