**The Golden Ratio (phi)**

There are several well known “irrational numbers”, which are numbers that cannot be expressed as a ratio between two numbers and it cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal. When irrational numbers are written as decimals the decimal would go on forever without repeating. An irrational number you are probably very familiar with is pi or Π which is equal to 3.14159265359 …. it is the irrational number that helps us understand everything that is circular. Another irrational number you may have heard of is “Euler’s number” e = 2.71828182845 …. it is the irrational number that helps us describe natural growth. There is a third irrational number named “phi” Φ also known as the “golden ratio”, Φ=1.618033987…it is the irrational number that helps us make beautiful art.

The Greeks were fascinated by phi. It was discovered in the five pointed star (pentagram)

Phi or the golden ratio is a special number found by dividing a line into two parts so that the value you get when you divide the longer part by the smaller part is equal to the whole length divided by the long part.

In the five pointed start a whole length LL is split into a longer piece L and a shorter piece M. It turns out that LL divided by L is equal to L divided by M which are equal to 1.618 … or phi. The longer piece L can also be split up into two parts as well a longer part M and a shorter part S which also divides to phi.

Phi or the Golden Ratio can be seen in the architecture of many ancient structures like the Great Pyramids and the Parthenon. The Great Pyramid of Giza has a square base that has a length of approximately 756 feet and a height of about 481 feet. When you divide the base length and the height you get 1.5717 a value close to phi.

Phi was referred to the “Divine Proportion” by Italian mathematician Lucia Pacioli in 1509. Artist Leonardo da Vinci illustrated phi in “Vitruvian Man” , he also used the golden ratio to define many of the proportions in his famous painting “The Last Supper” including the dimensions of the table and the proportions of the walls and the background. The Golden Ratio was used to achieve balance and beauty in many Renascence paintings and sculptures. Artists that employed the Golden ratio included Michelangelo, Raphael, and Salvador Dali.

Vitruvian Man

The last supper

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