The Golden Ratio is a absorbing figure which can be found about everyplace –from nature to architecture to art. To 18 denary topographic points. it has a value of 1. 618033988749894848 but is normally shortened to 1. 618 much like? is normally rounded off to 3. 1416 ( Powis. n. d. ) . Signified by the missive Phi ( ? ) . the Golden Ratio can be merely defined as “to square it. you merely add 1” ( Knott. 2007 ) . Written in mathematical equation. this definition becomes? 2 = ? + 1. When the ensuing quadratic equation? 2- ? – 1=0 is solved. there are two solutions: 1.

6180339887… and -0. 6180339887… . Notice that the two solutions have indistinguishable decimal parts. The positive figure is the 1 considered to be the Golden Ratio. Another definition for? is “the figure which when you take off 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 merely one less than the Golden Ratio. The Origins of the Golden Ratio Euclid of Alexandria ( ca. 300 BC ) in the Elementss. defines a proportion derived from the division of a line into sections ( Livio. 2002 ) .

His definition is as follows: A consecutive line is said to hold been cut in extreme and intend ratio when. as the whole line is to the greater section. so is the greater to the lesser. In order to be more apprehensible. let’s take Figure 1 as an illustration. In the diagram. point C divides the line in such a manner 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 1 of the first of all time documented definitions of the Golden Ratio although Euclid did non name it such at that clip.

A C B Figure 1. Indicate C divides line section AB harmonizing 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 oculus ( Blacker. Polanski & A ; Schwach. n. d. ) . Rectangles whose ratio of its length and width equal the Golden Ratio are called aureate rectangles. The exterior dimensions of the Parthenon in Athens. sculpted by Phidias. organize a perfect aureate rectangle.

Phidias besides used the Golden Ratio extensively in his other plants of sculpture. The Egyptians. who lived before Phidias. were believed to hold used the? in the design and building of the Pyramids ( Blacker. Polanski & A ; Schwach. n. d. ) . This belief nevertheless has both protagonists and critics. Theories that support or reject the thought of the Golden Ratio being used in the building of the Pyramids do be – it is up to the reader to make up one’s mind which 1s are more sensible ( Knott. 2007 ) . Many books besides claim that the celebrated painter Leonardo district attorney Vinci used the Golden Ratio in painting the Mona Lisa ( Livio. 2002 ) .

These books province that if you draw a rectangle around the face of Mona Lisa. the ratio of the tallness to the breadth of the rectangle is equal to the Golden Ratio. There has been no documented grounds that points to da Vinci’s witting usage of the Golden Ratio but what can non be denied is that Leonardo is a close personal friend of Luca Paciolo. who wrote extensively about the Golden Ratio. Unlike district attorney Vinci. the surrealist painter Salvador Dali intentionally used the Golden Ratio in his picture Sacrament of the Last Supper.

The ratio of the dimensions of his picture is equal to? ( Livio. 2002 ) . The Golden Ratio in Nature The Golden Ratio can besides be found in nature. One of the most common illustrations is snail shells. If you draw a rectangle with proportions harmonizing to the Golden Ratio so accordingly draw smaller aureate rectangles within it. and so fall in the diagonal corners The Golden Ratio 4 with an discharge. the consequence is a perfect snail shell ( Singh. 2002 ) .

There have besides been ongoing arguments and conflicting research consequences sing the relationship of beauty and in worlds. Some argue that human faces whose dimensions follow the Golden Ratio are more physically attractive than those who don’t ( Livio. 2002 ) . With conflicting consequences aside. the being of the Golden Ratio merely 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 aureate ratio. Retrieved October 8. 2007 from hypertext transfer protocol: //www. geom. uiuc. edu/~demo5337/s97b/ . Knott. R. ( 2007 ) . The aureate subdivision ratio: Phi.

Retrieved October 8. 2007 from hypertext transfer protocol: //www. megahertz. Surrey. Ac. uk/Personal/R. Knott/Fibonacci/phi. hypertext markup language. Livio. M. ( 2002 ) . The aureate ratio and aesthetics. Plus Magazine. Retrieved October 8. 2007 from hypertext transfer protocol: //plus. maths. org/issue22/features/golden/index. hypertext markup language. Powis. A. ( n. vitamin D ) . The aureate ratio. Retrieved October 8. 2007 from hypertext transfer protocol: //people. bath. Ac. uk/ajp24/goldenratio. hypertext markup language. Singh. S. ( 2002 March ) . The aureate ratio. BBC Radio. Retrieved October 8. 2007 from hypertext transfer protocol: //www. bbc. co. uk/radio4/science/5numbers3. shtml.