Golden ratio

The Golden Ratio 1

Running Head : THE GOLDEN RATIO

The Golden Ratio

The Golden Ratio 2 ? is usually rounded off to 3 .1416 (Powis , n .d . Signified by the letter Phi , the Golden Ratio can be simply defined as to square it , you just add 1 (Knott , 2007 . Written in mathematical equation this definition becomes ?2 1 . When the resulting quadratic equation ?2- ? - 1 0 is solved , there are two solutions : 1 and -0 . Notice that the two solutions have identical decimal parts . The positive number is the one considered

to be the Golden Ratio . Another definition for ? is the number which when you take away one becomes the value of its reciprocal (Powis , n .d Notice that the value of the reciprocal of 1 .618 (1 /1 .618 ) is 0 .618 which is just one less than the Golden Ratio

The Origins of the Golden Ratio

Euclid of Alexandria (ca . 300 BC ) in the Elements , defines a proportion derived from the division of a line into segments (Livio , 2002 . His definition is as follows

A straight line is said to have been cut in extreme and mean ratio when as the whole line is to the greater segment , so is the greater to the lesser

In to be more understandable , let 's take Figure 1 as an example In the diagram , point C divides the line in such a way that the ratio of AC to CB is equal to the ratio of AB to AC (Livio , 2002 . When this happens , the ratio can be calculated as 1 .618 . This is the one of the first ever documented definitions of the Golden Ratio although Euclid did not call it such at that time A C B

Figure 1 . Point C divides line segment AB according to the Golden Ratio

The Golden Ratio 3

The Golden Ratio in Art and Architecture ? in the design and construction of the Pyramids (Blacker , Polanski Schwach , n .d . This belief however has both supporters and critics Theories that support or reject the idea of the Golden Ratio being used in the construction of the Pyramids do exist - it is up to the reader to decide which ones are more reasonable (Knott , 2007

Many books also claim that the famous painter Leonardo da Vinci used the Golden Ratio in painting the Mona Lisa (Livio , 2002 . These books state that if you draw a rectangle around the face of Mona Lisa , the ratio of the height to the width of the rectangle is equal to the Golden Ratio . There has been no documented evidence that points to da Vinci 's conscious use of the Golden Ratio but what cannot be denied is that Leonardo is a close personal friend of Luca Paciolo , who wrote extensively about the Golden Ratio

of his painting is equal to (Livio , 2002

The Golden Ratio in Nature

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ntly draw smaller golden rectangles within it , and then join the...