B-Cognizance

Indian Institute of Information Technology, Allahabad.

E- Magazine August - September 2006 Vol 2 Issue 10

Indian Institute of Information Technology, Allahabad.

E- Magazine August - September 2006 Vol 2 Issue 10

Welcome to the world of phi……..

Phi (=1.61803398874949……) most often is pronounced as fi is simply an irrational number like pi(=3.1415926…..) .and that is where the similarity and that is where the similarity between the two numbers ends, because it is the one with many unusual mathematical properties. The ratio or proportion determined by phi was known to the Greeks as the “dividing a line in the extreme and mean ratio” and to renaissance artists as the “divine proportion”. It is also called the golden section, golden ratio and golden mean.

What makes phi even more unusual is that it can be derived in many ways and shows up in relationships throughout the universe .It is there in the proportions of human body, proportions of many other animals, plants, DNA, the solar system, art and architecture , music, population growth, the stock market, the bible and in theology also.

Some of the glaring examples of the presence pf phi can be listed as follows To begin with lets see the importance of phi in art and architecture. The acropolis including the Parthenon is often claimed to have been constructed using golden ratio. The Parthenon shows various golden rectangles, which are used in its design.

Le co Busier also used the golden ratio as the basis of his modular system of architecture.

The ratio is used many times in modern human constructions such as stairs, buildings, woodwork and in paper size.

**Mathematics**

The construction of a pentagram is based on the golden ratio. It can be seen as a geometric shape consisting of 5 straight lines arranged as a star with 5 points. The intersection of the lines naturally divides each length into 3 parts .The smaller part is proportional to the longer length by a ratio of 1:1.1618 (golden ratio).

**Philosophy**

In the power of limits ,Gyorgy Doczi writes that Buddha’s teachings expressed the need to avoid excess by walking the middle path between self indulgence and self mortification. Doczi indicates the middle path reflects the harmony or golden ratio as it pertains to human behaviour.

Another interesting extension of the phi is the Fibonacci sequence. In the west. This sequence was first studied by Leanardo of Pisa.

**Applications**

1. It is important in the run time analysis of Euclids algorithm to determine the greatest common divisor of two integers. The worst case input for this algorithm is a pair of consecutive Fibonacci numbers.

2. Fibonacci numbers can occur in a formula about diagonals of Pascal’s triangle.

3. They are used in pseudorandom number generators.

4. In music Fibonacci sequence is used to determine timings and as in visual art to determine the length of content or formal elements.

Fibbonaci sequence have been noted to appear in biological settings, such as branching patterns of sunflowers, arrangement of pine in pine trees, seeds on rasp berry, and spiral pattern in horns and shells.

The relation of phi and Fibonacci sequences is that when you divide an element of the sequence with its preceding element you get THE GOLDEN RATIO

For eg. The sequence is 1,1,2,3,5,8,13,21……

Divide 13 by 8 and the result is 1.618 (the golden ratio!!!).

This sequence was recently used in the famous book THE DA VINCI CODE … Therefore all things which come within the purview of the golden ratio or the divine proportion follow the Fibonacci sequence.

MBA 1st Semester

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