The Golden Ratio is a fascinating number which can be found almost everywhere from nature to architecture to art. To 18 decimal places, it has a value of 1. 618033988749894848 but is usually shortened to 1. 618 much like ? 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.

6180339887¦ and -0. 6180339887¦. 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 order to be more understandable, lets 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 Throughout history, the Golden Ratio, when used in architecture, has been found to be the most pleasing to the eye (Blacker, Polanski & Schwach, n. d. ). Rectangles whose ratio of its length and width equal the Golden Ratio are called golden rectangles. The exterior dimensions of the Parthenon in Athens, sculpted by Phidias, form a perfect golden rectangle.

Phidias also used the Golden Ratio extensively in his other works of sculpture. The Egyptians, who lived before Phidias, were believed to have used the ? 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 Vincis 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. Unlike da Vinci, the surrealist painter Salvador Dali deliberately used the Golden Ratio in his painting Sacrament of the Last Supper.

The ratio of the dimensions of his painting is equal to ? (Livio, 2002). The Golden Ratio in Nature The Golden Ratio can also be found in nature. One of the most common examples is snail shells. If you draw a rectangle with proportions according to the Golden Ratio then consequently draw smaller golden rectangles within it, and then join the diagonal corners The Golden Ratio 4 with an arc, the result is a perfect snail shell (Singh, 2002).

There have also been ongoing debates and conflicting research results regarding the relationship of beauty and in humans. Some argue that human faces whose dimensions follow the Golden Ratio are more physically attractive than those who dont (Livio, 2002). With conflicting results aside, the existence of the Golden Ratio just shows that beauty (whether in art, architecture or in nature) can be linked to mathematics.

The Golden Ratio 5 References Blacker, S. , Polanski, J. and Schwach, M. (n. d. ). The golden ratio. Retrieved October 8, 2007 from http://www. geom. uiuc. edu/~demo5337/s97b/. Knott, R. (2007). The golden section ratio: Phi.

Retrieved October 8, 2007 from http://www. mcs. surrey. ac. uk/Personal/R. Knott/Fibonacci/phi. html. Livio, M. (2002). The golden ratio and aesthetics. Plus Magazine. Retrieved October 8, 2007 from http://plus. maths. org/issue22/features/golden/index. html. Powis, A. (n. d). The golden ratio. Retrieved October 8, 2007 from http://people. bath. ac. uk/ajp24/goldenratio. html. Singh, S. (2002 March). The golden ratio. BBC Radio. Retrieved October 8, 2007 from http://www. bbc. co. uk/radio4/science/5numbers3. shtml.

6180339887¦ and -0. 6180339887¦. 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 order to be more understandable, lets 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 Throughout history, the Golden Ratio, when used in architecture, has been found to be the most pleasing to the eye (Blacker, Polanski & Schwach, n. d. ). Rectangles whose ratio of its length and width equal the Golden Ratio are called golden rectangles. The exterior dimensions of the Parthenon in Athens, sculpted by Phidias, form a perfect golden rectangle.

Phidias also used the Golden Ratio extensively in his other works of sculpture. The Egyptians, who lived before Phidias, were believed to have used the ? 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 Vincis 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. Unlike da Vinci, the surrealist painter Salvador Dali deliberately used the Golden Ratio in his painting Sacrament of the Last Supper.

The ratio of the dimensions of his painting is equal to ? (Livio, 2002). The Golden Ratio in Nature The Golden Ratio can also be found in nature. One of the most common examples is snail shells. If you draw a rectangle with proportions according to the Golden Ratio then consequently draw smaller golden rectangles within it, and then join the diagonal corners The Golden Ratio 4 with an arc, the result is a perfect snail shell (Singh, 2002).

There have also been ongoing debates and conflicting research results regarding the relationship of beauty and in humans. Some argue that human faces whose dimensions follow the Golden Ratio are more physically attractive than those who dont (Livio, 2002). With conflicting results aside, the existence of the Golden Ratio just shows that beauty (whether in art, architecture or in nature) can be linked to mathematics.

The Golden Ratio 5 References Blacker, S. , Polanski, J. and Schwach, M. (n. d. ). The golden ratio. Retrieved October 8, 2007 from http://www. geom. uiuc. edu/~demo5337/s97b/. Knott, R. (2007). The golden section ratio: Phi.

Retrieved October 8, 2007 from http://www. mcs. surrey. ac. uk/Personal/R. Knott/Fibonacci/phi. html. Livio, M. (2002). The golden ratio and aesthetics. Plus Magazine. Retrieved October 8, 2007 from http://plus. maths. org/issue22/features/golden/index. html. Powis, A. (n. d). The golden ratio. Retrieved October 8, 2007 from http://people. bath. ac. uk/ajp24/goldenratio. html. Singh, S. (2002 March). The golden ratio. BBC Radio. Retrieved October 8, 2007 from http://www. bbc. co. uk/radio4/science/5numbers3. shtml.