Fibonacci Numbers and the Pascal Triangle
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Golden Section

The Golden Proportion

What is common for the beauty of a sculpture, picture, simphony, poem, etc? Is it possible to compare the beauty of the temple to the beauty of a nocturne? It appears it is, provided the unified criteria of beauty are found. Among many proportions there is one having unique properties. This proportion is called: "golden section", "golden ratio", "golden number" and "golden mean"..

The golden section- a precise way of dividing a line, music or anything else-is showed up early in mathematics. It goes back at least as far as 300 B.C., when Euclid described it in his major work, the Elements. Moreover, the Pythagoreans apparently knew about the golden section around 500 B.C. The oldest examples of this principle, however, appear in nature's proportions, including the morphology of pine cones and starfish. Further more, "The golden section is thought by some people to offer the aesthetically most pleasing proportion."

Euclid, a Greek mathematician wrote the Elements which is a collection of 13 books . It was the most important mathematical work up to present days. In Book 6, Proposition 30, Euclid shows how to divide a line in mean and extreme ratio which we would call "finding the golden section point on the line." Euclid used this phrase to mean the ratio of the smaller part of the line, to the larger part is the SAME as the ratio of the larger part, to the whole line. Euclid in Elements called dividing a line at the ratio 0.6180399.. : 1 , dividing a line in the extreme and mean ratio. This later gave rise to the name golden mean, golden ratio and even the divine proportion.

In order to describe the golden section, imagine a line that is one unit long. Then divide the line in two unequal segments, such that the shorter one equals x, the longer one equals (1 - x) and the ratio of the shorter segment to the longer one equals the ratio of the longer segment to the overall line; that is, x/(1 - x) = (1 - x)/1. That equality leads to a quadratic equation that can be used to solve for x, and substituting that value back into the equality yields a common ratio of approximately 0.618. We shall use the Greek letter phi for this Golden Proportion . Also, we shell use Phi for the closely related value 1.6180339887...

Phi to a few thousands decimal places:

Phi has the value (sqrt(5)+1)/2 and phi is (sqrt(5)-1)/2 .Both of them have identical fractional parts after the decimal point. The value of phi is the same but begins with 0.6... instead of 1.6...

The Golden Proportion, phi, has been observed to evoke emotion or aesthetic feelings within us. The ancient Egyptians used it in the construction of the great pyramids and in the design of hieroglyphs found on tomb walls. At another time, thousands of miles away, the ancients of Mexico embraced phi while building the Sun Pyramid at Teotihuacan. The Greeks studied phi closely through their mathematics and used it in their architecture. The Parthenon at Athens is a classic example of the use of the Golden Rectangle. Plato in his Timaeus considered it the most binding of all mathematical relations and makes it the key to the physics of the cosmos. During the Renaissance, phi served as the "hermetic" structure on which great masterpieces were composed. Renowned artists such as Michelangelo, Raphael, and Leonardo da Vinci made use of it for they knew of its appealing qualities. Evidence suggests that classical music composed by Mozart, Beethoven, and Bach embraces phi.